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Scilab Textbook Companion for Numerical Methods: Principles, Analysis, And Algorithms by S. Pal1 Created by Saurav Suman B.Tech Others NIT Jamshedpur College Teacher NA Cross-Checked by

May 23, 2016

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Funded by a grant from the National Mission on Education through ICT, http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilab codes written in it can be downloaded from the ”Textbook Companion Project” section at the website http://scilab.in

Book Description Title: Numerical Numerical Methods: Principles, Principles, Analysis, Analysis, And Algorithms Algorithms Author: S. Pal Publisher: Oxford University Press Edition: 1 Year: 2009 ISBN: 9780195693751

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Scilab numbering policy used in this document and the relation to the above book. Exa Example (Solved example) Eqn Equation (Particular equation of the above book) AP Appendix to Example(Scilab Code that is an Appednix to a particular

Example of the above book) For example, Exa 3.51 means solved example 3.51 of this book. Sec 2.3 means a scilab code whose theory is explained in Section 2.3 of the book.

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Contents Listt of Scilab Codes Lis

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1 Bac Backgr kgroun ound d to Numeri Numerical cal Method Methodss

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2 Sco Scope pe of Numerical Numerical and Mathe Mathemat matica icall Methods Methods

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3 Erro Errors rs and and Their Their Prop Propaga agatio tion n

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4 Progra Programming mming Tools and Techn Techniques iques

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5 Solut Solutions ions of Algebrai Algebraic c and Transcen Transcendent dental al Equations Equations

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6 Num Numeri erical cal Methods Methods of Linear Linear Equation Equationss Direct Direct Methods Methods

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7 Num Numeri erical cal Soluti Solutions ons for Matrix Matrix Inversi Inversion on

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8 Numeri Numerical cal Soluti Solutions ons of Linear Linear Syst Systems ems of Equations Equations Iter Iteraative Methods 107 9 Lin Linear ear Least Least Squa Squares res Prob Problem lem

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10 Numerical Solutions of System of Non Linear Equations

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11 Eigenvalues and Eigenvectors

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12 Interpolation and Extrapolation

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13 Numerical Differentiation

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14 Numerical Integration

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15 Numerical Solutions of Ordinary Differential Equations Initial Value Problem 195 16 Numerical Solutions of Ordinary Differential Equations Boundary Value Problems 216 18 Numerical Solutions of Parabolic Partial Differential Equations 225 19 Numerical Solutions of Hyperbolic Partial Differential Equations 235 20 Numerical Solutions of Elliptical Partial Differential Equations 254 21 Advances in Numerical Methods Using Parallel Computing Paradigm 265 22 Numerical Methods Using Neural Networks

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List of Scilab Codes Exa 1.1 Exa 1.2 Exa 1.3 Exa 1.4 Exa 1.5 Exa 1.6 Exa 1.7 Exa 1.8 Exa 1.9 Exa 1.10 Exa 1.13 Exa 1.14 Exa 1.15 Exa 1.16 Exa 1.17 Exa 1.18 Exa 1.19 Exa 1.20 Exa 1.23 Exa 1.24 Exa 1.25 Exa 1.26 Exa 1.29 Exa 1.30 Exa 1.31 Exa 1.32 Exa 1.33 Exa 1.34

Conversion to Decimal System . . . . . . . . Conversion Using Shortcut Method . . . . . . Conversion to Base B from Decimal System . Conversion to Binary System . . . . . . . . . Conversion to Binary System . . . . . . . . . Conversion to Decimal Number . . . . . . . . Conversion to Decimal Number . . . . . . . . Conversion to Base B from Binary System . . Conversion to Binary System . . . . . . . . . Conversion to Binary System and to Base N . 1s compliment and 2s compliment . . . . . . 1s compliment . . . . . . . . . . . . . . . . . Addition and Subtraction . . . . . . . . . . . Addition . . . . . . . . . . . . . . . . . . . . Addition . . . . . . . . . . . . . . . . . . . . Addition . . . . . . . . . . . . . . . . . . . . Addition . . . . . . . . . . . . . . . . . . . . Subtraction . . . . . . . . . . . . . . . . . . . Multiplication . . . . . . . . . . . . . . . . . Multiplication . . . . . . . . . . . . . . . . . Division . . . . . . . . . . . . . . . . . . . . . Multiplication . . . . . . . . . . . . . . . . . Normalized Floating Point Representation . . Add . . . . . . . . . . . . . . . . . . . . . . . Add . . . . . . . . . . . . . . . . . . . . . . . Add . . . . . . . . . . . . . . . . . . . . . . . Add . . . . . . . . . . . . . . . . . . . . . . . Subtraction . . . . . . . . . . . . . . . . . . . 5

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5 6 7 7 8 9 9 10 11 11 12 14 15 17 19 20 22 23 25 26 27 29 31 32 32 33 33 34

Exa 1.35 Exa 1.36 Exa 2.4 Exa 2.6 Exa 3.1 Exa 3.2 Exa 3.3 Exa 3.4 Exa 3.5 Exa 4.1 Exa 4.2 Exa 5.1 Exa 5.2 Exa 5.3 Exa 5.4 Exa 5.5 Exa 5.6 Exa 5.7 Exa 5.8 Exa 5.9 Exa 5.10 Exa 5.11 Exa 5.12 Exa 5.13 Exa 5.14 Exa 5.15 Exa 5.19 Exa 5.20 Exa 5.21 Exa 5.22 Exa 5.23 Exa 5.24 Exa 5.25 Exa 5.26 Exa 5.27 Exa 5.28 Exa 5.29 Exa 5.31

Multiplication . . . . . . . . . . . . . . . Division . . . . . . . . . . . . . . . . . . . Solving Simultaneous Linear Equation . . Integration . . . . . . . . . . . . . . . . . Limiting Error . . . . . . . . . . . . . . . Known Error . . . . . . . . . . . . . . . . Absolute Relative and Percentage Errors . Absolute Relative and Percentage Errors . Absolute Relative and Percentage Errors . Quadratic Equation . . . . . . . . . . . . Database Management . . . . . . . . . . . Bisection Method . . . . . . . . . . . . . Bisection Method . . . . . . . . . . . . . Regula Falsi Method . . . . . . . . . . . . Ridders Method . . . . . . . . . . . . . . General Iterative Method . . . . . . . . . Linear Iterative Method . . . . . . . . . . Aitkens Method . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . Modified Newton Raphson Method . . . . Newton Raphson Method . . . . . . . . . Newton Raphson Method . . . . . . . . . Newton Raphson Method . . . . . . . . . Secant Method . . . . . . . . . . . . . . . Kizner Method . . . . . . . . . . . . . . . Brent Method . . . . . . . . . . . . . . . Horner Method . . . . . . . . . . . . . . . Laguerre Method . . . . . . . . . . . . . . Mullers Method . . . . . . . . . . . . . . Mullers Method . . . . . . . . . . . . . . Bairstow Hitchcock Method . . . . . . . . Bernoulli Method . . . . . . . . . . . . . Graeffe Method . . . . . . . . . . . . . . . QD Method . . . . . . . . . . . . . . . . . Linear Iteration Method . . . . . . . . . . Aitkens Method . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . Secant Method . . . . . . . . . . . . . . . 6

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34 35 36 37 38 38 39 39 40 42 42 45 46 47 48 49 49 50 51 52 52 53 54 55 56 57 57 58 59 62 65 67 67 69 70 71 71 72

Exa 5.32 Exa 5.33 Exa 5.34 Exa 5.35 Exa 5.36 Exa 5.37 Exa 5.38 Exa 5.39 Exa 5.40 Exa 6.1 Exa 6.2 Exa 6.3 Exa 6.4 Exa 6.5 Exa 6.6 Exa 6.7 Exa 6.8 Exa 6.9 Exa 6.10 Exa 7.1 Exa 7.2 Exa 7.3 Exa 7.4 Exa 7.5 Exa 7.6 Exa 8.1 Exa 8.2 Exa 8.3 Exa 8.4 Exa 8.5 Exa 8.6 Exa 8.7 Exa 8.8 Exa 9.1 Exa 9.2 Exa 9.3 Exa 9.4

Regula Falsi Newton Raphson and Mullers Method . . Newton Raphson and Mullers Method . . . . . . . . . QD Method . . . . . . . . . . . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . . . . . . . . Secant Method . . . . . . . . . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . . . . . . . . Newton Raphson Method . . . . . . . . . . . . . . . . Gaussian Elimination Method . . . . . . . . . . . . . . Gaussian Elimination Method for TriDiagonal System Gauss Jordan Method . . . . . . . . . . . . . . . . . . Gaussian Elimination Method without Pivoting . . . . Dolittle Factorization Method . . . . . . . . . . . . . . Trangularization Method . . . . . . . . . . . . . . . . Wilkinson Method . . . . . . . . . . . . . . . . . . . . Choleskys Factorization . . . . . . . . . . . . . . . . . Complex System of Linear Equation . . . . . . . . . . Solving Matrices . . . . . . . . . . . . . . . . . . . . . Gauss Jordan Two Array Method . . . . . . . . . . . Inverse in Place without Pivoting . . . . . . . . . . . . Inverse in Place with Pivoting . . . . . . . . . . . . . . Inverse of Triangular Matrices . . . . . . . . . . . . . . Inverse of Complex Matrices . . . . . . . . . . . . . . Iterative Procedure . . . . . . . . . . . . . . . . . . . . Jacobi Method . . . . . . . . . . . . . . . . . . . . . . Gauss Seidel Method . . . . . . . . . . . . . . . . . . . SOR Method . . . . . . . . . . . . . . . . . . . . . . . Gauss Seidel Point Iterative Method . . . . . . . . . . Gauss Seidel Point Iterative Method . . . . . . . . . . Block Jacobi Method . . . . . . . . . . . . . . . . . . Block Gauss Seidel Method . . . . . . . . . . . . . . . Block SOR Method . . . . . . . . . . . . . . . . . . . Moore Penrose Generalized Inverse . . . . . . . . . . . Curve Fitting . . . . . . . . . . . . . . . . . . . . . . . Gram Schmidt Orthogonalization or Orthonormalization Process . . . . . . . . . . . . . . . . . . . . . . . QR Decomposition . . . . . . . . . . . . . . . . . . . . 7

73 77 79 80 81 82 83 84 85 86 87 88 89 90 92 93 94 95 95 97 101 102 104 105 106 107 108 109 110 112 115 116 120 124 124 127 128


Vector Computation . . . . . . . . . . . . . . . House Holder Transformation . . . . . . . . . . Givens QR Method . . . . . . . . . . . . . . . Recursive Least Square Method . . . . . . . . . System of Non Linear Equations . . . . . . . . Contraction Method and Seidel Method . . . . Non Linear System of Equation . . . . . . . . . Newton Method . . . . . . . . . . . . . . . . . Newton Raphshon Method . . . . . . . . . . . Newton Method . . . . . . . . . . . . . . . . . Iterative Method . . . . . . . . . . . . . . . . . Steepest Descent . . . . . . . . . . . . . . . . . Eigenvalues and Eigenvectors . . . . . . . . . . Leverriers Method . . . . . . . . . . . . . . . . Danilevsky Method . . . . . . . . . . . . . . . Power Method . . . . . . . . . . . . . . . . . . Inverse Power Method . . . . . . . . . . . . . . Rayleigh Quotient . . . . . . . . . . . . . . . . Jacobi Method . . . . . . . . . . . . . . . . . . Recursive Formula . . . . . . . . . . . . . . . . QR Method . . . . . . . . . . . . . . . . . . . . LU Method . . . . . . . . . . . . . . . . . . . . Generalized Eigenvalue Problem . . . . . . . . Linear Interpolation Technique . . . . . . . . . Lagarangian Method . . . . . . . . . . . . . . . Aitken Nevilles Method . . . . . . . . . . . . . Newtons Divided Difference Interpolation . . . Interpolation Methods . . . . . . . . . . . . . . Chebyshev Interpolating Polynomial . . . . . . Double Interpolation . . . . . . . . . . . . . . . Spline Interpolation . . . . . . . . . . . . . . . Differentiation . . . . . . . . . . . . . . . . . . Calculation of x coordinate of Minimum Point Newton Forward Difference Formula . . . . . . Newton Backward Difference Formula . . . . . Stirlings Central Difference Derivatives . . . . . Extrapolation . . . . . . . . . . . . . . . . . . . Richardson Extrapolation . . . . . . . . . . . . 8

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Application . . . . . . . . . . . . . . . . . . . . . . . . Simpsons 1 3rd Rule . . . . . . . . . . . . . . . . . . . Trapezoidal Rule and Simpsons Rule . . . . . . . . . . Romberg Method . . . . . . . . . . . . . . . . . . . . . Gaussian Quadrature Formula . . . . . . . . . . . . . Gauss Legendre Two Point Rule . . . . . . . . . . . . Gauss Legendre Three Point Rule . . . . . . . . . . . Spline Integration Method . . . . . . . . . . . . . . . . Trapezoidal Rule . . . . . . . . . . . . . . . . . . . . . Trapezoidal and Simpsons Rule . . . . . . . . . . . . . Trapezoidal and Simpsons Rule . . . . . . . . . . . . . Multiple Integration with Variable Limits . . . . . . . Integration . . . . . . . . . . . . . . . . . . . . . . . . Integration . . . . . . . . . . . . . . . . . . . . . . . . Ordinary Differential Equation . . . . . . . . . . . . . Taylor Method . . . . . . . . . . . . . . . . . . . . . . Picard Method . . . . . . . . . . . . . . . . . . . . . . Euler Method . . . . . . . . . . . . . . . . . . . . . . . Trapezium Method . . . . . . . . . . . . . . . . . . . . Heun Method . . . . . . . . . . . . . . . . . . . . . . . Midpoint Method . . . . . . . . . . . . . . . . . . . . Modified Midpoint Method . . . . . . . . . . . . . . . Single Step Method . . . . . . . . . . . . . . . . . . . Second Order Runge Kutta Method . . . . . . . . . . Third Order Runge Kutta Method . . . . . . . . . . . Fourth Order Runge Kutta Method . . . . . . . . . . New Variant of Runge Kutta Method . . . . . . . . . Runge Kutta Merson Method . . . . . . . . . . . . . . Runge Kutta Fehlberg Method . . . . . . . . . . . . . Carp Karp Runge Kutta Method . . . . . . . . . . . . Implicit Runge Kutta Method . . . . . . . . . . . . . . Linear Multi Step Method . . . . . . . . . . . . . . . . Milne Simpson Predictor Corrector Method . . . . . . Improved Milne Simpson Predictor Corrector Method Hamming Predictor Corrector Method . . . . . . . . . Multi Valued Method . . . . . . . . . . . . . . . . . . First order ODE . . . . . . . . . . . . . . . . . . . . . Differential Equation . . . . . . . . . . . . . . . . . . . 9

181 183 184 186 187 187 188 188 189 190 191 193 194 194 195 195 196 197 197 198 199 199 200 201 202 202 203 204 204 205 206 207 208 208 210 211 213 214

Exa 16.1 Exa 16.2 Exa 16.3 Exa 16.4 Exa 16.5 Exa 16.6 Exa 18.4 Exa 18.5 Exa 18.6 Exa 18.7 Exa 18.8 Exa 19.3 Exa 19.4 Exa 19.5 Exa 19.6 Exa 19.7 Exa 19.8 Exa 19.9 Exa 19.10 Exa 19.11 Exa 19.12 Exa 19.13 Exa 20.1 Exa 20.2 Exa 20.3 Exa 20.4 Exa 20.5 Exa 20.6 Exa 20.7 Exa 20.8 Exa 21.1 Exa 21.2 Exa 21.3 Exa 21.4 Exa 21.5 Exa 22.1

Outline of Linear Shooting Method . . . . . . . . . . . Linear Shooting Method . . . . . . . . . . . . . . . . . Multiple Shooting Method . . . . . . . . . . . . . . . . Finite Difference Method . . . . . . . . . . . . . . . . Non Linear Problem . . . . . . . . . . . . . . . . . . . Collocation Method . . . . . . . . . . . . . . . . . . . Forward Difference Method . . . . . . . . . . . . . . . Bender Schmidt Method . . . . . . . . . . . . . . . . . Crank Nicolson Method . . . . . . . . . . . . . . . . . Gauss Seidel Method . . . . . . . . . . . . . . . . . . . ADI Method . . . . . . . . . . . . . . . . . . . . . . . Simple Explicit Method . . . . . . . . . . . . . . . . . Simple Implicit Method . . . . . . . . . . . . . . . . . Lax Wendroff Method . . . . . . . . . . . . . . . . . . Wendroff Method . . . . . . . . . . . . . . . . . . . . Leapfrog Method . . . . . . . . . . . . . . . . . . . . . Variable Coefficients . . . . . . . . . . . . . . . . . . . Inhomogeneous 1st Order Hyperboolic Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non Linear 1st Order Hyperboolic Differential Equation Finite Difference Method . . . . . . . . . . . . . . . . Hyperbolic Partial Differential Equations . . . . . . . Hyperbolic Differential Equations in 2D or 3D . . . . . Direct Method . . . . . . . . . . . . . . . . . . . . . . Five Point Formula . . . . . . . . . . . . . . . . . . . Finite Difference Method . . . . . . . . . . . . . . . . Seven Point Formula . . . . . . . . . . . . . . . . . . . Nine Point Formula . . . . . . . . . . . . . . . . . . . Five Point Formula . . . . . . . . . . . . . . . . . . . Laplace Distribution . . . . . . . . . . . . . . . . . . . Spherical Coordinate System . . . . . . . . . . . . . . Parallel Bisection Method . . . . . . . . . . . . . . . . Lagrange Interpolation in Parallel Computing . . . . . Trapezoidal Rule and Simpsons Rule in Parallel Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . Parallel Gauss Seidel Method . . . . . . . . . . . . . . Poissons Partial Differential Equation . . . . . . . . . MLP Algorithm . . . . . . . . . . . . . . . . . . . . . 10

216 217 219 221 222 223 225 226 227 229 230 235 236 237 238 239 241 244 247 248 250 250 254 255 256 257 258 260 261 262 265 266 267 269 270 273

Exa 22.2 Exa 22.3 Exa 22.4 Exa 22.5 Exa 22.7

MLP . . . . . . . . . . . Bisection Method . . . . Hopfield Neural Network RBF Network . . . . . . . First Order ODE . . . . .

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Chapter 1 Background to Numerical Methods

Scilab code Exa 1.1 Conversion to Decimal System

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E xa mp le 1 . 1 / / C o n v e r s i o n t o D e ci m a l S ys te m / / Pa ge no . 4 clc ; close ; clear ; function [ s ] = b a s 2 d e c ( x , b ) xi = int ( x ) x d = x - int ( x ) s =0 for i = 1 : 1 0 xi=xi/10 s = s + ( 1 0 * ( x i -fix ( x i ) ) * b ^ ( i - 1 ) ) xi = int ( x i ) if ( x i = = 0 ) break end end for i = 1 : 1

12

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

xd=xd*10; s = s + ( ceil ( x d ) / b ^ ( i ) ) x d = x d - fix ( x d ) if ( x d = = 0 ) break end

end endfunction

/ / c o n v e r s i o n f ro m h e xa d ec i ma l t o d e c im a l s ys te m // i n b u i t f u n c t i o n disp ( hex2dec ( ’ 1A2C ’ ) , ’ 1A2C= ’ ) ; / / c o n v e r s i o n f ro m h e xa d ec i ma l t o d e c im a l s ys te m // i n l i n e f u n c t i o n disp ( b a s 2 d e c ( 4 2 8 . 5 , 8 ) , ’ 428 .5= ’ ) / / c o n v e r s i o n f ro m h e xa d ec i ma l t o d e c im a l s ys te m disp ( b a s 2 d e c ( 1 2 0 . 1 , 3 ) , ’ 120 .1= ’ ) // i n l i n e function

Scilab code Exa 1.2 Conversion Using Shortcut Method

1 2 3 4 5 6 7

/ / E xa mp le 1 . 2 / / C o n v e r s i o n U s i n g S h o r t c u t Meth od / / Pa ge no . 4 clc ; close ; clear ; A=10;C=12; d=(((1)*16+A)*16+2)*16+C; disp (d , ’ D ec im a l f or m o f 1A2C i s = ’ ) ;

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Scilab code Exa 1.3 Conversion to Base B from Decimal System

1 2 3 4 5 6

/ / E xa mp le 1 . 3 / / C o n v er s i o n t o B as e B f ro m D ec im a l S ys te m / / Pa ge no . 5 clc ; close ; clear ;

/ / c o n v e rs i o n fro m b i na r y t o o c t a l disp ( d e c 2 o c t ( b i n 2 d e c ( ’ 1 0 1 0 1 1 0 1 1 1 0 ’ ) ) , ’ O c ta l f or m o f 1 01 01 10 11 10 i s = ’ ) ; // i n b u i l t f u n c t i o n

7 8 / / c o n v e r s i o n f ro m b i n a r y t o h e xa d ec i ma l 9 disp ( dec2hex ( b i n 2 d e c ( ’ 1 0 1 0 1 1 0 1 1 1 0 ’ ) ) , ’ H e x a d e c i m a l form o f 1 01 01 10 11 10 i s = ’ ) ; // i n b u i l t f u n c t i o n 10 11 / / c o n v e rs i o n fro m b i na r y t o o c t a l 12 s = d e c 2 o c t ( b i n 2 d e c ( ’ 101 1 ’ ) ) ; 13 s 1 = d e c 2 o c t ( b i n 2 d e c ( ’ 1 1 0 0 1 1 0 1 0 1 0 0 ’ ) ) ; // i n b u i l t

function 14 printf ( ’ \n O ct al form o f 1 0 1 1 . 11 0 0 1 1 0 1 0 1 i s = \n\n %s . %s ’ , s , s 1 ) 15 16 / / c o n v e r s i o n f ro m b i n a r y t o h e xa d ec i ma l 17 s = dec2hex ( b i n 2 d e c ( ’ 101 1 ’ ) ) ; 18 s1 = dec2hex ( b i n 2 d e c ( ’ 1 1 0 0 1 1 0 1 0 1 0 0 ’ ) ) ; // i n b u i l t

function 19 printf ( ’ \n\n H ex ad ec im al fo rm o f 1 0 11 . 11 0 01 1 01 0 1 i s = \n\n %s . %s ’ , s , s 1 )

Scilab code Exa 1.4 Conversion to Binary System

1 / / E xa mp le 1 . 4 2 / / C o n v er s i o n t o B i n ar y S ys te m 3 / / Pa ge no . 6

14

4 clc ; close ; clear ; 5 / / c o n v e rs i o n fro m o c t a l t o b i na r y 6 disp ( d e c 2 b i n ( o c t 2 d e c ( ’ 175 3 ’ ) ) , ’ B i na ry f or m o f 1 75 3 i s = ’ ); // i n b u i l t f u n c t i o n 7 8 / / c o n v e rs i o n fro m o c t a l t o b i na r y 9 disp ( d e c 2 b i n ( hex2dec ( ’ A478 ’ ) ) , ’ B i n ar y f or m o f A478 i s = ’ ); // i n b u i l t f u n c t i o n 10 11 / / c o n v e rs i o n fro m o c t a l t o b i na r y 12 s = d e c 2 b i n ( o c t 2 d e c ( ’ 3 ’ ) ) ; 13 s 1 = d e c 2 b i n ( o c t 2 d e c ( ’ 154 ’ ) ) ; // i n b u i l t f u n c ti o n 14 printf ( ’ \n O c t a l form o f 3 .1 5 4 i s = \n\n %s . 0 0 %s ’ ,s , s1 )


1 2 3 4 5 6 7 8 9 10 11 12 13

/ / E xa mp le 1 . 5 / / C o n v er s i o n t o B i n ar y S ys te m / / Pa ge no . 6 clc ; close ; clear ;

/ / c o n v e rs i o n fro m o c t a l t o b i na r y b = d e c 2 b i n ( o c t 2 d e c ( ’ 175 3 ’ ) ) disp (b , ’ B in ar y fo rm o f 1 75 3 i s = ’ ) ; // i n b u i l t function b = dec2hex ( o c t 2 d e c ( ’ 175 3 ’ ) ) // disp (b , ’ H ex ad ec im al f or m o f 1 75 3 i s = ’ ) ; i n b u i l t f u nc t io n / / c o n v e rs i o n fro m o c t a l t o b i na r y b = d e c 2 b i n ( hex2dec ( ’ A478 ’ ) ) disp (b , ’ B in ar y fo rm o f A478 i s = ’ ) ; // i n b u i l t function b = d e c 2 o c t ( hex2dec ( ’ A478 ’ ) ) 15

14 disp (b , ’ O ct al form o f A478 i s = ’ ) ;

// i n b u i l t

function 15 / / c o n v e rs i o n fro m o c t a l t o b i na r y 16 s = d e c 2 b i n ( o c t 2 d e c ( ’ 3 ’ ) ) ; 17 s 1 = d e c 2 b i n ( o c t 2 d e c ( ’ 154 ’ ) ) ; // i n b u i l t f u n c ti o n 18 printf ( ’ \n O c t a l form o f 3 .1 5 4 i s = \n\n %s . 0 0 %s ’ ,s , s1 ) 19 s = dec2hex ( o c t 2 d e c ( ’ 3 ’ ) ) ; 20 s1 = dec2hex ( o c t 2 d e c ( ’ 154 ’ ) ) ; // i n b u i l t f u n c ti o n 21 printf ( ’ \n\n H e xa de c i ma l fo rm o f 3 . 15 4 i s = \n\n % s . %s ’ , s , s 1 )

Scilab code Exa 1.6 Conversion to Decimal Number

1 2 3 4 5 6

/ / E xa mp le 1 . 6 / / C o n v e r s i o n t o D e c i m a l Number / / Pa ge no . 7 clc ; close ; clear ; disp ( d e c 2 b i n ( 1 8 2 ) , ’ B i n a r y o f 182= ’ )

function

Scilab code Exa 1.7 Conversion to Decimal Number

1 2 3 4 5

/ / E xa mp le 1 . 7 / / C o n v e r s i o n t o D e c i m a l Number / / Pa ge no . 7 clc ; close ; clear ;

16

// i n b u i l t

6 disp ( d e c 2 o c t ( 4 6 7 ) , ’ O c t a l o f 467= ’ )

//

i n b u i l t f u nc t io n

Scilab code Exa 1.8 Conversion to Base B from Binary System

1 2 3 4 5 6

/ / E xa mp le 1 . 8 / / C o n v er s i o n t o B as e B f ro m B i na r y S ys te m / / Pa ge no . 8 clc ; close ; clear ;

/ / c o n v e rs i o n fro m b i na r y t o o c t a l disp ( d e c 2 o c t ( b i n 2 d e c ( ’ 1 0 1 0 1 1 0 1 1 1 0 ’ ) ) , ’ O c ta l f or m o f 1 01 01 10 11 10 i s = ’ ) ; // i n b u i l t f u n c t i o n

7 8 / / c o n v e r s i o n f ro m b i n a r y t o h e xa d ec i ma l 9 disp ( dec2hex ( b i n 2 d e c ( ’ 1 0 1 0 1 1 0 1 1 1 0 ’ ) ) , ’ H e x a d e c i m a l form o f 1 01 01 10 11 10 i s = ’ ) ; // i n b u i l t f u n c t i o n 10 11 / / c o n v e rs i o n fro m b i na r y t o o c t a l 12 s = d e c 2 o c t ( b i n 2 d e c ( ’ 101 1 ’ ) ) ; 13 s 1 = d e c 2 o c t ( b i n 2 d e c ( ’ 1 1 0 0 1 1 0 1 0 1 0 0 ’ ) ) ; // i n b u i l t

function 14 printf ( ’ \n O ct al form o f 1 0 1 1 . 11 0 0 1 1 0 1 0 1 i s = \n\n %s . %s ’ , s , s 1 ) 15 16 / / c o n v e r s i o n f ro m b i n a r y t o h e xa d ec i ma l 17 s = dec2hex ( b i n 2 d e c ( ’ 101 1 ’ ) ) ; 18 s1 = dec2hex ( b i n 2 d e c ( ’ 1 1 0 0 1 1 0 1 0 1 0 0 ’ ) ) ; // i n b u i l t

function 19 printf ( ’ \n\n H ex ad ec im al fo rm o f 1 0 11 . 11 0 01 1 01 0 1 i s = \n\n %s . %s ’ , s , s 1 )

17


1 2 3 4 5 6

/ / E xa mp le 1 . 9 / / C o n v er s i o n t o B i n ar y S ys te m / / Pa ge no . 8 clc ; close ; clear ;

/ / c o n v e rs i o n fro m o c t a l t o b i na r y disp ( d e c 2 b i n ( o c t 2 d e c ( ’ 175 3 ’ ) ) , ’ B i na ry f or m o f 1 75 3 i s = ’ ); // i n b u i l t f u n c t i o n

7 8 / / c o n v e rs i o n fro m o c t a l t o b i na r y 9 disp ( d e c 2 b i n ( hex2dec ( ’ A478 ’ ) ) , ’ B i n ar y f or m o f A478 i s = ’ ); // i n b u i l t f u n c t i o n 10 11 / / c o n v e rs i o n fro m o c t a l t o b i na r y 12 s = d e c 2 b i n ( o c t 2 d e c ( ’ 3 ’ ) ) ; 13 s 1 = d e c 2 b i n ( o c t 2 d e c ( ’ 154 ’ ) ) ; // i n b u i l t f u n c ti o n 14 printf ( ’ \n O c t a l form o f 3 .1 5 4 i s = \n\n %s . 0 0 %s ’ ,s , s1 )

Scilab code Exa 1.10 Conversion to Binary System and to Base N

1 2 3 4 5 6

/ / E x am pl e 1 . 1 0 / / C o n v er s i o n t o B i n ar y S ys te m and t o B as e N / / Pa ge no . 9 clc ; close ; clear ; b = d e c 2 b i n ( o c t 2 d e c ( ’ 175 3 ’ ) )

18

7 disp (b , ’ B in ar y fo rm o f 1 75 3 i s = ’ ) ;

// i n b u i l t

function 8 b = dec2hex ( o c t 2 d e c ( ’ 175 3 ’ ) ) 9 disp (b , ’ H ex ad ec im al f or m o f 1 75 3 i s = ’ ) ; 10 11 12 13 14 15 16 17 18

//

i n b u i l t f u nc t io n / / c o n v e rs i o n fro m o c t a l t o b i na r y b = d e c 2 b i n ( hex2dec ( ’ A478 ’ ) ) disp (b , ’ B in ar y fo rm o f A478 i s = ’ ) ; // i n b u i l t function b = d e c 2 o c t ( hex2dec ( ’ A478 ’ ) ) // i n b u i l t disp (b , ’ O ct al form o f A478 i s = ’ ) ; function / / c o n v e rs i o n fro m o c t a l t o b i na r y s = d e c 2 b i n ( o c t 2 d e c ( ’ 3 ’ ) ) ; s 1 = d e c 2 b i n ( o c t 2 d e c ( ’ 154 ’ ) ) ; // i n b u i l t f u n c ti o n printf ( ’ \n O c t a l form o f 3 .1 5 4 i s = \n\n %s . 0 0 %s ’ ,s ,

s1 ) 19 s = dec2hex ( o c t 2 d e c ( ’ 3 ’ ) ) ; 20 s1 = dec2hex ( o c t 2 d e c ( ’ 154 ’ ) ) ; // i n b u i l t f u n c ti o n 21 printf ( ’ \n\n H e xa de c i ma l fo rm o f 3 . 15 4 i s = \n\n % s . %s ’ , s , s 1 )

Scilab code Exa 1.13 1s compliment and 2s compliment

1 2 3 4 5 6

/ / E x am pl e 1 . 1 3 / / 1 s c om p li me n t and 2 s c om p li m en t / / P ag e n o . 1 1 clc ; close ; clear ;

// f u n c ti o n f o r 1 s

function [ x 1 ] = c o m 1 ( x )

compliment 7 8

for i = 8 : - 1 : 1 x=x/10;

19

9 x d = x - fix ( x ) if ( floor ( ( x d * 1 0 ) + 0 . 1 ) = = 1 ) 10 11 x1(1,i)=0; else 12 13 x1(1,i)=1; end 14 15 x=x-xd; end 16 17 e n d f u n c t i o n 18 function [ x 1 ] = c o m 2 ( x ) // f u n c t i o n f o r 2 s

compliment () 19 for i = 8 : - 1 : 1 x=x/10; 20 21 x d = x - fix ( x ) if ( int ( ( x d * 1 0 ) + 0 . 1 ) = = 1 ) 22 23 x1(1,i)=0; else 24 25 x1(1,i)=1; end 26 27 end 28 for i = 8 : - 1 : 1 29 if ( x 1 ( 1 , i ) = = 0 ) then 30 x1(1,i)=1; break ; 31 32 else x1(1,i)=0; 33 34 end 35 36 end 37 e n d f u n c t i o n 38 a = [ 0 0 0 1 0 0 1 1 , 0 1 1 1 0 1 1 0 , 1 1 1 0 1 1 0 1 , 1 0 0 0 0 0 0 1 , 1 0 0 0 0 0 0 0 , 0 0 0 00 0 0 0 ] ; 39 for i = 1 : 6 40 printf ( ’ 1 s Co mp li me nt o f %. 8 i = ’ , a ( i ) ) ; 41 disp ( c o m 1 ( a ( i ) ) ) printf ( ’ 2 s Co mp li me nt o f %. 8 i = ’ , a ( i ) ) ; 42 43 disp ( c o m 2 ( a ( i ) ) )

20

44 printf ( ’ \n\n ’ ) 45 end

Scilab code Exa 1.14 1s compliment

1 2 3 4 5 6

/ / E x am pl e 1 . 1 4 / / 1 s c o mp l i me n t / / P ag e n o . 1 2 clc ; close ; clear ;

// f u n c ti o n f o r 1 s

function [ x 1 ] = c o m 1 ( x )

compliment for i = 8 : - 1 : 1 7 8 x=x/10; x d = x - fix ( x ) 9 10 if ( floor ( ( x d * 1 0 ) + 0 . 1 ) = = 1 ) 11 x1(1,i)=0; 12 else 13 x1(1,i)=1; end 14 15 x=x-xd; end 16 17 e n d f u n c t i o n 18 a = [ 0 0 0 1 0 0 1 1 , 0 1 1 1 0 1 1 0 , 1 1 1 0 1 1 0 1 , 1 0 0 0 0 0 0 1 , 1 0 0 0 0 0 0 0 , 0 0 0 00 0 0 0 ] ; 19 for i = 1 : 6 20 printf ( ’ 1 s Co mp li me nt o f %. 8 i = ’ , a ( i ) ) ; 21 disp ( c o m 1 ( a ( i ) ) ) 22 printf ( ’ \n\n ’ ) 23 end

21

Scilab code Exa 1.15 Addition and Subtraction

1 2 3 4 5

/ / E x am pl e 1 . 1 5 / / A d d i t i on and S u b t r a c t i o n / / P ag e n o . 1 3 clc ; clear ; close ; function [ x 1 ] = a d d ( x , y )

// fun ctio n

f o r a dd it io n o f b i n a ri e s 6 7 8 9 10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26

c=0; for i = 1 : 1 0 x1(1,i)=0 end for i = 1 0 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=1;c=0; else x1(1,i)=0;c=0; end else i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then

22

27 28

x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=0;c=0; else x1(1,i)=1;c=0; end

29 30 31 32 end 33 34 end 35 disp ( x 1 , ’ A d d it i o n o f 1 73 and 141= ’ ) 36 e n d f u n c t i o n 37 function [ x 1 ] = s u b ( x , y ) // f u n c t i o n f o r

s u bt r ac t io n o f b i n a r i e s 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

c=0; for i = 1 : 1 0 x1(1,i)=0 end for i = 1 0 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd*10)+0.1)==0 x1(1,i)=1;c=0; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 1 & floor ((yd*10)+0.1)==1 x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 1 & floor ((yd*10)+0.1)==0 x1(1,i)=0;c=0; end

23

59 60

else

61 62

63 64

65 66

i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ((yd * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd*10)+0.1)==0 x1(1,i)=0;c=0; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 1 & floor ((yd*10)+0.1)==0 x1(1,i)=1;c=0; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd*10)+0.1)==1 x1(1,i)=1;c=1; end

67 68 end 69 70 end disp ( x 1 , ’ S u b t ra c t i o n o f 45 fro m 228= ’ ) 71 72 e n d f u n c t i o n 73 a d d ( 1 0 1 0 1 1 0 1 , 1 0 0 0 1 1 0 1 ) 74 s u b ( 1 1 1 0 0 1 0 0 , 0 0 1 0 1 1 0 1 )

Scilab code Exa 1.16 Addition

1 2 3 4 5 6

/ / E x am pl e 1 . 1 6 // Additi on / / P ag e n o . 1 4 clc ; close ; clear ;

// fun ctio n

function [ x 1 ] = a d d ( x , y )

f o r a dd it io n o f b i n a ri e s 7 8 9

c=0; printf ( ’ A d d i t i on o f %. 4 i and %. 4 i = ’ , x , y ) for i = 1 : 4

24

10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

x1(1,i)=0 end for i = 4 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=1;c=0; else x1(1,i)=0;c=1; end else i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=0;c=0; else x1(1,i)=1;c=0; end end end

disp ( x 1 ) endfunction add(0010,0101); add(1110,1011); add(1110,0101); add(0010,1011);

25

44 a d d ( 1 1 1 0 , 0 0 1 0 ) ; 45 a d d ( 0 0 0 0 , 0 0 0 0 ) ;


1 2 3 4 5 6


// fun ctio n


f o r a dd it io n o f b i n a ri e s 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

c=0; printf ( ’ A d d i t i on o f %. 4 i and %. 4 i = ’ , x , y ) for i = 1 : 5 x1(1,i)=0 end for i = 5 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=1;c=0; else x1(1,i)=0;c=1;

26

end

26 27 28

else

29 30

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=0;c=0; else x1(1,i)=1;c=0; end

end end disp ( x 1 ) endfunction

add(0010,0101); add(1101,1010); add(1101,0101); add(0010,1010); add(1101,0010); add(1111,0000);


1 2 3 4 5 6


// fun ctio n


f o r a dd it io n o f b i n a ri e s 27

7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

c=0; printf ( ’ A d d i t i on o f %. 4 i and %. 4 i = ’ , x , y ) for i = 1 : 5 x1(1,i)=0 end for i = 5 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=1;c=0; else x1(1,i)=0;c=1; end else i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=0;c=0; else x1(1,i)=1;c=0; end end end

31 32 33 34 35 36 37 38 disp ( x 1 ) 39 e n d f u n c t i o n 40

28

41 a d d ( 0 1 0 0 , 0 1 0 1 ) ; 42 a d d ( 1 1 0 0 , 1 0 1 1 ) ; 43 a d d ( 1 0 0 0 , 1 0 0 0 ) ;


1 2 3 4 5 6


// fun ctio n


f o r a dd it io n o f b i n a ri e s 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

c=0; printf ( ’ A d d i t i on o f %. 4 i and %. 4 i = ’ , x , y ) for i = 1 : 5 x1(1,i)=0 end for i = 5 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=1;c=0; else

29

25 26 27 28

x1(1,i)=0;c=1; end

else

29 30

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=0;c=0; else x1(1,i)=1;c=0; end

end end disp ( x 1 ) endfunction

add(0010,0101); add(11110,11011); add(1000,0101); add(00010,11011); add(11110,00010); add(11111,0000);

Scilab code Exa 1.20 Subtraction

1 2 3 4 5

/ / E x am pl e 1 . 2 0 // Subtra ction / / P ag e n o . 1 6 clc ; close ; clear ; function [ x 1 ] = a d d ( x , y )

// fun ctio n

f o r a dd it io n o f b i n a ri e s 30

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

c=0; for i = 1 : 5 x1(1,i)=0 end for i = 5 : - 1 : 1 x=x/10; x d = x - fix ( x ) x=x-xd; y=y/10; y d = y - fix ( y ) y=y-yd; if c = = 1 then i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=1;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=1;c=0; else x1(1,i)=0;c=0; end else i f f lo or ( ( x d * 10 ) + 0 . 1) = = 1 & floor ( ( y d * 1 0 ) + 0 . 1 ) = = 1 then x1(1,i)=0;c=1; e l se i f f l oo r( ( x d * 10 ) + 0 . 1) = = 0 & floor ((yd *10)+0.1)==0 x1(1,i)=0;c=0; else x1(1,i)=1;c=0; end end end disp ( x 1 , ’ A d d it i o n o f 1 73 and 141= ’ ) endfunction

29 30 31 32 33 34 35 36 37 38 a d d ( 0 1 0 0 , 1 0 1 1 ) ; 39 a d d ( 1 1 0 0 , 0 1 0 1 ) ;

31

40 a d d ( 1 0 0 0 , 1 0 0 0 ) ;

Scilab code Exa 1.23 Multiplication

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

/ / E x am pl e 1 . 2 3 // M u l t i p l i c a t i o n / / P ag e n o . 1 8 clc ; clear ; close ; function [ x 1 ] = m u l ( x , y ) for i = 1 : 8 x1(1,i)=0 end printf ( ’ M u l t i p l i c a t i o n o f %. 4 i and %. 4 i = ’ , x , y ) x=x*y; c=0; for i = 8 : - 1 : 1 x=x/10; xd = floor ( ( x - fix ( x ) ) * 1 0 + 0 . 1 ) if c = = 1 then if x d = = 0 then x1(1,i)=1;c=0 elseif x d = = 1 x1(1,i)=0; c=1; elseif x d = = 2 x1(1,i)=1;c=1; end else if x d == 0 | xd = =1 then x1(1,i)=xd;c=0 elseif x d = = 2 x1(1,i)=0;

32

30 i=i-1;c=1; end 31 32 end end 33 34 disp ( x 1 ) 35 e n d f u n c t i o n 36 m u l ( 1 1 1 0 , 1 0 1 1 ) ;


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

/ / E x am pl e 1 . 2 4 // M u l t i p l i c a t i o n / / P ag e n o . 1 8 clc ; clear ; close ; function [ x 1 ] = m u l ( x , y ) for i = 1 : 8 x1(1,i)=0 end printf ( ’ M u l t i p l i c a t i o n o f %. 4 i and %. 4 i = ’ , x , y ) x=x*y; c=0; for i = 8 : - 1 : 1 x=x/10; xd = floor ( ( x - fix ( x ) ) * 1 0 + 0 . 1 ) if c = = 1 then if x d = = 0 then x1(1,i)=1;c=0 elseif x d = = 1 x1(1,i)=0; c=1; elseif x d = = 2 x1(1,i)=1;c=1;

33

24 end else 25 26 if x d == 0 | xd = =1 then x1(1,i)=xd;c=0 27 28 elseif x d = = 2 x1(1,i)=0; 29 30 i=i-1;c=1; end 31 32 end 33 end 34 disp ( x 1 ) 35 e n d f u n c t i o n 36 m u l ( 1 1 1 0 , 1 0 1 1 ) ;

Scilab code Exa 1.25 Division

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 1 . 2 5 // Div isi on / / P ag e n o . 1 9 clc ; close ; clear ; function [ c o ] = c o m ( x , y ) co=1; for i = 1 : length ( x ) if x ( i ) > y ( i ) then break elseif x ( i ) = = y ( i ) continue else c o = 0 ; break end end endfunction function [ x 1 ] = s u b ( x , y )

// f u n c t i o n f o r 34

s u bt r ac t io n o f b i n a r i e s 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

c=0;m=0; for i = 1 : 5 x1(1,i)=0 end for i = 5 : - 1 : 1 if c = = 1 then if x ( i ) == 0 & y ( i ) == 1 then x1(1,i)=0;c=1; elseif x ( i ) == 0 & y ( i ) == 0 x1(1,i)=1;c=0; elseif x ( i ) == 1 & y ( i ) == 1 x1(1,i)=1;c=1; elseif x ( i ) == 1 & y ( i ) == 0 x1(1,i)=0;c=0; end else if x ( i ) == 1 & y ( i ) == 1 then x1(1,i)=0;c=0; elseif x ( i ) == 0 & y ( i ) == 0 x1(1,i)=0;c=0; elseif x ( i ) == 1 & y ( i ) == 0 x1(1,i)=1;c=0; elseif x ( i ) == 0 & y ( i ) == 1 x1(1,i)=1;c=1; end end end disp ( x 1 , ’ R e ma i nd e r = ’ ) endfunction d1=11011001;d2=01011;d22=[0,0,0,0,0] for i = 8 : - 1 : 1 d3=d1/10; d i v ( 1 , i ) =int ( 1 0 * ( d 3 - int ( d 3 ) ) ) d1=d1/10 end for i = 5 : - 1 : 1 d3=d2/10;

35

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84

d 2 1 ( 1 , i ) =int ( 1 0 * ( d 3 - int ( d 3 ) ) + 0 . 5 ) d2=d2/10 end div1(1,1)=0 for j = 1 : 4 div1(1,j+1)=div(1,j) end for i 1 = 1 : 5 printf ( ’ A f te r S te p %i : \n ’ , i 1 ) if c o m ( d i v 1 , d 2 1 ) = = 1 then dis(1,i1)=1 n=sub(div1,d21) else dis(1,i1)=0 n=sub(div1,d22) end disp ( d i s , ’ D i v i s o r = ’ ) if i 1 = = 5 then break end for j = 1 : 5 if j <5 then div1(1,j)=n(j+1) else div1(1,j)=div(1,i1+4) end end printf ( ’ \n\n\n\n ’ ) end


36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

/ / E x am pl e 1 . 2 6 // M u l t i p l i c a t i o n / / P ag e n o . 1 9 clc ; clear ; close ;

function [ x 1 ] = m u l ( x , y ) for i = 1 : 8 x1(1,i)=0 end printf ( ’ M u l t i p l i c a t i o n o f %. 4 i and %. 4 i = ’ , x , y ) x=x*y; c=0; for i = 1 0 : - 1 : 1 x=x/10; xd = floor ( ( x - fix ( x ) ) * 1 0 + 0 . 1 ) if c = = 1 then if x d = = 0 then x1(1,i)=1;c=0 elseif x d = = 1 x1(1,i)=0; c=1; elseif x d = = 2 x1(1,i)=1;c=1; end else if x d == 0 | xd = =1 then x1(1,i)=xd;c=0 elseif x d = = 2 x1(1,i)=0; i=i-1;c=1; end end end for i = 1 : 1 0 if x 1 ( 1 , i ) = = 1 then x1(1,i-1)=1; break

37

39 end end 40 41 disp ( x 1 ) 42 e n d f u n c t i o n 43 m u l ( 1 1 1 0 , 1 0 1 1 ) ;

Scilab code Exa 1.29 Normalized Floating Point Representation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 . 2 9 // N o r ma l iz ed F l o a t i n g P o in t R e p r e s e n t a t i o n / / P ag e n o . 2 3 clc ; clear ; close ; function [ ] = f p ( x ) x1=x; if x >0 then for i = 1 : 1 0 x=x/10 if i nt ( x ) = = 0 then break end end printf ( ’ \n

%i \ n No rm al iz ed F l o a t i n g P o in t R e p r e s e n t a t i o n o f %g = %. 4 f x 10 ’ , i , x 1 , x ) 16 17 18 19 20 21

else

for i = 1 : 1 0 x=x*10 i f c ei l( x ) ~ = 0 then break end

38

22 23 24

end x=x/10;i=i-1; printf ( ’ \n

−%i \ n No rm a li ze d F l o a t i n g P o in t R e p r e s e n t a t i o n o f %g = %. 4 f x 10 ’ , i , x 1 , x ) 25 26 27 28 29 30 31 32

end endfunction x=[25.12,-0.00287,87000]; for i = 1 : 3 fp(x(i)) end

Scilab code Exa 1.30 Add

1 2 3 4 5 6 7

/ / E x am pl e 1 . 3 0 //Add / / P ag e n o . 2 6 clc ; clear ; close ; a=0.4532e7;b=0.5427e7; c=a+b printf ( ’ A d d i t i o n o f %. 6 g a nd %. 6 g = %. 6 g ’ , a , b , c )


1 / / E x am pl e 1 . 3 1

39

2 3 4 5 6 7

//Add / / P ag e n o . 2 6 clc ; clear ; close ; a=0.4532e5;b=0.5427e7; c=a+b printf ( ’ A d d i t i o n o f %. 4 g a nd %. 6 g = %. 6 g ’ , a , b , c )


1 2 3 4 5 6 7

/ / E x am pl e 1 . 3 2 //Add / / P ag e n o . 2 6 clc ; clear ; close ; a=0.4532e3;b=0.5427e7; c=a+b printf ( ’ A d d i t i o n o f %. 2 g a nd %. 6 g = %. 4 g ’ , a , b , c )


1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 . 3 3 //Add / / P ag e n o . 2 7 clc ; clear ; close ; a=[0.4632e3,0.4632e99];b=[0.5427e3,0.5427e99]; for i = 1 : 2 c(i)=a(i)+b(i) printf ( ’ \ n A d d i t i o n o f %. 2 g a nd %. 2 g = %. 5 g\n ’ ,a ( i),b(i),c(i))

40

10 end

Scilab code Exa 1.34 Subtraction

1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 . 3 4 // Subtra ction / / P ag e n o . 2 7

clc ; clear ; close ; a=[0.5427e-3,0.9627e4,0.9627e-99];b=[0.9632e -4,0.9622e4,0.9622e-99]; for i = 1 : 3 c(i)=a(i)-b(i) printf ( ’ \ n S u b t r a c t i o n o f %. 2 g f ro m %. 3 g = %. 6 g \n ’ ,a(i),b(i),c(i)) end


1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 . 3 5 // M u l t i p l i c a t i o n / / P ag e n o . 2 8 clc ; clear ; close ; a=[ 0.9632 e12 ,0.1132 e12 ,0.1132 e52 ,0.1132e -52];b =[0.5427e-15,0.1027e15 ,0.1027e50 ,0.1027e-50]; for i = 1 : 4 c(i)=a(i)*b(i) printf ( ’ \ n M u l t i p l i c a t i o n o f %. 3 g and %. 2 g = %. 6 g \n ’ , a ( i ) , b ( i ) , c ( i ) ) end

41

Scilab code Exa 1.36 Division

1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 . 3 6 // Div isi on / / P ag e n o . 2 8

clc ; clear ; close ; a = [ 0 . 1 1 3 2 e 1 , 0 . 1 1 3 2 e - 6 , 0 . 1 1 3 2 e 6 ] ; b = [ 0 . 1 0 0 0 e - 9 9 , 0 . 1 00 0 e99 ,0.1000 e3]; for i = 1 : 3 c(i)=a(i)/b(i) printf ( ’ \ n D i v i s i o n o f %. 2 g by %. 3 g= %. 3 g \n ’ , a ( i ) ,b(i),c(i)) end

42

Chapter 2 Scope of Numerical and Mathematical Methods

Scilab code Exa 2.4 Solving Simultaneous Linear Equation

1 2 3 4 5 6 7 8 9 10

/ / E xa mp le 2 . 4 / / S o l v i n g S i m ul t a ne o u s L i n e a r E qu a ti o n / / P a ge 3 6 clc ; close ; clear ;

/ / e q 1 = 5 x −331y=3.5 / / e q 2 = 6 x −397y=5.2 A=[5,-331;6,-397]; B=[3.5;5.2]; // f i n d i n g C = inv ( A ) * B ;

i n v e r s e w it h v a l ue s 11 disp ( C ( 1 , 1 ) , ’ V al ue o f x= ’ ) ; 12 disp ( C ( 2 , 1 ) , ’ V al ue o f y= ’ ) ;

43

v a l u e by m u l t i p l y i n g

Scilab code Exa 2.6 Integration

1 2 3 4 5

/ / E xa mp le 2 . 6 // Int egra tio n / / P ag e n o . 3 6 clc ; clear ; close ; disp ( i n t e g r a t e( ’ 1 / x ’ , ’ x ’ , exp ( - 4 ) , 1 ) , ’ I n t e g r a t i o n Value=’ ); // p e rf o rm i ng i n t e g r a t i o n w it h

r e s p e c t t o dx

44

Chapter 3 Errors and Their Propagation

Scilab code Exa 3.1 Limiting Error

1 2 3 4 5 6 7

/ / E xa mp le 3 . 1 // Limiting Error / / P ag e n o . 4 5 clc ; clear ; close ; R=1000; // l i m i t i n g e r r o r c a l c u l a t i o n e=0.1*1000; printf ( ’ Mag ni tude o f t he R e s i s t o r r e s i s t e n c e (R) =\ n% i <= R <= %i ’ , R - e , R + e )

Scilab code Exa 3.2 Known Error

1 2 3 4

/ / E xa mp le 3 . 2 //Known Er ro r / / P ag e n o . 4 6 clc ; clear ; close ;

45

5 6 7 8 9

l=28;d=5; v=%pi*l*(d/2)^2; printf ( ’ \ n Vol ume o f C y l i n d e r = %f c u . cm ’ , v ) ; re_d=0.1;re_l=-0.5; // r e l a t i v e e r r o r re_v=2*re_d+re_l;

computation 10 printf ( ’ \n\ n R e l a t i v e e r r o r i n v ol um e= %f %% ’ , r e _ v ) ;

Scilab code Exa 3.3 Absolute Relative and Percentage Errors

1 2 3 4 5 6 7 8 9

/ / E xa mp le 3 . 3 // A b so lu te , R e l a t i v e and P e rc e ta g e E r r o rs / / P ag e n o . 4 8

clc ; clear ; close ; x=0.00006;x1=0.00005; // a b s o l u t e e r r o r ex=x-x1; Ex=ex/x1; // r e l a t i v e e r r o r //percentage erro r px=100*Ex; printf ( ’ \ n A b s ol u t e E r r o r= %f \ n R e l a t i v e E r ro r= %f \ n P e r c e n t a g e E r r o r = %f %% ’ , e x , E x , p x ) ;


1 2 3 4 5 6

/ / E xa mp le 3 . 4 // A b so lu te , R e l a t i v e and P e rc e ta g e E r r o rs / / P ag e n o . 4 8 clc ; clear ; close ; x=100500;x1=100000; // a b s o l u t e e r r o r ex=x-x1;

46

7 Ex=ex/x1; // r e l a t i v e e r r o r 8 px=100*Ex; //percentage erro r 9 printf ( ’ \ n A b s ol u t e E r r o r= %f \ n R e l a t i v e E r ro r= %f \ n P e r c e n t a g e E r r o r = %f %% ’ , e x , E x , p x ) ;


/ / E xa mp le 3 . 5 / / A b so l ut e , R e l a t i v e and P e r ce n t ag e E r r o r s / / P ag e n o . 5 2

1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17

18 19 20 21

clc ; clear ; close ; x=9.12345;y=7.654321; x1=9.1234;y1=7.6543; / / on a 5 d e c im a l c om pu te r // a b s o l u t e e r r o r o f x ex=x-x1; ey=y-y1; // a b s o l u t e e r r o r o f y z1=x1+y1; printf ( ’ \ n A b s ol u t e E r r o r i n x= %f ’ , e x ) ; printf ( ’ \ n A b s ol u t e E r r o r i n y= %f ’ , e y ) ; printf ( ’ \ n Ad di ti on on a 5 d ec i ma l co mp ut er y i e l d s = % . 5 g ’ ,z1); z2=16.777; printf ( ’ \ n A b s ol u t e T o t a l E r r o r= % f ’ , x + y - z 2 ) ; printf ( ’ \ n A b s o l u t e P r o p a g at e d E r r o r = %f ’ , x + y - z 1 ) ; printf ( ’ \ n A b s o l u t e Round− o f f E r r o r= %. 4 g ’ , z 1 - z 2 ) ; printf ( ’ \ n R e a l t i v e T o t a l E r r o r= %. 4 g ’ , ( x + y - z 2 ) / ( x + y ) ); printf ( ’ \ n R e l a t i v e P r op a g at e d E r r o r= %. 2 g ’ , ( x + y - z 1 ) /(x+y)); printf ( ’ \ n R e l a t i v e Round− o f f E r r o r= %. 3 g ’ , ( z1 - z 2 ) / ( x +y)); printf ( ’ \nBound on t he p ro pa ga te d r e l a t i v e e r r o r = %f ’ ,2*10^-4); printf ( ’ \nBound on t he t o t a l r e l a t i v e e r r o r= %f ’

47

,3*10^-4); 22 printf ( ’ \nAs we ca n s e e t h at b ot h t he p ro pa ga te d and

t o t a l r e l a t i v e e r r o r a re l e s s th a n t h e i r bound values ’)

48

Chapter 4 Programming Tools and Techniques

Scilab code Exa 4.1 Quadratic Equation

1 2 3 4 5 6 7 8 9 10 11

/ / E xa mp le 4 . 1 / / Q u a d r a t i c E q u a t io n / / P ag e n o . 9 6

clc ; clear ; close ; a = input ( ” E nt e r v a l u e o f a= ” ) ; b = input ( ” E nt er v l au e o f b= ” ) ; c = input ( ” E nt e r v a l u e o f c= ” ) ; x 1 = ( - 1 * b + sqrt ( ( b ^ 2 ) - 4 * a * c ) ) / ( 2 * a ) ; x 2 = ( - 1 * b - sqrt ( ( b ^ 2 ) - 4 * a * c ) ) / ( 2 * a ) ; printf ( ’ \ n 1 s t R oo t= %f ’ , x1 ); printf ( ’ \ n2nd Root= %f ’ , x2 );

Scilab code Exa 4.2 Database Management

49

//1st root / / 2nd r o o t

1 2 3 4 5

/ / E xa mp le 4 . 2 / / D a t a b a s e M an ag em en t / / P ag e n o . 1 1 2 clc ; clear ; close ; M = [ 1 2 , 2 5 , 2 1 , 3 5 ; 2 5 , 7 , 2 3 , 2 9 ; 1 0 , 2 7 , 7 , 3 6 ; 2 6 , 2 6 , 2 6 , 3 5 ; 2 9 ,0 , 2 3 , 3 0 ] ;

//marks 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

// c a l c u l a t i o n o f c om po si te s c o r e for i = 1 : 5 , j=1;k=0; max1=M(i,j); if ( m a x 1 < M ( i , j + 1 ) ) max1=M(i,j+1) else k=1; end , if ( M ( i , j + 2 ) > M ( i , j + k ) ) max2=M(i,j+2);

else

max2=M(i,j);

end , CS(i,1)=max1+max2+M(i,4); end I =[ ’ R eg . No . ’ , ’ Name o f S t u d e n t s ’ , ’ T e s t 1 ’ , ’ T e s t 2 ’ , ’ T e st 3 ’ , ’ F i n a l ’ ; ’CS/01 ’ , ’C.V. Rajan ’ , ’ 1 2 ’ , ’ 2 5 ’ , ’ 2 1 ’ , ’ 3 5 ’ ; ’CS/02 ’ , ’B. X. Roy ’ , ’ 2 5 ’ , ’ 0 7 ’ , ’ 2 3 ’ , ’ 2 9 ’ ; ’CS/03 ’ , ’ P . C . S a s i k u m a r ’ , ’ 1 0 ’ , ’ 2 7 ’ , ’ 0 7 ’ , ’ 3 6 ’ ; ’CS/04 ’ , ’B .D . Box ’ , ’ 2 6 ’ , ’ 2 6 ’ , ’ 2 6 ’ , ’ 3 5 ’ ; ’CS/05 ’ , ’K.K. Mukher jee ’ , ’ 2 9 ’ , ’ 0 ’ , ’ 2 3 ’ , ’ 3 0 ’ ;] printf ( ’ \n ’ ) for i = 1 : 6 for j = 1 : 6 if ( j > 2 ) printf ( ’ \ t ’ )

50

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

end

’ ,I(i,j)); printf ( ’ %s if ( i ~ = 1 ) if ( j > 2 ) printf ( ’ \ t ’ ) end printf ( ’ ’) end if ( i == 1 & j = =6 ) printf ( ’ C o mp o s it e S c o r e \n ’ ) end

end

end

if ( i ~ = 1 ) printf ( ’ %i \n ’ , C S ( i - 1 , 1 ) ) ;

end

/ / d i s p ( CS , ’ C o m p o s i t e S c o r e ’ , I ) ;

max1=CS(1,1);j=1; for i = 2 : 5 if ( m a x 1 < C S ( i , 1 ) ) max1=CS(i,1);j=i; end , end printf ( ’ \n\ n To pp er i s : \ n%s \ t%s \ t%s ’ , I ( 1 , 1 ) , I ( 1 , 2 ) , ’ C o mp o s it e S c o r e ’ ) 65 printf ( ’ \nCS/0%i\ t \ t% s \ t \ t \ t%i ’ , j , I ( j + 1 , 2 ) , C S ( j , 1 ) )

51

Chapter 5 Solutions of Algebraic and Transcendental Equations

Scilab code Exa 5.1 Bisection Method

1 2 3 4 5 6 7 8

/ / E xa mp le 5 . 1 / / B i s e c t i o n Metho d / / P ag e n o . 1 4 5

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=2ˆx−3∗x ’ ) x1=0;x2=2;e=0.001;i=0; printf ( ’ I t e r a t i o n \ t x 1 \ t \ t x 2 \ t \ t z \ t \ t f ( z ) \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 w h i le a bs ( x 1 - x 2 ) > e 10 z=(x1+x2)/2 11 %i \ t \ t%f \ t%f \ t%f \ t%f \n ’ , i , x 1 , x 2 , z , f printf ( ’ (z)) 12 if f ( z ) * f ( x 1 ) > 0 x1=z 13 14 else

52

15 x2=z end 16 17 i=i+1 18 end 19 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n i s %g a f t e r %i I t e r a t i o n s ’ , z , i - 1 )


1 2 3 4 5 6 7 8

/ / E xa mp le 5 . 2 / / B i s e c t i o n Metho d / / P ag e n o . 1 4 7

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ x−2∗x+2 ’ ) x1=0;x2=2;e=0.001;i=0; printf ( ’ I t e r a t i o n \ t x 1 \ t \ t x 2 \ t \ t z \ t \ t f ( z ) \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 w h i le a bs ( x 1 - x 2 ) > e 10 z=(x1+x2)/2 printf ( ’ 11 %i \ t \ t%f \ t%f \ t%f \ t%f \n ’ , i , x 1 , x 2 , z , f (z)) if f ( z ) * f ( x 1 ) > 0 12 13 x1=z else 14 15 x2=z 16 end 17 i=i+1 18 end 19 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n i s %g a f t e r %i I t e r a t i o n s ’ , z , i - 1 ) 20

53

21 printf ( ’ \n\n\ nNote : Th ere a r e c o mp u ta t io n al

erro rs i n t he a n sw e r g i ve n by t he book f o r t h i s ex a m pl e ’

)

Scilab code Exa 5.3 Regula Falsi Method

1 2 3 4 5 6 7

/ / E xa mp le 5 . 3 / / R e gu l a F a l s i Method / / P ag e n o . 1 4 9

8

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ3 −3∗x−5 ’ ) x1=2;x2=3;e=0.00001 printf ( ’ n\ t x 1 \ t \ t f ( x 1 ) \ t \ t x 2 \ t \ t f ( x 2 ) \ t \ t x 3 \ t \ t f ( x 3 ) ’) printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 for i = 0 : 1 9 10 x3=x2*f(x1)/(f(x1)-f(x2))+x1*f(x2)/(f(x2)-f(x1)) printf ( ’ %i \ t%f \ t%f \ t%f \ t%f \ t%f \ t%f \n ’ , i , x 1 , f ( x 1 11 ),x2,f(x2),x3,f(x3)) if f ( x 1 ) * f ( x 3 ) > 0 then 12 13 x1=x3 else 14 15 x2=x3 end 16 17 if a bs ( f ( x 3 ) ) < e then 18 break 19 end 20 end 21 printf ( ’ \n\ n T he r ef o re t he s o l u t i o n i s %. 1 0 g ’ , x 3 )

54

Scilab code Exa 5.4 Ridders Method

1 2 3 4 5 6 7

/ / E xa mp le 5 . 4 / / R i d d e r s M et ho d / / P ag e n o . 1 5 3

8

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ3 −3∗x−5 ’ ) x1=2;x2=3;e=0.00001 printf ( ’ n\ t x 1 \ t \ t f ( x 1 ) \ t \ t x 2 \ t \ t f ( x 2 ) \ t \ t x 3 \ t \ t f ( x 3 ) s i g n \ t x4 ’ ) \t printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 for i = 0 : 8 x3=(x1+x2)/2 10 11 a=f(x1)-f(x2); s = a * abs ( 1 / a ) 12 13 x 4 = x 3 + ( x 3 - x 2 ) * ( s * f ( x 3 ) ) /sqrt ( f ( x 3 ) - f ( x 1 ) * f ( x 2 ) ) printf ( ’ %i \ t%f \ t%f \ t%f \ t%f \ t%f \ t%f 14 %i \ t%f \n ’ ,i,x1,f(x1),x2,f(x2),x3,f(x3),s,x4) if f ( x 1 ) * f ( x 4 ) > 0 then 15 16 x1=x4 17 else 18 x2=x4 19 end if a bs ( f ( x 4 ) ) < e then 20 21 break end 22 23 end 24 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n i s %g a f t e r %i I t e r a t i o n s ’ , x 4 , i ) 25 printf ( ’ \n\n\ nThere a r e c om pu ta ti on e r r o r i n t he

55

a ns we rs g i ve n by t he book i n t h i s ex am ple \n\n ( v al ue o f x1 i s u s e d i n s t e a d o f x2 ) ’ )

Scilab code Exa 5.5 General Iterative Method

1 2 3 4 5 6 7 8 9 10 11 12 13

/ / E xa mp le 5 . 5 / / G e ne r al I t e r a t i v e Method / / P ag e n o . 1 5 4

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x= s q r t (3 + 5 / x ) ’ ) printf ( ’ n\ t x \ t \ t f ( x ) \n ’ ) printf ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’ ) x=2; for i = 1 : 8 printf ( ’ %i \t% . 1 0 f \t% . 1 0 f \n ’ ,i,x,f(x)) x=f(x); end printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i I t e r a t i o n s i s %. 1 0 f ’ , i , x )

Scilab code Exa 5.6 Linear Iterative Method

1 2 3 4 5 6 7

/ / E xa mp le 5 . 6 / / L i n e a r I t e r a t i v e Method / / P ag e n o . 1 5 9 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=1+s i n (x ) /10 ’ ) printf ( ’ n\ t x \ t \ t f ( x ) \n ’ ) printf ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’ )

56

8 9 10 11 12 13

x=0; for i = 1 : 7 . 1 0 f \n ’ ,i,x,f(x)) printf ( ’ %i \t% . 1 0 f \t% x=f(x); end printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i I t e r a t i o n s i s %. 1 0 f ’ , i , x )

Scilab code Exa 5.7 Aitkens Method

1 2 3 4 5 6 7

/ / E xa mp le 5 . 7 / / A i t k e n s M et ho d / / P ag e n o . 1 6 1 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=exp(−x ) ’ ) printf ( ’ n\ t x 0 \ t \ t x 1 \ t \ t x 2 \ t \ t x 3 \ t \ t y \ t \ t d x 0 \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 8 x0=0.5;e=0.0001 9 for i = 1 : 3 x1=f(x0);x2=f(x1);x3=f(x2); 10 11 y=x3-((x3-x2)^2)/(x3-2*x2+x1) dx0=y-x0; 12 13 printf ( ’ %i \t% . 1 0 f \t% 14 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \ t% . 1 0 f \n ’ ,i,x0,x1,x2,x3,y,dx0) 15 x0=y; 16 if a bs ( x 0 ) < e then 17 break ; end 18 19 end 20 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i

57

Iterations

i s %. 1 0 f ’ , i , y )

Scilab code Exa 5.8 Newton Raphson Method

1 2 3 4 5 6 7

/ / E xa mp le 5 . 8 //Newton Raphson Method / / P ag e n o . 1 6 3

8

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=x−exp(−x ) ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=1+exp (−x ) ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=0.5;e=0.00001 10 for i = 1 : 4 11 x1=x0-f(x0)/f1(x0) 12 e1 = abs ( x 0 - x 1 ) printf ( ’ %i \t% . 1 0 f \t% 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n ’ ,i-1,x0,f(x0),f1(x0),x1,e1) x0=x1; 14 15 if a bs ( x 0 ) < e then break ; 16 17 end 18 end 19 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i I t e r a t i o n s i s %. 1 0 f ’ , i , x 1 )

58

Scilab code Exa 5.9 Modified Newton Raphson Method

1 2 3 4 5 6 7

/ / E xa mp le 5 . 9 / / M o d i f i e d N ew to n R a ph s on M et ho d / / P ag e n o . 1 6 5

8

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=exp ( x ) −3∗x−s i n ( x ) ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=exp ( x )−3−c o s ( x ) ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=0;e=0.00001 10 for i = 1 : 4 11 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) 12 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n printf ( ’ %i \t% . 1 0 f \t% ’ ,i-1,x0,f(x0),f1(x0),x1,e1) 14 x0=x1; 15 if a bs ( x 0 ) < e then 16 break ; 17 end 18 end 19 printf ( ’ \n\ n Th er ef or e , t he r o o t i s %. 1 0 f ’ , x 1 )


1 2 3 4 5

/ / E x am pl e 5 . 1 0 //Newton Raphson Method / / P ag e n o . 1 6 7 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=x ∗ exp(−x ) ’ )

59

6 deff ( ’ x= f 1 ( x ) ’ , ’ x=exp(−x )−x ∗ exp(−x ) ’ ) 7 printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) 8 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=2;e=0.00001 10 for i = 1 : 1 1 11 x1=x0-f(x0)/f1(x0) 12 e1 = abs ( x 0 - x 1 ) 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n printf ( ’ %i \t% . 1 0 f \t% ’ ,i-1,x0,f(x0),f1(x0),x1,e1) x0=x1; 14 15 if a bs ( x 0 ) < e then break ; 16 17 end 18 end 19 printf ( ’ \n\ n Th er ef or e , t h i s i s n ot c o nv e rg e nt ( i . e . ) divergent ’)


1 2 3 4 5 6 7

/ / E x am pl e 5 . 1 1 //Newton Raphson Method / / P ag e n o . 1 6 7

8

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=xˆ3−x−3 ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=3∗xˆ2 −1 ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 60

9 x0=0;e=0.00001 10 for i = 1 : 1 1 11 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) 12 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n printf ( ’ %i \t% . 1 0 f \t% ’ ,i-1,x0,f(x0),f1(x0),x1,e1) 14 x0=x1; if a bs ( x 0 ) < e then 15 16 break ; 17 end 18 end 19 printf ( ’ \n\ n Th er ef o re , i t i s c y c l i c i n n at ur e ’ )


1 2 3 4 5 6 7


8

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=atan ( x ) ’ ) deff ( ’ x= f 1 ( x ) ’ , ’x=1/(1+x ˆ2 ) ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=1.45;e=0.00001 10 for i = 1 : 1 2 11 x1=x0-f(x0)/f1(x0) 12 e1 = abs ( x 0 - x 1 ) printf ( ’ %i \t% .5 g 13 \ t \t% . 5 g \ t \t% . 5 g \ t \t% . 5 g \ t \t% . 5 g \n ’ , i - 1 , x 0 , f ( x 0 ) , f 1 ( x 0 ) , x 1 , e 1 ) x0=x1; 14

61

15 if a bs ( x 0 ) < e then break ; 16 17 end 18 end 19 printf ( ’ \n\ n Th er ef or e ,

it

i s d i v e r g e nt ’ )

Scilab code Exa 5.13 Secant Method

1 2 3 4 5 6 7

/ / E x am pl e 5 . 1 3 / / S e c a n t M et ho d / / P ag e n o . 1 7 0

8

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=exp ( x ) −3∗x−s i n ( x ) ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=exp ( x )−3−c o s ( x ) ’ ) printf ( ’ n\ t x n \ t \ t f ( x n ) \ t \ tXn+1\ t \ t f (Xn+1)\ t \tXn+2\ t \ t E r r o r \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=0.567123008;x1=1;e=0.00001 10 for i = 1 : 9 x2=x1-f(x1)*(x1-x0)/(f(x1)-f(x0)) 11 12 e1 = abs ( x 0 - x 2 ) printf ( ’ %i \t% . 1 0 f \t% 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \ t% . 1 0 f \n ’ ,i-1,x0,f(x0),x1,f(x1),x2,e1) x0=x1; 14 15 x1=x2 16 if a bs ( x 0 ) < e then 17 break ; 18 end 19 end 20 printf ( ’ \n\ n Th er ef or e , t he r o o t i s %. 1 0 f ’ , x 2 )

62

Scilab code Exa 5.14 Kizner Method

1 2 3 4 5 6 7

/ / E x am pl e 5 . 1 4 / / K i z n e r M et ho d / / P ag e n o . 1 7 2 clc ; clear ; close ; h2=0.00001 deff ( ’x=f ( x ) ’ , ’ x=2∗x−3−c o s ( x ) ’ ) deff ( ’ y=f1 ( x , y ) ’ , ’ y=h2 /(−x+y) ’ )

// f u n c t i o n f o r

differentiation 8 printf ( ’ n\ th \ t c \ t xn \ t \ t f ( x n ) \ t \ t F ( x n ) \ t \ t k 1 \ t \ t v \ t \ tXn+1\n ’ ) 9 printf ( ’ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 x 0 = 2 ; e = 0 . 0 0 0 0 1 ; h = 0 . 5 ; c = 0 . 5 ; 11 for i = 1 : 1 1 12 h1=-f(x0); F=f1(f(x0),f(x0+h2)) 13 14 k1=h1*F/2; v=h*f(x0)/(c*(f(x0+c+h)-f(x0+c)))-k1/c; 15 16 a=0; 17 for j = 0 : 3 18 a=a+(v^j)/factorial(j+1) 19 end x1=x0+k1*a 20 21 printf ( ’ %i \t%g\t%g\t% . 6 f \t% . 6 f \t% . 6 f \t% . 8 f \ t % . 5 f \t% . 6 f \n ’ , i - 1 , h , c , x 0 , f ( x 0 ) , F , k 1 , v , x 1 ) 22 x0=x1; if a bs ( x 0 ) < e then 23 24 break ; end 25

63

26 end 27 printf ( ’ \n\ n Th er ef or e , t he s o l u t i o n

i s %. 1 0 f ’ , x 1 )

Scilab code Exa 5.15 Brent Method

1 2 3 4 5 6 7 8 9 10

/ / E x am pl e 5 . 1 5 / / B r e n t M et ho d / / P ag e n o . 1 7 3

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ2+x−2 ’ ) x1=0;x2=0.5;x3=2; r=f(x2)/f(x3);s=f(x2)/f(x1);t=f(x1)/f(x3); q=(t-1)*(r-1)*(s-1); p=r*t*(s-1)*(x2-x3)-s*(1-r)*(x2-x1)+(t*s-r)*x2 printf ( ’ R oot i s : %. 1 0 g ’ , x 2 + ( p / q ) )

Scilab code Exa 5.19 Horner Method

1 2 3 4 5 6 7 8 9 10

/ / E x am pl e 5 . 1 9 / / H o r n e r M et ho d / / P ag e n o . 1 7 7 clc ; clear ; close ; deff ( ’ y=f ( x , a1 , a2 , a3 , a4 ) ’ , ’ y=a1 ∗ xˆ3+a2 ∗ xˆ2+a3 ∗x+a4 ’ ) k=1;m=2; a=[4;-13;-31;-275]; for i = 1 : 1 0 s=1;

64

11

12 13

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

s i = f ( s , a ( 1 ) , a ( 2 ) , a ( 3 ) , a ( 4 ) ) *abs ( 1 / f ( s , a ( 1 ) , a ( 2 ) , a(3),a(4))) while 1 a 1 = f ( s , a ( 1 ) , a ( 2 ) , a ( 3 ) , a ( 4 ) ) *abs ( 1 / f ( s , a ( 1 ) , a (2),a(3),a(4))) if s i ~ = a 1 then d(i)=s-1 break end si=a1; s=s+1; end b(1)=a(1) for j = 1 : 3 for k = 1 : 4 - j b(k+1)=a(k+1)+b(k)*d(i) a(k+1)=b(k+1) end end for j = 1 : 3 a(j+1)=10^j*a(j+1) end

end printf ( ’ The p o s i t i v e r o ot for i = 2 : 1 0 printf ( ’ %i ’ , d ( i ) ) end

i s %i . ’ , d ( 1 ) )

Scilab code Exa 5.20 Laguerre Method

1 / / E x am pl e 5 . 2 0 2 / / L a g u e r r e M et ho d 3 / / P ag e n o . 1 8 0

65

4 5 6 7 8 9

10

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’y=xˆ3+xˆ2+10∗x−20 ’ ) deff ( ’ y= f 1 ( x ) ’ , ’ y=3∗xˆ2+2∗x+10 ’ ) deff ( ’ y= f 2 ( x ) ’ , ’ y=6∗x+2 ’ ) n=3; printf ( ’ i \ tn \ t x i \ t \ t P ( x ) \ t \ t P 1 ( x ) \ t \ t P 2 ( x ) \ t \ t P r o o t \ t \ t N r o o t \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 11 x i = 1 12 for i = 0 : 9 P r o o t = x i - ( n * f ( x i ) ) / ( f 1 ( x i ) +sqrt ( ( n - 1 ) * f 1 ( x i ) ^ 2 - n 13 *f(xi)*f2(xi))) N r o o t = x i - ( n * f ( x i ) ) / ( f 1 ( x i ) -sqrt ( ( n - 1 ) * f 1 ( x i ) ^ 2 - n 14 *f(xi)*f2(xi))) printf ( ’ %i \ t% i \ t%f \ t%f \ t%f \ t%f \ t%f \ t%f \n ’ , i , n , 15 xi,f(xi),f1(xi),f2(xi),Proot,Nroot) xi=Proot 16 17 end 18 printf ( ’ \n\ n P r o o t = % f \ n\ n N r o o t = %f ’ , P r o o t , N r o o t )

Scilab code Exa 5.21 Mullers Method

1 2 3 4 5 6 7 8

/ / E x am pl e 5 . 2 1 / / M u l l e r s Me th od / / P ag e n o . 1 8 2 clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ3−x−4 ’ ) zi=[1;2;3]; s =[ ” i ” , ” z 2 ” , ” z 0 ” , ” z 1 ” , ” f2 ” , ” f0 ” , ” f1 ” , ” a 0 ” , ” a 1 ” , ” a 2 ” , ” z r + ” , ” z r −” ]

66

9 li(1)=(zi(3,1)-zi(2,1))/(zi(2,1)-zi(1,1)) 10 h i ( 1 ) = z i ( 3 , 1 ) - z i ( 2 , 1 ) ; 11 for i = 2 : 6 for j = 1 : 3 12 13 fz(j,i-1)=f(zi(j,i-1)) end 14 15 di(i-1)=1+li(i-1) gi(i-1)=fz(1,i-1)*li(i-1)^2-fz(2,i-1)*di(i-1)^2+ 16 fz(3,i-1)*(li(i-1)+di(i-1)) 17 D 1 ( i - 1 ) = g i ( i - 1 ) +sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) D 2 ( i - 1 ) = g i ( i - 1 ) -sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i 18 -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) 19 if a bs ( D 1 ( i - 1 ) ) >abs ( D 2 ( i - 1 ) ) then li(i)=-2*fz(3,i-1)*di(i-1)/D1(i-1) 20 21 else li(i)=-2*fz(3,i-1)*di(i-1)/D2(i-1) 22 23 end 24 hi(i)=li(i)*hi(i-1); 25 z(i-1)=zi(3,i-1)+hi(i) 26 for j = 1 : 2 zi(j,i)=zi(j+1,i-1) 27 28 end zi(3,i)=z(i-1) 29 30 end 31 for i = 1 : 1 2 32 if i = = 1 then printf ( s ( i ) ) 33 34 for j = 1 : 5 35 printf ( ’ \ t \ t \ t%i ’ , j - 1 ) 36 end 37 elseif i < = 4 38 printf ( ’ \n %s ’ , s ( i ) ) 39 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f ’ , z i ( i - 1 , j ) ) 40 41 end

67

42 elseif i < = 7 printf ( ’ \n %s ’ , s ( i ) ) 43 44 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f ’ , f z ( i - 4 , j ) ) 45 46 end elseif i < = 8 47 48 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 49 50 printf ( ’ \ t \t% . 1 0 f ’ , l i ( j ) ) 51 end 52 elseif i < = 9 53 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 54 55 printf ( ’ \ t \t% . 1 0 f ’ , d i ( j ) ) end 56 57 elseif i < = 1 0 printf ( ’ \n %s ’ , s ( i ) ) 58 59 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f ’ , g i ( j ) ) 60 61 end 62 elseif i < = 1 1 63 printf ( ’ \n %s ’ , s ( i ) ) 64 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f ’ , z ( j ) ) 65 66 end elseif i < = 1 2 67 68 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 69 70 printf ( ’ \ t \t% . 1 0 f ’ , z i ( j ) ) end 71 72 end 73 end 74 printf ( ’ \n\nAt t he end o f t h e % i i t e r a t i o n , t h e r o o t o f t he e q u a t i on i s %. 1 0 f ’ , j - 2 , z ( j ) )

68

Scilab code Exa 5.22 Mullers Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18

19 20 21 22 23 24

/ / E x am pl e 5 . 2 2 / / M u l l e r s Me th od / / P ag e n o . 1 8 3 clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ3−x−4 ’ ) zi=[1;2;3]; s =[ ” i ” , ” z 0 ” , ” z 1 ” , ” z 2 ” , ” f0 ” , ” f1 ” , ” f2 ” , ” l i ” , ” d i ” , ” gi ” , ” l i +1” , ” h i ” , ” h i + 1 ” , ” z i +1” , ”D+” , ” D ” ] li(1)=(zi(3,1)-zi(2,1))/(zi(2,1)-zi(1,1)) hi(1)=zi(3,1)-zi(2,1); for i = 2 : 6 for j = 1 : 3 fz(j,i-1)=f(zi(j,i-1)) end di(i-1)=1+li(i-1) gi(i-1)=fz(1,i-1)*li(i-1)^2-fz(2,i-1)*di(i-1)^2+ fz(3,i-1)*(li(i-1)+di(i-1)) D 1 ( i - 1 ) = g i ( i - 1 ) +sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) D 2 ( i - 1 ) = g i ( i - 1 ) -sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) if a bs ( D 1 ( i - 1 ) ) >abs ( D 2 ( i - 1 ) ) then li(i)=-2*fz(3,i-1)*di(i-1)/D1(i-1) else li(i)=-2*fz(3,i-1)*di(i-1)/D2(i-1) end hi(i)=li(i)*hi(i-1);

69

25 z(i-1)=zi(3,i-1)+hi(i) for j = 1 : 2 26 27 zi(j,i)=zi(j+1,i-1) end 28 29 zi(3,i)=z(i-1) 30 end 31 for i = 1 : 1 6 if i = = 1 then 32 33 printf ( s ( i ) ) 34 for j = 1 : 5 35 printf ( ’ \ t \ t \ t%i ’ , j - 1 ) 36 end elseif i < = 4 37 38 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 39 40 printf ( ’ \ t \t% . 1 0 f ’ , z i ( i - 1 , j ) ) end 41 42 elseif i < = 7 printf ( ’ \n %s ’ , s ( i ) ) 43 44 for j = 1 : 5 45 printf ( ’ \ t \t% . 1 0 f ’ , f z ( i - 4 , j ) ) 46 end 47 elseif i < = 8 printf ( ’ \n %s ’ , s ( i ) ) 48 49 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f ’ , l i ( j ) ) 50 51 end elseif i < = 9 52 53 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 54 55 printf ( ’ \ t \t% . 1 0 f ’ , d i ( j ) ) 56 end 57 elseif i < = 1 0 58 printf ( ’ \n %s ’ , s ( i ) ) 59 for j = 1 : 5 60 printf ( ’ \ t \t% . 1 0 f ’ , g i ( j ) ) end 61 62 elseif i < = 1 1

70

63 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 64 65 printf ( ’ \ t \t% . 1 0 f end 66 67 elseif i < = 1 2 printf ( ’ \n %s ’ , s ( i ) ) 68 69 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f 70 71 end 72 elseif i < = 1 3 73 printf ( ’ \n %s ’ , s ( i ) ) 74 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f 75 76 end elseif i < = 1 4 77 78 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 79 80 printf ( ’ \ t \t% . 1 0 f end 81 82 elseif i < = 1 5 83 printf ( ’ \n %s ’ , s ( i ) ) 84 for j = 1 : 5 85 printf ( ’ \ t \t% . 1 0 f end 86 87 elseif i < = 1 6 printf ( ’ \n %s ’ , s ( i ) ) 88 89 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f 90 91 end end 92 93 end 94 printf ( ’ \n\nAt t he end o f t he

’ ,li(j+1))

’ ,hi(j))

’ ,hi(j+1))

’ ,z(j))

’ ,D1(j))

’ ,D2(j))

%i th i t e r a t i o n , t he r o o t o f t he e qu a ti o n i s %. 1 0 f ’ , j - 1 , z ( j ) )

71

Scilab code Exa 5.23 Bairstow Hitchcock Method

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 5 . 2 3 / / B a i r s t o w H i t c h c o c k Method / / P ag e n o . 1 8 7

clc ; clear ; close ; deff ( ’ y=f (x , p , q ) ’ , ’ y=xˆ2+p ∗ z+q ’ ) a=[1,-1,1,-1,1] a=a’;a=[a,a,a,a,a] printf ( ’ I t e r a t i o n −−> ’ ) for i = 1 : 5 printf ( ’ \ t%i \ t ’ ,i ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 13 p ( 1 , 1 ) = - 1 . 2 ; q ( 1 , 1 ) = 0 . 9 5 ; 14 s =[ ” b 1 ” , ” b 2 ” , ” b 3 ” , ” b 4 ” , ” c 1 ” , ” c 2 ” , ” c 3 ” , ” c 4 ” , ” c ” , ”dp” , ”dq”,”p”,”q”] 15 // s1 =[ b1 ; b2 ; b3 ; b4 ; c1 ; c2 ; c3 ; c 4 ; c ; dp ; dq ; p ; q ] 16 for i = 1 : 5 17 b(1,i)=0;b(2,i)=a(1,i);c(1,i)=0;c(2,i)=a(1,i); 18 for k = 1 : 4 b(k+2,i)=a(k+1,i)-p(1,i)*b(k+1,i)-q(1,i)* 19 b(k,i) c(k+2,i)=b(k+2,i)-p(1,i)*c(k+1,i)-q(1,i)* 20 c(k,i) end 21 22 cb(1,i)=c(6,i)-b(6,i); dq(1,i)=(b(6,i)*c(4,i)-b(5,i)*cb(1,i))/(c(4,i) 23 ^2-cb(1,i)*c(3,i)) 24 dp(1,i)=(b(5,i)*c(4,i)-b(6,i)*c(3,i))/(c(4,i) ^2-cb(1,i)*c(3,i)) 25 p(1,i+1)=p(1,i)+dp(1,i);q(1,i+1)=q(1,i)+dq(1,i) ; 26 end 27 for j = 1 : 1 3 28 printf ( ’ \n %s\ t \ t ’ , s ( j ) )

72

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

if j <5 then for i = 1 : 5 printf ( ’%. 9 f \ t ’ , b ( j + 2 , i ) ) end elseif j <9 then for i = 1 : 5 printf ( ’%. 9 f \ t ’ , c ( j - 2 , i ) ) end elseif j < 1 0 for i = 1 : 5 printf ( ’%. 9 f \ t ’ , c b ( 1 , i ) ) end elseif j < 1 1 for i = 1 : 5 printf ( ’%. 9 f \ t ’ , d p ( 1 , i ) ) end elseif j < 1 2 for i = 1 : 5 printf ( ’%. 9 f \ t ’ , d q ( 1 , i ) ) end elseif j < 1 3 for i = 1 : 5 printf ( ’%. 9 f \ t ’ , p ( 1 , i + 1 ) ) end else for i = 1 : 5 printf ( ’%. 9 f \ t ’ , q ( 1 , i + 1 ) ) end end end z = poly (0 , ’ z ’ ) ; a=f(z,p(1,i+1),q(1,i+1)); printf ( ’ \n\ nRo ot s f o r Q u ad ra ti c E qu at io n Q = ’ ) disp ( a ) a = roots ( a ) printf ( ’ \n\ t a r e \n ’ ) disp ( a ( 1 ) ) disp ( a ( 2 ) )

73

Scilab code Exa 5.24 Bernoulli Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 5 . 2 4 / / B e r n o u l l i Metho d / / P ag e n o . 1 8 9 clc ; clear ; close ; a=[1,-8,-15,10]; for i = 1 : 2 c(i)=0; end c(3)=1; for k = 4 : 1 3 c(k)=-(a(2)*c(k-1)+a(3)*c(k-2)+a(4)*c(k-3)) r(k-3)=c(k)/c(k-1) end disp (c , ’ Ck V a l u e s ’ ) disp (r , ’ Rk V a l u e s ’ ) disp ( r ( k - 3 ) , ’ T h e r e fo r e t h e e xa ct r o o t i s = ’ )

Scilab code Exa 5.25 Graeffe Method

1 2 3 4 5

/ / E x am pl e 5 . 2 5 / / G r a e f f e Me th od / / P ag e n o . 1 9 1 clc ; clear ; close ;

74

6 a=[1,-6,11,-6] 7 k=0; 8 for k = 2 : 6 for i = 1 : 4 9 10 a(k,i)=(-1)^(i-1)*(a(k-1,i))^2 j=1; 11 12 while i +j < 5 & i +j > 2 a(k,i)=a(k,i)+(-1)^(i-j-1)*2*(a(k-1,i-j) 13 )*a(k-1,i+j) 14 break 15 j=j+1; 16 end end 17 18 end 19 printf ( ’ \ t \ t \ t \ t a 1 \ t \ t \ t \ t a 2 \ t \ t \ t \ t a 3 \n k \ t a 0 \ t a 1 \ t

\ t −−\t \ t a 2 \ t \ t −−\t \ t a 3 \ t \ t −−\t \n\ t \ t \ t \ t a 0 \ t \ t \ t \ ta1 \ t \ t \ t \ ta2 ’ ) 20 printf ( ’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 21 for i = 1 : 4 22 printf ( ’ %i \t%g\t% . 4 g\ t \t% . 5 g\ t \t% .9 g \ t \t% . 8 g \ t%g\ t \t% . 1 0 g\n ’ , i - 1 , a ( i , 1 ) , a ( i , 2 ) , abs ( a ( i , 2 ) / a ( i , 1 ) ) ^ ( 1 / ( 2 ^ ( i - 1 ) ) ) , a ( i , 3 ) , abs ( a ( i , 3 ) / a ( i , 2 ) ) ^ ( 1 / ( 2 ^ ( i - 1 ) ) ) , a ( i , 4 ) , abs ( a ( i , 4 ) / a ( i , 3 ) ) ^(1/(2^(i-1)))) 23 end 24 for i = 5 : 6 25 printf ( ’ %i \t%g\t% . 4 g\t% .5 g \ t \t% . 9 g \t% . 8 g\t% . 7 g \ t% .1 0 g \n ’ , i - 1 , a ( i , 1 ) , a ( i , 2 ) , abs ( a ( i , 2 ) / a ( i , 1 ) ) ^ ( 1 / ( 2 ^ ( i - 1 ) ) ) , a ( i , 3 ) ,abs ( a ( i , 3 ) / a ( i , 2 ) ) ^ ( 1 / ( 2 ^ ( i - 1 ) ) ) , a ( i , 4 ) , abs ( a ( i , 4 ) / a ( i , 3 ) ) ^(1/(2^(i-1)))) 26 end 27 printf ( ’ \n\nThe A bs o lu te V al ue s o f t he r o o t s a r e %g , %.8 g and %g ’ , abs ( a ( i , 2 ) / a ( i , 1 ) ) ^ ( 1 / ( 2 ^ ( i - 1 ) ) ) , abs ( a ( i , 3 ) / a ( i , 2 ) ) ^ ( 1 / ( 2 ^ ( i - 1 ) ) ) , abs ( a ( i , 4 ) / a ( i ,3))^(1/(2^(i-1))))

75

Scilab code Exa 5.26 QD Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

/ / E x am pl e 5 . 2 6 / /QD Meth od / / P ag e n o . 1 9 4 clc ; clear ; close ; a=[32,-48,18,-1] for i = 1 : 5 e(i,1)=0; e(i,4)=0; end q(1,1)=-a(2)/a(1); q(1,2)=0;q(1,3)=0; e(1,2)=a(3)/a(2); e(1,3)=a(4)/a(3); for i = 2 : 1 6 for j = 1 : 3 q(i,j)=e(i-1,j+1)+q(i-1,j)-e(i-1,j) end for j = 1 : 2 e(i,j+1)=e(i-1,j+1)*q(i,j+1)/q(i,j) end end printf ( ’ e 0 \ t \ t q 1 \ t \ t e 1 \ t \ t q 2 \ t \ t e 2 \ t \ t q 3 \ t \ t e 3 \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 25 for i = 1 : 1 4 26 for j = 1 : 3 printf ( ’ \ t \t% . 1 0 f \ 27 t ’ ,q(i,j)) 28 end

76

29 printf ( ’ \n ’ ) for j = 1 : 4 30 31 printf ( ’%.1 0 f \ t \ t \ t ’ , e ( i , j ) ) end 32 33 printf ( ’ \n ’ ) 34 end 35 printf ( ’ \ t \t% . 1 0 f \ t \ t \t% . 1 0 f \ t \ t \t% . 1 0 f \n ’ ,q(15,1),q (15,2),q(15,3)) 36 printf ( ’ \nThe e x ac t r o o t s a r e \t% . 1 0 f and %. 1 0 f ’ ,q(15,1),q(15,3))

Scilab code Exa 5.27 Linear Iteration Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 5 . 2 7 / / L i n e a r I t e r a t i o n Method / / P ag e n o . 1 9 8 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’x =20/(xˆ2+2∗ x+10) ’ ) printf ( ’ n\ t x \ t \ t f ( x ) \n ’ ) printf ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’ ) x=1; for i = 1 : 1 9 printf ( ’ %i \t% . 1 0 f \t% . 1 0 f \n ’ ,i,x,f(x)) x1=x; x=f(x); end printf ( ’ \n\ n x = % . 1 0 f ’ , x 1 )

77

Scilab code Exa 5.28 Aitkens Method

1 2 3 4 5 6 7 8

/ / E x am pl e 5 . 2 8 / / A i t k e n s M et ho d / / P ag e n o . 1 9 9 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’x =20/(xˆ2+2∗ x+10) ’ ) printf ( ’ n\ t x 0 \ t \ t x 1 \ t \ t x 2 \ t \ t x 3 \ t \ t y \ t \ t d x 0 \n ’ ) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=1;e=0.0001 10 for i = 1 : 3 x1=f(x0);x2=f(x1);x3=f(x2); 11 12 y=x3-((x3-x2)^2)/(x3-2*x2+x1) dx0=y-x0; 13 14 printf ( ’ %i \t% . 1 0 f \t% 15 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \ t% . 1 0 f \n ’ ,i,x0,x1,x2,x3,y,dx0) 16 x0=y; 17 if a bs ( x 0 ) < e then 18 break ; end 19 20 end 21 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i I t e r a t i o n s i s %. 1 0 f ’ , i , y )


1 / / E x am pl e 5 . 2 9 2 //Newton Raphson Method 3 / / P ag e n o . 1 9 9

78

4 5 6 7

8

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=xˆ3+2 ∗ xˆ2+10∗x−20 ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=3∗xˆ2+4∗x+10 ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=01;e=0.00001 10 for i = 1 : 4 11 x1=x0-f(x0)/f1(x0) 12 e1 = abs ( x 0 - x 1 ) printf ( ’ %i \t% . 1 0 f \t% 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n ’ ,i-1,x0,f(x0),f1(x0),x1,e1) x0=x1; 14 15 if a bs ( x 0 ) < e then break ; 16 17 end 18 end 19 printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i I t e r a t i o n s i s %. 1 0 f ’ , i , x 1 )


1 2 3 4 5

/ / E x am pl e 5 . 3 1 / / S e c a n t M et ho d / / P ag e n o . 2 0 0

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=(x − 0 . 6 ) ∗ ( x − 1.3 ) ˆ2 ∗ ( x −2 ) ˆ 3 + 0 . 0 1 2 3 4∗ log (x) ’ ) 6 printf ( ’ n\ t x n \ t \ t f ( x n ) \ t \ tXn+1\ t \ t f (Xn+1)\ t \tXn+2\ t \ t E r r o r \n ’ ) 7 printf ( ’

79

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 8 x0=0.1;x1=1.2;e=0.00001 9 for i = 1 : 7 10 x2=x1-f(x1)*(x1-x0)/(f(x1)-f(x0)) e1 = abs ( x 0 - x 2 ) 11 12 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \ printf ( ’ %i \t% . 1 0 f \t% t% . 1 0 f \n ’ ,i-1,x0,f(x0),x1,f(x1),x2,e1) 13 x0=x1; 14 x1=x2 15 if a bs ( x 0 ) < e then 16 break ; end 17 18 end 19 printf ( ’ \n\ n Th er ef or e , t he r o o t i s %. 1 0 f ’ , x 2 )

Scilab code Exa 5.32 Regula Falsi Newton Raphson and Mullers Method

1 2 3 4 5

/ / E x am pl e 5 . 3 2 / / R e gu l a F a l s i , Newton R ap hs on a nd M u l l e r s Method / / P ag e n o . 2 0 1

clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=x ˆ5 −3.7∗ x ˆ 4 + 7 . 4∗ x ˆ 3 −1 0 . 8∗ x ˆ 2 + 1 0 . 8∗ x −6.8 ’ ) 6 deff ( ’ x= f 1 ( x ) ’ , ’ x=5∗x ˆ 4 −4∗3 .7∗ x ˆ 3 + 3∗7 . 4 ∗ x ˆ 2 −21.6 ∗ x +10.8 ’ ) 7 / / n ew to n r a p h s o n 8 printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) 9 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 x 0 = 1 . 5 ; e = 0 . 0 0 0 0 1

80

11 for i = 1 : 4 x1=x0-f(x0)/f1(x0) 12 13 e1 = abs ( x 0 - x 1 ) printf ( ’ %i \t% . 1 0 f \t% 14 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n ’ ,i-1,x0,f(x0),f1(x0),x1,e1) x0=x1; 15 16 if a bs ( x 0 ) < e then break ; 17 18 end 19 end 20 printf ( ’ \n\nThe s o l u t i o n o f t h i s e q u a ti o n by newton r a p h s h o n a f t e r %i I t e r a t i o n s i s %. 1 0 f \n\n\n ’ ,i,x1 ) 21 22 / / r e g u l a f a l s i 23 x 1 = 1 ; x 2 = 2 ; e = 0 . 0 0 0 0 1 24 printf ( ’ n\ t x 1 \ t \ t f ( x 1 ) \ t \ t x 2 \ t \ t f ( x 2 ) \ t \ t x 3 \ t \ t f ( x 3 ) ’) 25 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 26 for i = 0 : 7 27 x3=x2*f(x1)/(f(x1)-f(x2))+x1*f(x2)/(f(x2)-f(x1)) printf ( ’ %i \ t%f \ t%f \ t%f \ t%f \ t%f \ t%f \n ’ , i , x 1 , f ( x 1 28 ),x2,f(x2),x3,f(x3)) if f ( x 1 ) * f ( x 3 ) > 0 then 29 30 x1=x3 else 31 32 x2=x3 end 33 34 if a bs ( f ( x 3 ) ) < e then 35 break 36 end 37 end 38 printf ( ’ \n\ n T he r ef o re t he s o l u t i o n by r e g u l a f a l s i method a f t e r %i i t e r a t i o n s i s %. 1 0 g ’ , i , x 3 ) 39 40 / / m u l l e r s m et ho d

81

41 z i = [ 1 ; 2 ; 3 ] ; 42 s =[ ” i ” , ” z 0 ” , ” z 1 ” , ” z 2 ” , ” f0 ” , ” f1 ” , ” f2 ” , ” l i ” , ” d i ” , ” gi ” , ” l i +1” , ” h i ” , ” h i + 1 ” , ” z i +1” , ”D+” , ” D ” ] 43 l i ( 1 ) = ( z i ( 3 , 1 ) - z i ( 2 , 1 ) ) / ( z i ( 2 , 1 ) - z i ( 1 , 1 ) ) 44 h i ( 1 ) = z i ( 3 , 1 ) - z i ( 2 , 1 ) ; 45 for i = 2 : 6 46 for j = 1 : 3 fz(j,i-1)=f(zi(j,i-1)) 47 48 end 49 di(i-1)=1+li(i-1) 50 gi(i-1)=fz(1,i-1)*li(i-1)^2-fz(2,i-1)*di(i-1)^2+ fz(3,i-1)*(li(i-1)+di(i-1)) D 1 ( i - 1 ) = g i ( i - 1 ) +sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i 51 -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) 52 D 2 ( i - 1 ) = g i ( i - 1 ) -sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) if a bs ( D 1 ( i - 1 ) ) >abs ( D 2 ( i - 1 ) ) then 53 54 li(i)=-2*fz(3,i-1)*di(i-1)/D1(i-1) 55 else 56 li(i)=-2*fz(3,i-1)*di(i-1)/D2(i-1) 57 end hi(i)=li(i)*hi(i-1); 58 59 z(i-1)=zi(3,i-1)+hi(i) for j = 1 : 2 60 61 zi(j,i)=zi(j+1,i-1) end 62 63 zi(3,i)=z(i-1) 64 end 65 printf ( ’ \n\n ’ ) 66 for i = 1 : 1 6 67 if i = = 1 then 68 printf ( s ( i ) ) 69 for j = 1 : 5 70 printf ( ’ \ t \ t \ t %i ’ ,j-1) end 71 72 printf ( ’ \n

82

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

elseif i < = 4 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 7 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 8 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 9 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 1 0 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 1 1 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 1 2 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f end elseif i < = 1 3

83

’ ,zi(i-1,j))

’ ,fz(i-4,j))

’ ,li(j))

’ ,di(j))

’ ,gi(j))

’ ,li(j+1))

’ ,hi(j))

109 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 110 111 printf ( ’ \ t \t% . 1 0 f end 112 113 elseif i < = 1 4 printf ( ’ \n %s ’ , s ( i ) ) 114 115 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f 116 117 end 118 elseif i < = 1 5 119 printf ( ’ \n %s ’ , s ( i ) ) 120 for j = 1 : 5 printf ( ’ \ t \t% . 1 0 f 121 122 end elseif i < = 1 6 123 124 printf ( ’ \n %s ’ , s ( i ) ) for j = 1 : 5 125 126 printf ( ’ \ t \t% . 1 0 f end 127 128 end 129 end 130 printf ( ’ \n\nAt t he end o f t h e

’ ,hi(j+1))

’ ,z(j))

’ ,D1(j))

’ ,D2(j))

%i th i t e r a t i o n by m u l l e r s method , t he r o o t o f t he e q ua t io n i s %. 1 0 f ’ ,j-1,z(j))

Scilab code Exa 5.33 Newton Raphson and Mullers Method

1 2 3 4 5 6

/ / E x am pl e 5 . 3 3 / / N e wt on R ap hs on a nd M u l l e r s M et ho d / / P ag e n o . 2 0 2 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=xˆ4 −8∗xˆ3+18∗ x ˆ 2 + 0 . 1 2∗ x − 24.24 ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=4∗x ˆ3 −24∗ xˆ2+36∗ x + 0 . 1 2 ’ )

84

7 8 9 10 11 12 13 14 15 16 17 18 19 20

/ / n ew to n r a p h s o n

x9=[1.5,2.5,2.7,3.1;4,5,14,10] for h = 1 : 4 x0=x9(1,h);e=0.00001 for i = 1 : x 9 ( 2 , h ) x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) x0=x1; if a bs ( x 0 ) < e then break ; end end printf ( ’ \nThe s o l u t i o n o f t h i s e q u a ti o n by newton r a p h s h o n a f t e r %i I t e r a t i o n s i s %. 5 f \n ’ ,i,x1) 21 end 22 23 / / m u l l e r s m et ho d 24 z x = [ 1 , 2 , 2 . 7 , 3 . 1 ; 2 , 3 , 3 . 7 , 4 . 1 ; 3 , 4 , 4 . 7 , 5 . 1 ] 25 z i = [ 1 ; 2 ; 3 ] ; 26 s =[ ” i ” , ” z 0 ” , ” z 1 ” , ” z 2 ” , ” f0 ” , ” f1 ” , ” f2 ” , ” l i ” , ” d i ” , ” gi ” , ” l i +1” , ” h i ” , ” h i + 1 ” , ” z i +1” , ”D+” , ” D ” ] 27 l i ( 1 ) = ( z i ( 3 , 1 ) - z i ( 2 , 1 ) ) / ( z i ( 2 , 1 ) - z i ( 1 , 1 ) ) 28 h i ( 1 ) = z i ( 3 , 1 ) - z i ( 2 , 1 ) ; 29 for i = 2 : 4 for j = 1 : 3 30 31 fz(j,i-1)=f(zi(j,i-1)) end 32 33 di(i-1)=1+li(i-1) gi(i-1)=fz(1,i-1)*li(i-1)^2-fz(2,i-1)*di(i-1)^2+ 34 fz(3,i-1)*(li(i-1)+di(i-1)) 35 D 1 ( i - 1 ) = g i ( i - 1 ) +sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) 36 D 2 ( i - 1 ) = g i ( i - 1 ) -sqrt ( g i ( i - 1 ) ^ 2 - 4 * f z ( 3 , i - 1 ) * d i ( i -1)*li(i-1)*(fz(1,i-1)*li(i-1)-fz(2,i-1)*di(i -1)+fz(3,i-1))) 37 if a bs ( D 1 ( i - 1 ) ) >abs ( D 2 ( i - 1 ) ) then

85

38 li(i)=-2*fz(3,i-1)*di(i-1)/D1(i-1) else 39 40 li(i)=-2*fz(3,i-1)*di(i-1)/D2(i-1) end 41 42 hi(i)=li(i)*hi(i-1); z(i-1)=zi(3,i-1)+hi(i) 43 44 for j = 1 : 2 zi(j,i)=zi(j+1,i-1) 45 46 end 47 zi(3,i)=z(i-1) 48 end 49 printf ( ’ \n\nAt t he end o f t h e %i th i t e r a t i o n

by m u l l e r s method , t he r o o t o f t he e q ua t io n i s %. 1 0 f ’ ,j+2,z(j))

Scilab code Exa 5.34 QD Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

/ / E x am pl e 5 . 3 4 / /QD Meth od / / P ag e n o . 2 0 2

clc ; clear ; close ; a=[1,2,10,-20] for i = 1 : 5 e(i,1)=0; e(i,4)=0; end q(1,1)=-a(2)/a(1); q(1,2)=0;q(1,3)=0; e(1,2)=a(3)/a(2); e(1,3)=a(4)/a(3); for i = 2 : 7 for j = 1 : 3 q(i,j)=e(i-1,j+1)+q(i-1,j)-e(i-1,j)

86

17 end for j = 1 : 2 18 19 e(i,j+1)=e(i-1,j+1)*q(i,j+1)/q(i,j) end 20 21 end 22 printf ( ’ e 0 \ t \ t q 1 \ t \ t e 1 \ t \ t q 2 \ t \ t e 2 \ t \ t q 3 \ t \ t e 3 \n ’ ) 23 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 24 for i = 1 : 7 25 for j = 1 : 3 26 t ’ ,q(i,j)) printf ( ’ \ t \t% . 1 0 f \ end 27 28 printf ( ’ \n ’ ) for j = 1 : 4 29 30 printf ( ’%.1 0 f \ t \ t \ t ’ , e ( i , j ) ) end 31 32 printf ( ’ \n ’ ) 33 end 34 printf ( ’ \ t \t% . 1 0 f \ t \ t \t% . 1 0 f \ t \ t \t% . 1 0 f \n ’ ,q(7,1),q (7,2),q(7,3)) 35 printf ( ’ \nThe e x ac t r o o t s a r e \t% . 1 0 f and %. 1 0 f ’ ,q(7,1),q(7,3))


1 2 3 4 5 6 7

/ / E x am pl e 5 . 3 5 //Newton Raphson Method / / P ag e n o . 2 0 3 clc ; clear ; close ; deff ( ’x=f ( x ) ’ , ’ x=xˆ3 −30∗ xˆ2+2552 ’ ) deff ( ’ x= f 1 ( x ) ’ , ’ x=3∗x ˆ2 −60∗ x ’ )

/ / n ew to n r a p h s o n 87

8 printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) 9 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 x 0 = 1 0 ; e = 0 . 0 0 0 0 1 11 for i = 1 : 4 x1=x0-f(x0)/f1(x0) 12 13 e1 = abs ( x 0 - x 1 ) 14 printf ( ’ %i \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n ’ ,i-1,x0,f(x0),f1(x0),x1,e1) 15 x0=x1; if a bs ( x 0 ) < e then 16 17 break ; end 18 19 end 20 printf ( ’ \n\ nThus t he b a l l i s su bmerg ed u pt o h e i g h t o f %. 1 0 f cm\n\n\n ’ , x 1 )


1 2 3 4 5 6

/ / E x am pl e 5 . 3 6 / / S e c a n t M et ho d / / P ag e n o . 2 0 4 clc ; clear ; close ; a=8670;c=10^-8;t2=1.4*10^-4; deff ( ’x=f ( x ) ’ , ’ x=−t2+lo g ((1 −2∗ x / a ) / ( 2 − x / a ) ) ∗ ( a ∗ x ∗ c ) /( a+x ) ’ )

7 8 printf ( ’ n\ t x n \ t \ t f ( x n ) \ t \ tXn+1\ t \ t f (Xn+1)\ t \tXn+2\ t \ t E r r o r \n ’ ) 9 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 88

n ’) 10 x 0 = 2 0 0 0 0 ; x 1 = 2 5 0 0 0 ; e = 0 . 0 0 0 0 1 11 for i = 1 : 8 x2=x1-f(x1)*(x1-x0)/(f(x1)-f(x0)) 12 13 e1 = abs ( x 0 - x 2 ) printf ( ’ %i \ t%f \t% . 1 0 f \ 14 t%f \t% . 1 0 f \ t%f \t% . 1 0 f \n ’, i-1,x0,f(x0),x1,f(x1),x2,e1) x0=x1; 15 16 x1=x2 17 if a bs ( x 0 ) < e then 18 break ; 19 end 20 end 21 printf ( ’ \n\ n T h e r e f o r e , Rb = %. 1 0 f ohm ’ , x 2 )


1 2 3 4 5 6 7 8


9

clc ; clear ; close ; p=1.1;T=250;R=0.082;a=3.6;b=0.043; deff ( ’y=f ( v ) ’ , ’ y=p∗ vˆ3 −(b ∗p+R∗T) ∗ vˆ2+a ∗v−a ∗b ’ ) deff ( ’ y= f 1 ( v ) ’ , ’ y=3∗p∗ v ˆ2 −2∗( b ∗p+R∗T) ∗ v ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 x 0 = 0 . 1 ; e = 0 . 0 0 0 0 1 11 for i = 1 : 1 0 12 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) 13

89

14

. 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n printf ( ’ %i \t% . 1 0 f \t% ’ ,i-1,x0,f(x0),f1(x0),x1,e1) x0=x1; if a bs ( x 0 ) < e then break ; end

15 16 17 18 19 end 20 printf ( ’ \n\ n T h e re f o r e , Volume v = %. 1 0 f

l t r ’ ,x1)


1 2 3 4 5 6 7


8

clc ; clear ; close ; deff ( ’ y=f ( p) ’ , ’ y=pˆ3 −9∗pˆ2+33∗p−65 ’ ) deff ( ’y=f1 (p ) ’ , ’ y=3∗pˆ2 −18∗p+33 ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=6;e=0.00001 10 for i = 1 : 1 0 11 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) 12 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n printf ( ’ %i \t% . 1 0 f \t% ’ ,i-1,x0,f(x0),f1(x0),x1,e1) 14 x0=x1; 15 if a bs ( x 0 ) < e then break ; 16 17 end 18 end

90

19 printf ( ’ \n\ n Th er ef or e , Market P r i c e a t e q u i l i b r i u m = Rs . %. f ’ , x 1 )


1 2 3 4 5 6 7


8

clc ; clear ; close ; deff ( ’y=f ( v ) ’ , ’ y=vˆ3 −20∗ v+30 ’ ) deff ( ’ y= f 1 ( v ) ’ , ’ y=3∗vˆ2 −20 ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=10;e=0.00001 10 for i = 1 : 1 0 11 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) 12 13 . 1 0 f \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n printf ( ’ %i \t% . 1 0 f \t% ’ ,i-1,x0,f(x0),f1(x0),x1,e1) 14 x0=x1; if a bs ( x 0 ) < e then 15 16 break ; end 17 18 end 19 printf ( ’ \n\ n Th er ef or e , s i d e s a r e = %. 5 f m x %. 5 f m x %. 5 f m ’ , x 1 , x 1 , 2 0 / x 1 ^ 2 )

91


1 2 3 4 5 6 7


8

clc ; clear ; close ; deff ( ’ y=f (F) ’ , ’ y=−10∗Fˆ 3 −21 ∗F+10 ’ ) deff ( ’ y=f1 (F) ’ , ’ y=−21−30∗Fˆ2 ’ ) printf ( ’ n\ t x n \ t \ t \ f ( xn ) \ t \ t f 1 ( x n ) \ t \tXn+1\ t \ t E r r o r \n ’) printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 x0=1;e=0.00001 10 for i = 1 : 1 0 11 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) 12 13 . 6 f \t% . 5 f \t% . 1 0 f \t% . 1 0 f \n ’, printf ( ’ %i \t% . 1 0 f \t% i-1,x0,f(x0),f1(x0),x1,e1) 14 x0=x1; if a bs ( x 0 ) < e then 15 16 break ; 17 end 18 end 19 printf ( ’ \n\ t \ t \ t \ t \ t 2 \n T h er e fo r e , M ag ne ti c F lu x = % . 5 f Wb m’ , x 1 )

92

Chapter 6 Numerical Methods of Linear Equations Direct Methods

Scilab code Exa 6.1 Gaussian Elimination Method

1 2 3 4 5 6

/ / E xa mp le 6 . 1 / / G a u s s i an E l i m i n a t i o n Method / / P ag e n o . 2 2 0 clc ; clear ; close ; A=[5,10,1,28;1,1,1,6;4,8,3,29];

a u gm e nt ed m a t r i x 7 8 9 10 11 12 13 14 15 16

// t r i a n g u l a r i z a t i o n for i = 1 : 4 B(1,i)=A(1,i) B(2,i)=A(2,i)-(A(2,1)/A(1,1))*A(1,i) B(3,i)=A(3,i)-(A(3,1)/A(1,1))*A(1,i) end disp (A , ’ A ug me nt ed M a t r i x = ’ ) disp (B , ’ T r i a n g u l a t e d M at ri x= ’ )

/ / ba ck s u b s t i t u t i o n 93

//

17 18 19 20 21 22 23 24 25 26

x(3)=B(3,4)/B(3,3); printf ( ’ \ nx ( 3 )=%f \ n ’ , x ( 3 ) ) for i = 2 : - 1 : 1 k =0 for j = i + 1 : 3 k=k+B(i,j)*x(j) end x(i)=(1/B(i,i))*(B(i,4)-k) printf ( ’ \ nx(%i)=%f \ n ’ , i , x ( i ) ) end

Scilab code Exa 6.2 Gaussian Elimination Method for TriDiagonal Sys-

tem 1 / / E xa mp le 6 . 2 2 / / G a u ss i an E l i m i n a t i o n Method f o r T ri −D i a g o n a l

System 3 / / P ag e n o . 2 2 2 4 5 6 7 8 9 10 11 12 13

clc ; clear ; close ;

/ / e q u a t i o n m a t ri x A=[1,2,0,0;2,3,-1,0;0,4,2,3;0,0,2,-1]; K=[5;5;11;10];i=1;

// i n i t i a l i z a t i o n w(1)=A(1,2)/A(1,1); g(1)=K(1)/A(1,1); printf ( ’ \nw(%i)=%f ’ , i , w ( i ) ) ; printf ( ’ \ ng ( %i)=%f ’ , i , g ( i))

14 15 / / c o m p u t a t i o n 16 for i = 2 : 3 w(i)=(A(i,i+1))/(A(i,i)-A(i,i-1)*w(i-1)) 17

94

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

g(i)=(K(i)-A(i,i-1)*g(i-1))/(A(i,i)-A(i,i-1)*w(i -1)) printf ( ’ \nw(%i)=%f ’ , i , w ( i ) ) printf ( ’ \ ng ( %i)=%f ’ , i , g ( i ) )

end i =4 m=-2 g(i)=m*(K(i)-A(i,i-1)*g(i-1))/(A(i,i)-A(i,i-1)*w(i -1)) x(i)=g(i) printf ( ’ \ ng ( %i)=%f ’ , i , g ( i ) ) printf ( ’ \n\ nx ( %i)=%f ’ , i , x ( i ) )

// sol uti on for i = 3 : - 1 : 1 x(i)=g(i)-w(i)*x(i+1) printf ( ’ \n\ nx ( %i)=%f ’ , i , x ( i ) ) end

Scilab code Exa 6.3 Gauss Jordan Method

1 2 3 4 5 6 7

/ / E xa mp le 6 . 3 //Gauss−J o r d a n M et ho d / / P ag e n o . 2 2 4 clc ; clear ; close ; A=[5,10,1,28;4,8,3,29;1,1,1,6];

matrix 8 9 for i = 1 : 3 10 j=i while ( A (i , i ) == 0 & j < = 3) 11

95

//augmented

12 for k = 1 : 4 B(1,k)=A(j+1,k) 13 14 A(j+1,k)=A(i,k) A(i,k)=B(1,k) 15 16 end disp ( A ) 17 18 j=j+1 end 19 20 disp ( A ) 21 for k = 4 : - 1 : i 22 A(i,k)=A(i,k)/A(i,i) 23 end disp ( A ) 24 25 for k = 1 : 3 if ( k ~ = i ) then 26 27 l=A(k,i)/A(i,i) for m = i : 4 28 29 A(k,m)=A(k,m)-l*A(i,m) end 30 31 end 32 33 end 34 disp ( A ) 35 end 36 37 for i = 1 : 3 38 printf ( ’ \ n x ( % i ) = % g\n ’ , i , A ( i , 4 ) ) 39 end

Scilab code Exa 6.4 Gaussian Elimination Method without Pivoting

1 / / E xa mp le 6 . 4 2 / / G a u ss i an E l i m i n a t i o n Method w i th o ut P i v o t i n g

96

3 / / P ag e n o . 2 2 7 4 clc ; clear ; close ; 5 6 A=[0.3*10^-11,1,0.7;1,1,0.9];

matrix 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

// t r i a n g u l a r i z a t i o n for i = 1 : 3 B(1,i)=A(1,i) B(2,i)=A(2,i)-(A(2,1)/A(1,1))*A(1,i) end disp (A , ’ A ug me nt ed M a t r i x = ’ ) disp (B , ’ T r i a n g u l a t e d M at ri x= ’ )

/ / ba ck s u b s t i t u t i o n x(2)=B(2,3)/B(2,2); printf ( ’ \ nx ( 2 )=%f \ n ’ , x ( 2 ) ) for i = 1 : - 1 : 1 k =0 for j = i + 1 : 2 k=k+B(i,j)*x(j) end x(i)=(1/B(i,i))*(B(i,3)-k) printf ( ’ \ nx(%i)=%f \ n ’ , i , x ( i ) ) end

Scilab code Exa 6.5 Dolittle Factorization Method

1 2 3 4 5

/ / E xa mp le 6 . 5 // D o l i t t l e F a c t o r i z a t i o n Method / / P ag e n o . 2 3 3 clc ; clear ; close ;

97

//augmented

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

A=[2,1,1;1,3,1;1,1,4]; printf ( ’ \ tL \ t \ t ∗\ t \tU \ t \ t =\t \tA ’ ) U(2,1)=0;U(3,1)=0;U(3,2)=0; L(1,2)=0;L(1,3)=0;L(2,3)=0; for i = 1 : 3 L(i,i)=1 end for i = 1 : 3 U(1,i)=A(1,i) end L(2,1)=1/U(1,1); for i = 2 : 3 U(2,i)=A(2,i)-U(1,i)*L(2,1); end L(3,1)=1/U(1,1); L(3,2)=(A(3,2)-U(1,2)*L(3,1))/U(2,2); U(3,3)=A(3,3)-U(1,3)*L(3,1)-U(2,3)*L(3,2); printf ( ’ \n ’ ) for i = 1 : 3 for j = 1 : 3 printf ( ’%. 2 f \ t ’ , L ( i , j ) ) end

if ( i = = 2 ) printf ( ’ ∗ else printf ( ’ \ t ’ ) end

’)

for j = 1 : 3 printf ( ’%. 2 f \ t ’ , U ( i , j ) ) end if ( i = = 2 ) printf ( ’ = ’) else printf ( ’ \ t ’ ) end for j = 1 : 3

98

44 printf ( ’%. 2 f \ t ’ , A ( i , j ) ) end 45 46 printf ( ’ \n ’ ) 47 end

Scilab code Exa 6.6 Trangularization Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

/ / E xa mp le 6 . 6 / / T r a n g u l a r i z a t i o n Method / / P ag e n o . 2 3 6 clc ; clear ; close ;

A=[2,1,1;1,3,1;1,1,4]; B=[7;10;15]; printf ( ’A c an be f a c t o r i z a e d a s f o l l o w s : \ n ’ ) printf ( ’ \ tL \ t \ t ∗\ t \tU \ t \ t =\t \tA ’ ) U(2,1)=0;U(3,1)=0;U(3,2)=0; L(1,2)=0;L(1,3)=0;L(2,3)=0; for i = 1 : 3 L(i,i)=1 end for i = 1 : 3 U(1,i)=A(1,i) end L(2,1)=1/U(1,1); for i = 2 : 3 U(2,i)=A(2,i)-U(1,i)*L(2,1); end L(3,1)=1/U(1,1); L(3,2)=(A(3,2)-U(1,2)*L(3,1))/U(2,2); U(3,3)=A(3,3)-U(1,3)*L(3,1)-U(2,3)*L(3,2); printf ( ’ \n ’ ) for i = 1 : 3

99

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

for j = 1 : 3 printf ( ’%. 2 f \ t ’ , L ( i , j ) ) end

if ( i = = 2 ) printf ( ’ ∗ else printf ( ’ \ t ’ ) end

’)

for j = 1 : 3 printf ( ’%. 2 f \ t ’ , U ( i , j ) ) end if ( i = = 2 ) printf ( ’ = ’) else printf ( ’ \ t ’ ) end for j = 1 : 3 printf ( ’%. 2 f \ t ’ , A ( i , j ) ) end printf ( ’ \n ’ )

end printf ( ’ \nY=U∗X ’ ) Y = inv ( L ) * B X = inv ( U ) * Y printf ( ’ \n\nX= ’ ) for i = 1 : 3 %i ’ , X ( i , 1 ) ) printf ( ’ \n end

Scilab code Exa 6.7 Wilkinson Method

100

1 2 3 4 5 6

/ / E xa mp le 6 . 7 / / W i l k i n s o n M et ho d / / P ag e n o . 2 4 0 clc ; clear ; close ; A = [ 0 . 3 * 1 0 ^ 5 , 0 . 2 1 2 , 0 . 3 3 2 ; 0 . 2 1 6 , 0 . 3 7 6 , 0 . 4 7 7 ; 0 . 1 7 3 , 0 . 6 63 , 0 . 6 2 6 ] ;

7 B=[0.235;0.128;0.285]; 8 X = inv ( A ) 9 disp ( X * B , ’ F i na l S o l u ti o n = ’ )

Scilab code Exa 6.8 Choleskys Factorization

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

/ / E xa mp le 6 . 8 / / Ch ol es ky ’ s F a c t o r i z a t i o n / / P ag e n o . 2 4 3 clc ; clear ; close ; A=[1,2,3;2,5,8;3,8,22]; U(2,1)=0;U(3,1)=0;U(3,2)=0; for i = 1 : 3 for j = 1 : 3 if ( i = = j ) k=0; for m = 1 : i - 1 k=k+U(m,i)^2; end U ( i , j ) = sqrt ( A ( i , j ) - k ) end if ( j > i ) k=0; for m = 1 : i - 1

101

20 k=k+U(m,j)*U(m,i); end 21 22 U(i,j)=(A(i,j)-k)/U(i,i) end 23 24 end 25 end 26 disp (U , ’ R e q u i r e d M a t ri x (U)= ’ )

Scilab code Exa 6.9 Complex System of Linear Equation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E xa mp le 6 . 9 / / C omplex S ys te m o f L i n e a r E q ua t io n / / P ag e n o . 2 4 4 clc ; clear ; close ; for i = 1 : 7 s=0; for j = 1 : 7 A(i,j)=jî s=s+(-1)^(j+1)*A(i,j) end B(i,1)=s; end X = inv ( A ) * B disp (X , ’ The S o l u ti o n = ’ )

Scilab code Exa 6.10 Solving Matrices

1 / / E x am pl e 6 . 1 0

102

2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / S o l v i n g M a t r i ce s / / P ag e n o . 2 4 4 clc ; close ; clear ; warning ( ’ o f f ’ ) for i = 1 : 7 s=0; for j = 1 : 7 A(i,j)=360360/(i+j) end B(i,1)=1; end X = inv ( A ) * B disp ( 3 6 0 3 6 0 * X , ’ The S o l u t i o n by 3 60 36 0∗X= ’ ) disp (X , ’ F i na l S o l u ti o n = ’ )

103

Chapter 7 Numerical Solutions for Matrix Inversion

Scilab code Exa 7.1 Gauss Jordan Two Array Method

1 2 3 4 5 6 7 8

/ / E xa mp le 7 . 1 //Gauss−J o r d a n Two A r r a y M et ho d / / P ag e n o . 2 5 4 clc ; clear ; close ; A=[2,6,1;3,9,2;0,-1,3]; C = eye ( 3 , 3 ) ; for i = 1 : 3

// matr ix / / U n it M a tr i x // i n t e r ch a n g e o f row 1

a nd 2 9 B(1,i)=A(1,i); 10 A(1,i)=A(2,i); 11 A(2,i)=B(1,i); 12 B(2,i)=C(1,i); C(1,i)=C(2,i); 13 14 C(2,i)=B(2,i); 15 end 16 printf ( ’ \n ’ )

104

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

// p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ;

for i = 1 : 3 A(1,i)=A(1,i)/3; C(1,i)=C(1,i)/3; end

// p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ; for i = 1 : 3 A(2,i)=A(2,i)-2*A(1,i); C(2,i)=C(2,i)-2*C(1,i); end

105

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

// p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ;

// i n t e r ch a n g e o f row 2 and 3 for i = 1 : 3 B(1,i)=A(2,i); A(2,i)=A(3,i); A(3,i)=B(1,i); B(2,i)=C(2,i); C(2,i)=C(3,i); C(3,i)=B(2,i); end // p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ; for i = 1 : 3 A(2,i)=-1*A(2,i); C(2,i)=-1*C(2,i);

106

93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

end for i = 1 : 3 A(1,i)=A(1,i)-3*A(2,i); C(1,i)=C(1,i)-3*C(2,i); end

// p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ; for i = 1 : 3 A(3,i)=-3*A(3,i); C(3,i)=-3*C(3,i); end

// p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ; for i = 1 : 3

107

131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152

A(1,i)=A(1,i)-A(3,i)*(29/3); C(1,i)=C(1,i)-29*C(3,i)/3; end for i = 1 : 3 A(2,i)=A(2,i)+A(3,i)*3; C(2,i)=C(2,i)+C(3,i)*3; end

// p r i n t i n g o f m a t r i c e s A and C for i = 1 : 3 for j = 1 : 3 printf ( ’ %f \ t ’ , A ( i , j ) ) end printf ( ’ | \ t ’ ) ; for j = 1 : 3 printf ( ’ %f \ t ’ , C ( i , j ) ) end printf ( ’ \n ’ ) end printf ( ’ \n\n ’ ) ; disp (C , ’ I n v e r s e M at ri x o f A ’ )

Scilab code Exa 7.2 Inverse in Place without Pivoting

1 2 3 4 5 6 7 8

/ / E xa mp le 7 . 2 / / I n v e r s e i n P l ac e w it ho ut P i v o ti n g / / P ag e n o . 2 5 6 clc ; clear ; close ;

// matr ix / / c o p i e d m a t ri x

A=[3,-6,7;9,0,-5;5,-8,6]; B=[3,-6,7;9,0,-5;5,-8,6]; for i = 1 : 3

108

9 printf ( ’ \n\ n S t a g e %i ’ , i ) ; for j = 1 : 3 10 11 if ( i = = j ) B(i,j)=1/B(i,j); 12 13 else B(i,j)=A(i,j)/A(i,i); 14 15 end , end 16 17 disp ( B ) 18 for j = 1 : 3 19 for k = 1 : 3 20 if ( i ~ = j ) B(j,k)=A(j,k)-A(j,i)*B(i,k); 21 22 end , end 23 24 end disp ( B ) 25 26 for j = 1 : 3 if ( i ~ = j ) 27 28 B(j,i)=-1*A(j,i)*B(i,i); 29 end , 30 31 end disp ( B ) 32 33 A=B; 34 end 35 disp (B , ’ I n v e r s e o f M at ri x A= ’ )

Scilab code Exa 7.3 Inverse in Place with Pivoting

1 / / E xa mp le 7 . 3 2 / / I n v e r s e i n P l ac e w it h P i v ot i n g 3 / / P ag e n o . 2 5 8

109

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41


// matr ix / / c o p i e d m a t ri x

A=[3,-6,7;9,0,-5;5,-8,6]; B=[3,-6,7;9,0,-5;5,-8,6];

for i = 1 : 3 printf ( ’ \n\ n S t a g e %i ’ ,i ) if ( i < 3 ) / / i n t e r c h a n g e o f r ow s for j = 1 : 3 C(i,j)=A(i,j); A(i,j)=A(i+1,j); A(i+1,j)=C(i,j); C(i,j)=B(i,j); B(i,j)=B(i+1,j); B(i+1,j)=C(i,j); end end disp ( B ) for j = 1 : 3 if ( i = = j ) B(i,j)=1/B(i,j); else B(i,j)=A(i,j)/A(i,i); end , end for j = 1 : 3 for k = 1 : 3 if ( i ~ = j ) B(j,k)=A(j,k)-A(j,i)*B(i,k); end , end end for j = 1 : 3 if ( i ~ = j ) B(j,i)=-1*A(j,i)*B(i,i); end , end

110

42 43 44 45 46 47 48 49 50 51 52 53 54 55

disp ( B ) A=B; end for j = 1 : 3 / / i n t e r c h a n g e o f c olu mn 2 and 3 C(j,1)=A(j,2); A(j,2)=A(j,3); A(j,3)=C(j,1); end / / i n t e r c h a n g e o f c olu mn 2 and 1 for j = 1 : 3 C(j,1)=A(j,2); A(j,2)=A(j,1); A(j,1)=C(j,1); end disp (A , ’ I n v e r s e o f M at ri x A= ’ )

Scilab code Exa 7.4 Inverse of Triangular Matrices

1 2 3 4 5 6

/ / E xa mp le 7 . 4 / / I n v e r s e o f T r i an g u l ar M a tr i ce s / / P ag e n o . 2 6 0 clc ; clear ; close ; R=[2,4,-4,0;0,3,-3,-3;0,0,4,2;0,0,0,3];

R 7 for i = 4 : - 1 : 1 for j = 4 : - 1 : 1 8 9 if ( i > j ) 10 Y(i,j)=0; 11 end 12 if ( i = = j ) Y(i,j)=1/R(i,j); 13 14 end if ( i < j ) 15

111

// matr ix

16 l=0; for k = i + 1 : j 17 18 l=l-R(i,k)*Y(k,j); end 19 20 Y(i,j)=l/R(i,i); end 21 22 end 23 end 24 disp (Y , ’ I n v e r s e o f M at ri x R= ’ )

Scilab code Exa 7.5 Inverse of Complex Matrices

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

/ / E xa mp le 7 . 5 / / I n v e r s e o f Complex M a t r i c e s / / P ag e n o . 2 6 2 clc ; clear ; close ;

17 18 19

A=[1,-1,0;2,3,4;0,1,2]; B=[1,1,3;1,3,-3;-2,-4,-4]; P=A+%i*B; disp (P , ’ Matri x P= ’ ) disp (A , ’ Matri x A= ’ ) ; disp (B , ’ Matri x B= ’ ) ; A1 = inv ( A ) ; B 1 =inv ( B ) ; disp ( A 1 , ’ I n v e r s e o f M at ri x A= ’ ) ; disp ( B 1 , ’ I n v e r s e o f M at ri x B= ’ ) ; B 1 A = B 1 * A ; disp ( B 1 A , ’ I n v e r s e ( B) ∗A= ’ ) ; A B 1 A _ B = A * B 1 A + B ;disp (AB1A_B , ’ A∗ I n v e r s e ( B) ∗A+B= ’ ) ; A B 1 A _ B 1 = inv ( A B 1 A _ B ) ;disp (AB1 A_B1 , ’ I n v e r s e ( A∗ I n v e r s e ( B) ∗A+B)= ’ ) ; X = B 1 A * A B 1 A _ B 1 ;disp (X , ’X= ’ ) ; Y = - 1 * A B 1 A _ B 1 ;disp (Y , ’Y= ’ ) ; Q = X + % i * Y ; disp (Q , ’ I n v e r s e o f M at ri x P= ’ )

112

Scilab code Exa 7.6 Iterative Procedure

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

/ / E xa mp le 7 . 6 // I t e r a t i v e P ro ce du re / / P ag e n o . 2 6 5 clc ; clear ; close ;

A=[3,1,3/2;-5/4,-1/4,-3/4;-1/4,-1/4,-1/4]; disp (A , ’ Matri x A= ’ ) ; B=[1,1,3.5;1,3,-3;-2,-3,-4]; disp (B , ’ A ss um ed M a t r i x B= ’ ) ; e=0.1;

// i t e r a t i o n s E1=e;k=1; while ( E 1 > = e ) printf ( ’ \n\n\ n I t e r a t i o n %i \n ’ ,k ) C = B * ( 2 * eye ( 3 , 3 ) - A * B ) ; disp (C , ’ Matri x C= ’ ) ; E = A * C - eye ( 3 , 3 ) ; disp (E , ’ Matri x E= ’ ) ; B = C ; printf ( ’ \ n I n v e rs e o f M a tr i x A a f t e r %i i t e r a t i o n s = ’ , k ) ; disp ( B ) ; E1=0; for i = 1 : 3 for j = 1 : 3 E1=E1+E(i,j)^2; end end E1 = sqrt ( E 1 ) ; k=k+1; end

113

Chapter 8 Numerical Solutions of Linear Systems of Equations Iterative Methods

Scilab code Exa 8.1 Jacobi Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E xa mp le 8 . 1 / / J a c o b i M et ho d / / P ag e n o . 2 7 3 clc ; clear ; close ; A=[8,-3,2;4,11,-1;6,3,12]; B=[20;33;36] for i = 0 : 1 9 X(i+1,1)=i; end for i = 2 : 4 X(1,i)=0; end for r = 1 : 1 9 for i = 1 : 3

114

/ / e q u a t i o n m a t ri x / / s o l u t i o n m a tr i x

16 17 18 19 20 21 22 23 24 25 26 27 28

k=0; for j = 1 : 3 if ( i ~ = j ) k=k-A(i,j)*X(r,j+1); end end X(r+1,i+1)=(k+B(i,1))/A(i,i);

end

end printf ( ’ r\t x ( r ) \ t \ ty ( r ) \ t z ( r ) ’ ); printf ( ’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) disp ( X ) printf ( ’ \n\ n A ft e r 18 i t e r a t i o n s e x ac t s o l u t i o n i s : \ nx=%i\ ty=%i\ t z=%i ’ , X ( 1 9 , 2 ) , X ( 1 9 , 3 ) , X ( 1 9 , 4 ) )

Scilab code Exa 8.2 Gauss Seidel Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

/ / E xa mp le 8 . 2 //Gauss−S e i d e l Metho d / / P ag e n o . 2 7 4 clc ; clear ; close ; A=[8,-3,2;4,11,-1;6,3,12]; B=[20;33;36] for i = 0 : 1 0 X(i+1,1)=i; end for i = 2 : 4 X(1,i)=0; end for r = 1 : 1 0 for i = 1 : 3 k1=0;

115

/ / e q u a t i o n m a t ri x / / s o l u t i o n m a tr i x

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

for j = 1 : i - 1

k1=k1-A(i,j)*X(r+1,j+1); end k2=0; for j = i + 1 : 3

k2=k2-A(i,j)*X(r,j+1); end X(r+1,i+1)=(k1+k2+B(i,1))/A(i,i);

end

end printf ( ’ r\t x ( r ) \ t \ ty ( r ) \ t z ( r ) ’ ); printf ( ’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) disp ( X ) printf ( ’ \n\ n A ft e r 9 i t e r a t i o n s e xa ct s o l u t i o n i s : \ nx =%i \ ty=%i\ tz=%i ’ , X ( 1 0 , 2 ) , X ( 1 0 , 3 ) , X ( 1 0 , 4 ) )

Scilab code Exa 8.3 SOR Method

1 2 3 4 5 6 7 8 9 10 11

/ / E xa mp le 8 . 3 //SOR Method / / P ag e n o . 2 7 5 clc ; clear ; close ; A=[5,2,1;-1,4,2;2,-3,10]; B=[-12;20;3]; w=0.9; for i = 0 : 1 3 X(i+1,1)=i; end

116

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

X(1,2)=-2.4; X(1,3)=5; X(1,4)=0.3; for r = 1 : 1 3 for i = 1 : 3 k1=0; for j = 1 : i - 1

end k2=0; for j = i + 1 : 3

k1=k1-A(i,j)*X(r+1,j+1);

k2=k2-A(i,j)*X(r,j+1); end X(r+1,i+1)=(1-w)*X(r,i+1)+(w*k1+w*k2+w*B(i ,1))/A(i,i);

30 end 31 end 32 printf ( ’ r\t x ( r ) \ t \ ty ( r ) \ t z ( r ) ’ ); 33 printf ( ’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’ ) ; 34 disp ( X ) ; 35 printf ( ’ \n\ n A ft e r 12 i t e r a t i o n s e x ac t s o l u t i o n i s : \ nx=%i\ ty=%i\ t z=%i ’ , X ( 1 3 , 2 ) , X ( 1 3 , 3 ) , X ( 1 3 , 4 ) ) ;

Scilab code Exa 8.4 Gauss Seidel Point Iterative Method

1 / / E xa mp le 8 . 4 2 //Gauss−S e i d e l P oi nt 3 / / P ag e n o . 2 7 8

I t e r a t i v e Method

117

4 clc ; clear ; close ; 5 6 7 A = [ 1 0 , 1 , 0 , 0 , 0 , - 1 ; 1 , 1 0 , 1 , 0 , 0 , 0 ; 2 , 0 , 2 0 , 1 , 0 , 0 ; 0 , 0 , 1 , 1 0 ,- 1 , 0 ; 0 , 3 , 0 , 0 , 3

/ / e q u a t i o n m a t ri x // sol uti on

8 B=[5;10;10;0;0;5]

matrix 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

for i = 1 : 6 for j = 1 : 6 if ( A ( j , j ) = = 0 ) for k = 1 : 6 C(j,k)=A(j,k); A(j,k)=A(j+1,k); A(j+1,k)=C(j,k); end end end end for i = 0 : 7 X(i+1,1)=i; end for i = 2 : 7 X(1,i)=0; end for r = 1 : 7 for i = 1 : 6 k1=0; for j = 1 : i - 1

k1=k1-A(i,j)*X(r+1,j+1); end k2=0; for j = i + 1 : 6

k2=k2-A(i,j)*X(r,j+1);

118

39 40 41 42 43 44 45 46 47

end X(r+1,i+1)=(k1+k2+B(i,1))/A(i,i);

end

end r printf ( ’ for i = 1 : 6 printf ( ’ x%i end printf ( ’ \n

’ ); ’ ,i);

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 48 disp ( X ) 49 printf ( ’ \n\ n A ft e r 6 i t e r a t i o n s e xa ct s o l u t i o n ); 50 for i = 1 : 6 51 ’ ,i,X(7,i+1)); printf ( ’ x%i=%f 52 end

is :\n’

Scilab code Exa 8.5 Gauss Seidel Point Iterative Method

1 2 3 4 5 6

/ / E xa mp le 8 . 5 //Gauss−S e i d e l P oi nt I t e r a t i v e Method / / P ag e n o . 2 7 9 clc ; clear ; close ; A=[2,3,-4,1;1,-2,-5,1;5,-3,1,-4;10,2,-1,2];

e q u a t i on m a tr i x 7 B=[3;2;1;-4]; / / s o l u t i o n m a tr i x 8 9 / / t r a n s f o rm a t i o n o f t he e q u a t io n s 10 for i = 1 : 4 11 A1(1,i)=A(4,i); B1(1,1)=B(4,1); 12

119

//

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

end for i = 1 : 4 A1(3,i)=A(2,i); B1(3,1)=B(2,1); end for i = 1 : 4 A1(2,i)=A(1,i)-A(2,i); B1(2,1)=B(1,1)-B(2,1); end for i = 1 : 4 A1(4,i)=2*A(1,i)-A(2,i)+2*A(3,i)-A(4,i); B1(4,1)=2*B(1,1)-B(2,1)+2*B(3,1)-B(4,1); end

/ / p r i n t i n g o f t r an s fo r me d e q u a t io n s printf ( ’ \ n T ra n sf or m ed E q u a t io n s a r e =\n\n ’ ) for i = 1 : 4 for j = 1 : 4 printf ( ’ ( %ix( %i) ) ’ , A 1 ( i , j ) , j ) ; if ( j < 4 ) printf ( ’ + ’ ) end end printf ( ’ = % i\n ’ , B 1 ( i , 1 ) ) ; end for i = 1 : 4 for j = 1 : 4 if ( A ( j , j ) = = 0 ) for k = 1 : 4 C(j,k)=A(j,k); A(j,k)=A(j+1,k); A(j+1,k)=C(j,k); end end end end for i = 0 : 1 2

120

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77

X(i+1,1)=i; end for i = 2 : 5 X(1,i)=0; end for r = 1 : 1 2 for i = 1 : 4 k1=0; for j = 1 : i - 1

k1=k1-A1(i,j)*X(r+1,j+1); end k2=0; for j = i + 1 : 4

k2=k2-A1(i,j)*X(r,j+1); end X(r+1,i+1)=(k1+k2+B1(i,1))/A1(i,i);

end

end printf ( ’ \n\n r for i = 1 : 4 printf ( ’ x%i end printf ( ’ \n

’ ); ’ ,i);

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 78 disp ( X ) 79 printf ( ’ \n\ n A ft e r 11 i t e r a t i o n s e x ac t s o l u t i o n i s : \ n ’ ); 80 for i = 1 : 4 81 printf ( ’ x%i=%f ’ ,i,X(12,i+1)); 82 end

121

Scilab code Exa 8.6 Block Jacobi Method

1 2 3 4 5 6

/ / E xa mp le 8 . 6 / / B l o c k J a c o b i Metho d / / P ag e n o . 2 8 1 clc ; clear ; close ; A = [ 1 0 , 1 , 0 , 0 , 0 , - 1 ; 1 , 1 0 , 1 , 0 , 0 , 0 ; 2 , 0 , 2 0 , 1 , 0 , 0 ; 0 , 0 , 1 , 1 0 ,- 1 , 0 ; 0 , 3 , 0 , 0 , 3


7 B=[5;10;10;0;0;5]

matrix 8 disp (B , ’B= ’ ,A , ’A= ’ ) 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

for i = 1 : 3 for j = 1 : 3 A11(i,j)=A(i,j); end B1(i,1)=B(i,1); end for i = 1 : 3 for j = 1 : 3 A12(i,j)=A(i,j+3); end end for i = 1 : 3 for j = 1 : 3 A21(i,j)=A(i+3,j); end end for i = 1 : 3 for j = 1 : 3 A22(i,j)=A(i+3,j+3);

122

28 end B2(i,1)=B(i+3,1); 29 30 end 31 disp ( B 2 , ’ B2= ’ , B 1 , ’ B1= ’ , A 2 2 , ’ A22= ’ , A 2 1 , ’ A21= ’ , A 1 2 , ’ A12=’ , A 1 1 , ’ A11= ’ ) ; 32 A 1 1 _ 1 = inv ( A 1 1 ) ; A 2 2 _ 1 = inv ( A 2 2 ) ; 33 disp ( A 2 2 _ 1 , ’ I n v e r s e o f A22= ’ , A 1 1 _ 1 , ’ I n v e r s e o f A11= ’ ) 34 for i = 1 : 3 35 X1(i,1)=0; 36 X2(i,1)=0; 37 end 38 for r = 1 : 2 39 X11=A11_1*(-1*A12*X2+B1); X22=A22_1*(-1*A21*X1+B2); 40 41 X1=X11; X2=X22; 42 43 disp ( X 1 , ’ X1= ’ ) disp ( X 2 , ’ X2= ’ ) 44 45 end 46 for i = 1 : 6 47 if ( i < 4 ) 48 X(i,1)=X1(i,1); else 49 50 X(i,1)=X2(i-3,1); end 51 52 end 53 disp (X , ’X= ’ ) 54 printf ( ’ \n\n\ nNote : Th ere i s a c o mp ut at io n e r r o r i n c a l c u l a t i o n o f X1 ( 2 ) ’ )

Scilab code Exa 8.7 Block Gauss Seidel Method

123

1 2 3 4 5 6

/ / E xa mp le 8 . 7 / / B l o c k G au ss−S e i d e l Method / / P ag e n o . 2 8 3 clc ; clear ; close ; A = [ 1 0 , 1 , 0 , 0 , 0 , - 1 ; 1 , 1 0 , 1 , 0 , 0 , 0 ; 2 , 0 , 2 0 , 1 , 0 , 0 ; 0 , 0 , 1 , 1 0 ,- 1 , 0 ; 0 , 3 , 0 , 0 , 3


7 B=[5;10;10;0;0;5]

matrix 8 disp (B , ’B= ’ ,A , ’A= ’ ) 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

for i = 1 : 2 for j = 1 : 2 A11(i,j)=A(i,j); end B1(i,1)=B(i,1); end for i = 1 : 2 for j = 1 : 2 A12(i,j)=A(i,j+2); end B2(i,1)=B(i+2,1); end for i = 1 : 2 for j = 1 : 2 A13(i,j)=A(i,j+4); end B3(i,1)=B(i+4,1); end for i = 1 : 2 for j = 1 : 2 A21(i,j)=A(i+2,j); end end for i = 1 : 2 for j = 1 : 2 A22(i,j)=A(i+2,j+2);

124

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

end

59 60 61 62 63 64 65 66 67 68 69 70

end for i = 1 : 2 for j = 1 : 2 A23(i,j)=A(i+2,j+4); end end for i = 1 : 2 for j = 1 : 2 A31(i,j)=A(i+4,j); end end for i = 1 : 2 for j = 1 : 2 A32(i,j)=A(i+4,j+2); end end for i = 1 : 2 for j = 1 : 2 A33(i,j)=A(i+4,j+4); end end disp ( B 3 , ’ B3= ’ , B 2 , ’ B2= ’ , B 1 , ’ B1= ’ , A 3 3 , ’ A33= ’ , A 3 2 , ’A32= ’ , A 3 1 , ’ A31= ’ , A 2 3 , ’ A23= ’ , A 2 2 , ’ A22= ’ , A 2 1 , ’ A21= ’ ,A13 , ’ A13= ’ , A 1 2 , ’ A12= ’ , A 1 1 , ’ A11= ’ ) ; A 1 1 _ 1 = inv ( A 1 1 ) ; A 2 2 _ 1 = inv ( A 2 2 ) ; A 3 3 _ 1 = inv ( A 3 3 ) ; disp ( A 3 3 _ 1 , ’ I n v e r s e o f M a tr i x A33= ’ , A 2 2 _ 1 , ’ I n v e r s e o f M a t ri x A22= ’ , A 1 1 _ 1 , ’ I n v e r s e o f M at r ix A11= ’ ) ; for i = 1 : 2 X1(i,1)=0; X2(i,1)=0; X3(i,1)=0; end for i = 1 : 6 X(i,1)=i-1; end for i = 2 : 7 X(1,i)=0;

125

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

end for r = 1 : 5 X11=A11_1*(-1*A12*X2+(-1)*A13*X3+B1); X22=A22_1*(-1*A21*X11+(-1)*A23*X3+B2); X33=A33_1*(-1*A31*X11+(-1)*A32*X22+B3); X1=X11; X2=X22; X3=X33; disp ( X 3 , ’ X3= ’ , X 2 , ’ X2= ’ , X 1 , ’ X1= ’ ) for i = 2 : 7 if ( i < 4 ) X(r+1,i)=X1(i-1,1); end if ( i<6 & i > 3) X(r+1,i)=X2(i-3,1); end if ( i<8 & i > 5) X(r+1,i)=X3(i-5,1); end end end printf ( ’ \n\ n I t e r a t i o n ’ ) ; for i = 1 : 6 printf ( ’ x%i ’ ,i); end printf ( ’ \n


126

is :\n’

Scilab code Exa 8.8 Block SOR Method

1 2 3 4 5 6 7

/ / E xa mp le 8 . 8 // Blo ck SOR Method / / P ag e n o . 2 8 4 clc ; clear ; close ;

A = [ 1 0 , 1 , 0 , 0 , 0 , - 1 ; 1 , 1 0 , 1 , 0 , 0 , 0 ; 2 , 0 , 2 0 , 1 , 0 , 0 ; 0 , 0 , 1 , 1 0 ,- 1 , 0 ; 0 , 3 , 0 , 0 , 3


8 B=[5;10;10;0;0;5]

matrix 9 disp (B , ’B= ’ ,A , ’A= ’ ) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

w=0.8 for i = 1 : 2 for j = 1 : 2 A11(i,j)=A(i,j); end B1(i,1)=B(i,1); end for i = 1 : 2 for j = 1 : 2 A12(i,j)=A(i,j+2); end B2(i,1)=B(i+2,1); end for i = 1 : 2 for j = 1 : 2 A13(i,j)=A(i,j+4); end B3(i,1)=B(i+4,1);

127

28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

60 61 62

end for i = 1 : 2 for j = 1 : 2 A21(i,j)=A(i+2,j); end end for i = 1 : 2 for j = 1 : 2 A22(i,j)=A(i+2,j+2); end end for i = 1 : 2 for j = 1 : 2 A23(i,j)=A(i+2,j+4); end end for i = 1 : 2 for j = 1 : 2 A31(i,j)=A(i+4,j); end end for i = 1 : 2 for j = 1 : 2 A32(i,j)=A(i+4,j+2); end end for i = 1 : 2 for j = 1 : 2 A33(i,j)=A(i+4,j+4); end end disp ( B 3 , ’ B3= ’ , B 2 , ’ B2= ’ , B 1 , ’ B1= ’ , A 3 3 , ’ A33= ’ , A 3 2 , ’A32= ’ , A 3 1 , ’ A31= ’ , A 2 3 , ’ A23= ’ , A 2 2 , ’ A22= ’ , A 2 1 , ’ A21= ’ ,A13 , ’ A13= ’ , A 1 2 , ’ A12= ’ , A 1 1 , ’ A11= ’ ) ; A 1 1 _ 1 = inv ( A 1 1 ) ; A 2 2 _ 1 = inv ( A 2 2 ) ; A 3 3 _ 1 = inv ( A 3 3 ) ; disp ( A 3 3 _ 1 , ’ I n v e r s e o f M a tr i x A33= ’ , A 2 2 _ 1 , ’ I n v e r s e o f M a t ri x A22= ’ , A 1 1 _ 1 , ’ I n v e r s e o f M at r ix A11= ’ ) ; for i = 1 : 2

128

63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

X1(i,1)=0; X2(i,1)=0; X3(i,1)=0; end for i = 1 : 7 X(i,1)=i-1; end for i = 2 : 7 X(1,i)=0; end for r = 1 : 6 X11=A11_1*((1-w)*X1+(-1)*w*A12*X2+(-1)*w*A13*X3+ w*B1); X22=A22_1*((1-w)*X2+(-1)*w*A21*X11+(-1)*w*A23*X3 +w*B2); X33=A33_1*((1-w)*X3+(-1)*w*A31*X11+(-1)*w*A32* X22+w*B3); X1=X11; X2=X22; X3=X33; disp ( X 3 , ’ X3= ’ , X 2 , ’ X2= ’ , X 1 , ’ X1= ’ ) for i = 2 : 7 if ( i < 4 ) X(r+1,i)=X1(i-1,1); end if ( i<6 & i > 3) X(r+1,i)=X2(i-3,1); end if ( i<8 & i > 5) X(r+1,i)=X3(i-5,1); end end end printf ( ’ \n\ n I t e r a t i o n ’ ) ; for i = 1 : 6 x%i ’ ,i); printf ( ’ end printf ( ’ \n

129


130

is :\n’

Chapter 9 Linear Least Squares Problem

Scilab code Exa 9.1 Moore Penrose Generalized Inverse

1 2 3 4 5 6 7 8 9

/ / E xa mp le 9 . 1 //Moore−P en ro se G e n e ra l i z ed I n v e r s e / / P ag e n o . 2 9 2 clc ; clear ; close ;

AT=[3,0,3;0,3,3]; // trans pose A=AT’; I = inv ( A T * A ) ; // inv ers e disp (I , ’ I n v e r s e o f AT∗A= ’ , A T * A , ’AT∗A= ’ ,A , ’A= ’ , A T , ’AT = ’ ); 10 A # = I * A T ; 11 disp ( A # , ’ Moore−P en ro se G e n er a l i z ed I n v e r s e o f A= ’ )

Scilab code Exa 9.2 Curve Fitting

131

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

/ / E xa mp le 9 . 2 / / Cu rve F i t t i n g / / P ag e n o . 2 9 3 clc ; clear ; close ; x(1)=0.25; for i = 2 : 6 x(1,i)=x(1,i-1)+0.25; end // x v a l u e s y(1,1)=3.1;y(1,2)=1.7;y(1,3)=1;y(1,4)=0.68;y(1,5) = 0 . 4 2 ; y ( 1 , 6 ) = 0 . 2 6 ; // y v a l u e s

// c o n s t r u c t i o n o f n orm al e q u a ti o n s

32

for i = 1 : 6 Y ( 1 , i ) = log10 ( y ( 1 , i ) ) ; end Ex=0; for i = 1 : 6 Ex=Ex+x(1,i); end EY=0; for i = 1 : 6 EY=EY+Y(1,i); end Ex2=0; for i = 1 : 6 Ex2=Ex2+x(1,i)^2; end ExY=0; for i = 1 : 6 ExY=ExY+x(1,i)*Y(1,i); end printf ( ’E x ( k ) \ t \ t y ( k ) \ t \ t E Y ( k ) \ t \ t E x 2 ( k ) \ t \ t E x ( k ) ∗Y ( k ) ’ ) printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 33 for i = 1 : 6 34 printf ( ’ \ n%f \ t%f \ t%f \ t%f \ t%f ’ , x ( 1 , i ) , y ( 1 , i ) , Y ( 1 ,

132

i),x(1,i)^2,x(1,i)*Y(1,i)) 35 end 36 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 37 printf ( ’ \n% f \ t%f \ t%f \ t%f \ t%f ’ , E x , 0 , E Y , E x 2 , E x Y ) 38 printf ( ’ \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n\n ’ ) 39 A = [ 6 , E x ; E x , E x 2 ] ; // s y s te m o f n or ma l e q u a t i o n s 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

B=[EY;ExY]; X = inv ( A ) * B ; a = exp ( X ( 1 , 1 ) ) ; b=-1*X(2,1); for i = 1 : 2 for j = 1 : 2 printf ( ’ %f ’ ,A(i,j)) end if ( i = = 1 ) printf ( ’ ∗ ’ ) end

printf ( ’ \ ta%i ’ , i ) ; if ( i = = 1 ) printf ( ’ = ’ ) end

printf ( ’ \ t%f \n ’ , B ( i , 1 ) ) end printf ( ’ \n\ na1=%f \ na2=%f \ n\ na=%f \ nb=%f \ n\n ’ , X ( 1 , 1 ) , X (2,1),a,b) 60 printf ( ’ The f i t t e d c ur ve i s : \ n %fx \ ny=%f e ’ ,b,a)

133

Scilab code Exa 9.3 Gram Schmidt Orthogonalization or Orthonormal-

ization Process 1 / / E xa mp le 9 . 3 2 //Gram−S ch mi dt O r t h o g o n a l i z a t i o n / O r t h o n o r m a l i z a t i o n

Process 3 / / P ag e n o . 2 9 4 4 clc ; clear ; close ; 5 deff ( ’ y=f (x , a ) ’ , ’ y= s q r t ( x ( 1 , a ) ˆ 2+ x ( 2 , a ) ˆ 2+ x ( 3 , a ) ˆ 2+ x (4 , a ) ˆ2 ) ’ ) ; 6 deff ( ’ y=f1 ( g , a , h , b ) ’ , ’ y=g ( 1 , a ) ∗h (1 , b)+g (2 , a ) ∗h ( 2 , b )+ g (3 , a ) ∗h (3 , b )+g (4 , a ) ∗h ( 4 , b ) ’ ) ; 7 8 U = [ 1 / sqrt ( 3 ) , - 2 / sqrt (7) ,1,0 ,0,0;0,1/ sqrt (7) ,0,1,0,0;1/ sqrt ( 3 ) , 1 / sqrt (7) ,0,0,1 ,0;-1/sqrt (3) ,-1/ sqrt (7) ,0,0,0,1]; 9 for i = 1 : 4 10 V(i,1)=U(i,1); 11 end 12 for i = 1 : 4 13 if ( f ( V , 1 ) ~ = 0 ) 14 W(i,1)=V(i,1)/f(V,1); 15 else W(i,1)=0; 16 17 end 18 end 19 for j = 2 : 6 for i = 1 : 4 20 21 for l = 1 : 4 k(l,1)=0; 22 23 end 24 for l = 1 : j - 1 25 for m = 1 : 4 26 w(m,1)=W(m,l); 27 end 28 k=k-(f1(U,j,W,l))*w; end 29 30 V(i,j)=U(i,j)+k(i,1);

134

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

end for i = 1 : 4 if ( j ~ = 4 ) if ( f ( V , j ) ~ = 0 ) W(i,j)=V(i,j)/f(V,j); else W(i,j)=0; end else W(i,j)=0; end end

end disp (U , ’U= ’ ) disp ( ’W= ’ ) printf ( ’ \n ’ ) for i = 1 : 4 for j = 1 : 6 printf ( ’%. 4 f \ t \ t ’ , W ( i , j ) ) end printf ( ’ \n ’ ) end disp ( ’V= ’ ) printf ( ’ \n ’ ) for i = 1 : 4 for j = 1 : 6 printf ( ’%. 4 f \ t \ t ’ , V ( i , j ) ) end printf ( ’ \n ’ ) end

Scilab code Exa 9.4 QR Decomposition

135

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

/ / E xa mp le 9 . 4 //QR Decomposition / / P ag e n o . 2 9 6 clc ; clear ; close ; A=[2,1,1;1,3,1;1,1,4]; B=A*A’; disp (B , ’AT∗A= ’ )

/ / c ho le sk y f a c t o r i z a t i o n to f i nd R R(2,1)=0;R(3,1)=0;R(3,2)=0; for i = 1 : 3 for j = 1 : 3 if ( i = = j ) k=0; for m = 1 : i - 1 k=k+R(m,i)^2; end R ( i , j ) = sqrt ( B ( i , j ) - k ) end if ( j > i ) k=0; for m = 1 : i - 1 k=k+R(m,j)*R(m,i); end R(i,j)=(B(i,j)-k)/R(i,i) end end end

// c h o l e s k y f a c t o r i z a t i o n end disp (R , ’ U pp er T r i a n g u l a r M a tr i x (R)= ’ )

R _ 1 = inv ( R ) ; disp ( R _ 1 , ’ I n v e r s e o f R ’ ) Q=A*R_1; disp (Q , ’ O r t h o g o n a l M a t r i x Q= ’ )

136

Scilab code Exa 9.5 Vector Computation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

/ / E xa mp le 9 . 5 / / V e c t o r C o m p u t a ti o n / / P ag e n o . 2 9 9 clc ; clear ; close ; X=[2,3,0,1]; n=X(1); for i = 2 : 4 if ( n < X ( i ) ) n=X(i); end end printf ( ’ \nMaximum Valu e ( n )=%i\n ’ ,n ) for i = 1 : 4 X(i)=X(i)/n; end disp (X , ’ N o r m a l i z e d X= ’ ) k=0; for i = 1 : 4 k=k+X(i)^2; end s i g m a = X ( 1 ) * abs ( 1 / X ( 1 ) ) * sqrt ( k ) ; printf ( ’ \ nsigma=%f \ n ’ , s i g m a ) ; X(1)=X(1)+sigma; printf ( ’ \ n M o d i fi e d x1 = %g\n ’ , X ( 1 ) ) for i = 1 : 4 U(1,i)=X(i); end disp (U , ’U= ’ ) p=sigma*X(1);sigma=n*sigma;

137

31 printf ( ’ \n p = %f \ n\n s ig m a = %f ’ , p , s i g m a ) ; 32 printf ( ’ \n\n\ nNote : Th ere i s a c o mp ut at io n e r r o r i n c a l c u l a t i o n o f U1 ’ )

Scilab code Exa 9.6 House Holder Transformation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

/ / E xa mp le 9 . 6 / / H ou se H o l d er T r a n s f o r ma t i o n / / P ag e n o . 3 0 0 clc ; clear ; close ; A=[4,2,1;2,5,-2;1,-2,7] disp (A , ’A= ’ ) k=0; for j = 2 : 3 k=k+A(j,1)^2; end a = A ( 2 , 1 ) * abs ( 1 / A ( 2 , 1 ) ) * sqrt ( k ) ; disp (a , ’ a l p h a = ’ ) U=[0;a+A(2,1);A(3,1)]; disp (U , ’U= ’ ) U1=U’*U; disp ( U 1 , ’UT∗U= ’ ) U2=U*U’; disp ( U 2 , ’U∗UT= ’ ) P = eye (3,3) -(2*U2)/U1 ; disp (P , ’P=’ ) ; B=P*A*P; disp (B , ’B= ’ ) ; printf ( ’ \n\n\ nThere a r e c om pu ta ti on e r r o r

i n t he a ns we rs g i ve n by t he book i n t h i s ex am ple \n\n ( a22 v al ue e r r o r i n U∗UT) ’ )

138

Scilab code Exa 9.7 Givens QR Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

/ / E xa mp le 9 . 7 //Givens QR Method / / P ag e n o . 3 0 3 clc ; clear ; close ; A=[4,2,1;2,5,-2;1,-2,7] deff ( ’ y=c ( i , j ) ’ , ’ y=A( j , j ) / s q r t ( (A( i , j )ˆ2+A( j , j ) ˆ2 ) ) ’ ) deff ( ’ y=s ( i , j ) ’ , ’ y=A( i , j ) / s q r t ( (A( i , j )ˆ2+A( j , j ) ˆ2 ) ) ’ ) disp (A , ’A= ’ ) R = A ; Q = eye ( 3 , 3 ) ; m=1; for j = 1 : 2 for i = j + 1 : 3 / /C m a tr i x e v a l u a t i o n for k = 1 : 3 for l = 1 : 3 if ( k = = l ) if ( k == i | k == j ) C(k,l)=c(i,j) else C(k,l)=1 end end if ( k > l ) if ( k == i & l == j ) C(k,l)=-1*s(i,j) else C(k,l)=0 end

139

29 end if ( k < l ) 30 31 if ( k == j & l == i ) C(k,l)=s(i,j) 32 33 else C(k,l)=0 34 35 end end 36 37 end 38 end 39 printf ( ’ \n\n I t e r a t i o n %i ’ ,m ) 40 m=m+1 disp (C , ’C=’ ) ; 41 42 R=C*R; Q=Q*C’; 43 44 disp (Q , ’Q= ’ ,R , ’R=’ ) end 45 46 end 47 disp ( Q * R , ’Q∗R=A= ’ ) // v e r i f i c a t i o n

Scilab code Exa 9.8 Recursive Least Square Method

1 2 3 4 5 6 7 8 9 10 11

/ / E xa mp le 9 . 8 / / R e c u r s i v e L e as t −S q u a r e M et ho d / / P ag e n o . 3 0 8 clc ; clear ; close ;

A0=[3,0;0,3;3,3]; B0=[2;2;2]; A1=[6,3];B1=[6]; A0T=A0’; G0=A0T*A0; disp ( G 0 , ’G0= ’ )

140

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

G 0 _ 1 = inv ( G 0 ) ; disp ( G 0 _ 1 , ’ I n v e r s e X0=G0_1*A0T*B0; disp ( X 0 , ’ X0= ’ )

o f G0= ’ )

/ /by r e c u r s i v e l e a s t s qu ar e a l g o r it h m

G1=G0+A1’*A1; disp ( G 1 , ’G1= ’ ) ; G 1 _ 1 = inv ( G 1 ) ; disp ( G 1 _ 1 , ’ I n v e r s e o f G1 ’ ) X1=X0+G1_1*A1’*(B1-A1*X0); disp ( X 1 , ’ X1= ’ )

// v e r i f i c a t i o n

A=[3,0;0,3;3,3;6,3]; B=[2;2;2;6]; AT=A’; G=AT*A; disp (G , ’G= ’ ) G _ 1 = inv ( G ) ; disp ( G _ 1 , ’ I n v e r s e o f G= ’ ) X=G_1*AT*B; disp (X , ’X= ’ ) disp ( ’ T hu s X a nd X1 a r e Same ’ )

141

Chapter 10 Numerical Solutions of System of Non Linear Equations

Scilab code Exa 10.1 System of Non Linear Equations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 1 0 . 1 / / S ys te m o f Non L i n e a r E q u a t io n s / / P ag e n o . 3 1 1 clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ2− exp (2 ∗ x ) −4 ’ ) deff ( ’ y= f 1 ( x ) ’ , ’ y=2∗x−2∗ exp (2 ∗ x ) ’ ) x0=0;e=0.00001 for i = 1 : 1 0 x1=x0-f(x0)/f1(x0) e1 = abs ( x 0 - x 1 ) x0=x1; if a bs ( x 0 ) < e then break ; end end printf ( ’ \n\nThe s o l u t i o n o f t h i s e qu a t io n a f t e r %i

142

I t e r a t i o n s by newto n r ap hs ho n method i s %. 1 0 f ’ ,i , x1 )

Scilab code Exa 10.2 Contraction Method and Seidel Method

1 2 3 4 5 6

/ / E x am pl e 1 0 . 2 / / C o n t r a c t i o n Method a nd S e i d e l Metho d / / P ag e n o . 3 1 5 clc ; clear ; close ; x(1)=0;y(1)=0 printf ( ’ ( a ) C o n t r a c t i o n Mapping \n\nn \ t x n \ t \ ty n \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n 0\ t%f \ t%f \n ’ ,x 7 8 9 10 11 12

(1),y(1)) for i = 2 : 9 x ( i ) = sin ( x ( i - 1 ) + y ( i - 1 ) ) y ( i ) = cos ( x ( i - 1 ) - y ( i - 1 ) ) printf ( ’ %i \ t%f \ t%f \n ’ , i - 1 , x ( i ) , y ( i ) ) end printf ( ’ \n\n\n ( b ) S e i d e l Method \n\nn \ t x n \ t \ ty n \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n 0\ t%f \ t%f \n ’ ,x 13 14 15 16 17

(1),y(1)) for i = 2 : 9 x ( i ) = sin ( x ( i - 1 ) + y ( i - 1 ) ) y ( i ) = cos ( x ( i ) - y ( i - 1 ) ) printf ( ’ %i \ t%f \ t%f \n ’ , i - 1 , x ( i ) , y ( i ) ) end

Scilab code Exa 10.3 Non Linear System of Equation

143

1 2 3 4 5 6 7

/ / E x am pl e 1 0 . 3 / / Non L i n e a r S ys te m o f E q ua t i on / / P ag e n o . 3 1 5 clc ; clear ; close ; x(1,1)=1;x(1,2)=0;x(1,3)=0; y(1,1)=0;y(1,2)=2;y(1,3)=2; printf ( ’ Case −−>\t \ t I \ t \ t \ t \ t \ I I \ t \ t \ t \ t I I I \ n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n I t e r a t i o n \ t x \ t \ t y \ t \ t x \ t \ t y \ t \ t x \ t \ t y \n\n 0\ t \ t%f \ t%f \ t%f \ t%f \ t%f \ t%f \n ’ , x ( 1 , 1 ) , y ( 1 , 1 ) , x ( 1 , 2 ) , 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

y(1,2),x(1,3),y(1,3)) for i = 2 : 9 printf ( ’ %i \ t ’ , i - 1 ) for j = 1 : 3 if j ==1 | j ==2 then x(i,j)=(-y(i-1,j)^2-4*x(i-1,j)^2+8*x(i -1,j)+4)/8 y(i,j)=(2*y(i-1,j)^2-2*x(i-1,j)+1)/4 printf ( ’ \ t%f \ t%f ’ , x ( i , j ) , y ( i , j ) ) end if j = = 3 then x(i,j)=(-y(i-1,j)^2-4*x(i-1,j)^2+11*x(i -1,j)+4)/11 y(i,j)=(-2*y(i-1,j)^2+8*y(i-1,j)-2*x(i-1,j) -1)/4 printf ( ’ \ t%f \ t%f ’ , x ( i , j ) , y ( i , j ) ) end end printf ( ’ \n ’ ) end printf ( ’ \n\n\n\ nNo te : T he re a r e c o m pu t a t io n a l e r r o r s i n t h i s e xa m p l e g i v en by t he book ’ )

144

Scilab code Exa 10.4 Newton Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

/ / E x am pl e 1 0 . 4 // Newton Method / / P ag e n o . 3 1 7 clc ; clear ; close ;

deff ( ’ y=f1 ( x1 , x2 ) ’ , ’ y=x1+3∗ l o g 1 0 ( x 1 )−x2ˆ2 ’ ) deff ( ’ y=f2 ( x1 , x2 ) ’ , ’ y=2∗x1ˆ2−x1 ∗ x2 −5∗x1 +1 ’ ) deff ( ’ y=f1 1 ( x1 , x2 ) ’ , ’ y = 1+ 3/ ( l o g ( 1 0 ) ∗ x 1 ) ’ ) deff ( ’ y=f1 2 ( x1 , x2 ) ’ , ’ y=−2∗x2 ’ ) deff ( ’ y=f2 1 ( x1 , x2 ) ’ , ’ y=4∗x1−x2 −5 ’ ) deff ( ’ y=f2 2 ( x1 , x2 ) ’ , ’ y=−x1 ’ ) x=[3.4;2.2]; disp (x , ’ x (0 ) = ’ ) for i = 1 : 3 fx=[f1(x(1),x(2));f2(x(1),x(2))] printf ( ’ \n f x ( %i ) = \n ’ ,i ) disp ( f x ) A=[f11(x(1),x(2)),f12(x(1),x(2));f21(x(1),x(2)), f22(x(1),x(2)),] disp (A , ’A = ’ ) A _ 1 = inv ( A ) disp ( A _ 1 , ’ I n v e r s e o f A = ’ ) x=x-A_1*fx printf ( ’ \n x ( % i ) = \n ’ ,i ) disp ( x ) end

Scilab code Exa 10.5 Newton Raphshon Method

1 / / E x am pl e 1 0 . 5 2 //Newton Raphshon Method

145

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

/ / P ag e n o . 3 2 0 clc ; clear ; close ;

deff ( ’ y=f1 ( x , y ) ’ , ’ y=xˆ3 −3∗x ∗ yˆ2+1 ’ ) deff ( ’ y=f2 ( x , y ) ’ , ’ y=3∗x ˆ2∗ y−yˆ3 ’ ) deff ( ’ y=f1 1 (x , y ) ’ , ’ y=3∗xˆ2 −6∗yˆ2 ’ ) deff ( ’ y=f1 2 (x , y ) ’ , ’ y=−6∗x ∗y ’ ) deff ( ’ y=f2 1 (x , y ) ’ , ’ y=6∗x ∗ y ’ ) deff ( ’ y=f2 2 (x , y ) ’ , ’ y=3∗xˆ2 −3∗yˆ2 ’ ) x=[0;1]; printf ( ’ \ n x ( 0 ) = %g\ n y ( 0 ) = %g\n ’ , x ( 1 ) , x ( 2 ) ) for i = 1 : 3 fx=[f1(x(1),x(2));f2(x(1),x(2))] printf ( ’ \n f x ( %i ) = \n ’ ,i ) disp ( f x ) J=[f11(x(1),x(2)),f12(x(1),x(2));f21(x(1),x(2)), f22(x(1),x(2)),] disp (J , ’ J = ’ ) d = det ( J ) ; if d = = 0 then dx1=0;dx2=0; else dx1=(fx(1)*J(2,2)-fx(2)*J(1,2))/d; dx2=(fx(2)*J(1,1)-fx(1)*J(2,1))/d; end x(1)=x(1)+dx1; x(2)=x(2)+dx2; printf ( ’ \ n x ( % i ) = % g\ n y ( % i ) = % g\n ’ , i , x ( 1 ) , i , x (2)) end

Scilab code Exa 10.6 Newton Method

146

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

/ / E x am pl e 1 0 . 6 // Newton Method / / P ag e n o . 3 2 2 clc ; clear ; close ;

deff ( ’ y=f 1 ( x , y , z ) ’ , ’ y=x −0.1∗ y ˆ 2 + 0 . 0 5∗ z ˆ2 −0.7 ’ ) deff ( ’ y=f 2 ( x , y , z ) ’ , ’ y=y+ 0.3 ∗ x ˆ2 −0.1∗ x ∗ z −0.5 ’ ) deff ( ’ y=f 3 ( x , y , z ) ’ , ’ y=z +0.4 ∗ y ˆ 2 + 0 . 1∗ x ∗y −1.2 ’ ) deff ( ’ y=f 1 1 ( x , y , z ) ’ , ’ y=1 ’ ) deff ( ’ y=f 1 2 ( x , y , z ) ’ , ’ y = −0. 2∗y ’ ) deff ( ’ y=f 1 3 ( x , y , z ) ’ , ’ y = 0 . 1∗ z ’ ) deff ( ’ y=f 2 1 ( x , y , z ) ’ , ’ y = 0 . 6∗ x −0.1∗ z ’ ) deff ( ’ y=f 2 2 ( x , y , z ) ’ , ’ y=1 ’ ) deff ( ’ y=f 2 3 ( x , y , z ) ’ , ’ y = −0. 1∗x ’ ) deff ( ’ y=f 3 1 ( x , y , z ) ’ , ’ y = 0 . 1∗ y ’ ) deff ( ’ y=f 3 2 ( x , y , z ) ’ , ’ y = 0 . 8∗ y + 0 . 1∗ x ’ ) deff ( ’ y=f 3 3 ( x , y , z ) ’ , ’ y=1 ’ ) x=[0;0;0]; printf ( ’ n\ tx n \ t \ t yn \ t \ t z n \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 20 for i = 1 : 6 21 fx=[f1(x(1),x(2),x(3));f2(x(1),x(2),x(3));f3(x (1),x(2),x(3))] 22 J=[f11(x(1),x(2),x(3)),f12(x(1),x(2),x(3)),f13(x (1),x(2),x(3));f21(x(1),x(2),x(3)),f22(x(1),x (2),x(3)),f23(x(1),x(2),x(3));f31(x(1),x(2),x (3)),f32(x(1),x(2),x(3)),f33(x(1),x(2),x(3))] 23 J _ 1 = inv ( J ) printf ( ’ %i \ t%f \ t%f \ t%f \n ’ , i - 1 , x ( 1 ) , x ( 2 ) , x ( 3 ) ) 24 25 x=x-J_1*fx 26 end 27 printf ( ’ \n\nThe s o l u t i o n i s x = %f , y = %f and z = %f ’ , x ( 1 ) , x ( 2 ) , x ( 3 ) ) 28 29 printf ( ’ \n\n\ nNote : Th ere a r e c om pu ta ti on e r r o r s i n c a l c u l a t i o n g i v en by t he book ’ )

147

Scilab code Exa 10.7 Iterative Method

1 2 3 4 5 6 7

/ / E x am pl e 1 0 . 7 / / I t e r a t i v e Method / / P ag e n o . 3 2 6 clc ; clear ; close ; x=[0;0;0]; printf ( ’ n\ t x n \ t \ t yn \ t \ t z n \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 8 for i = 1 : 7 9 . 1 0 f \t% . 1 0 f \n ’ ,i-1,x(1),x printf ( ’ %i \t% . 1 0 f \t% (2),x(3)) x(1)=0.7+0.1*x(2)^2-0.05*x(3)^2 10 11 x(2)=0.5-0.3*x(1)^2+0.1*x(1)*x(3) x(3)=1.2-0.4*x(2)^2-0.1*x(1)*x(2) 12 13 end 14 printf ( ’ \n\nThe s o l u t i o n i s x = %. 1 0 f , y = %. 1 0 f and z = %.10 f ’ ,x(1),x(2),x(3))

Scilab code Exa 10.8 Steepest Descent

1 2 3 4 5

/ / E x am pl e 1 0 . 8 / / S t e e p e s t D e s ce n t / / P ag e n o . 3 2 8 clc ; clear ; close ;

148

6 deff ( ’ y=f (x1 , x2 ) ’ , ’ y=(x1 −2)ˆ4+3∗(x2 +3)ˆ2 ’ ) 7 x=[1;-2]; 8 printf ( ’ n\ t x1 \ t \ t x2 \ t \ t F ( x1 , x2 ) \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 9 for i = 1 : 1 1 10 Fx=[f(x(1),x(2))]; J=[4*(x(1)-2)^3,6*(x(2)+3)]; 11 12 u=(Fx*J*J’*Fx)/(J*J’*Fx*J*J’*Fx) 13 printf ( ’ %i \t% . 1 0 f \t% . 1 0 f \t% . 1 0 f \n ’ ,i-1,x(1),x (2),Fx) 14 x=x-u*J’*Fx 15 end 16 printf ( ’ \n\ n T h i s s hows t ha t t he s o l u t i o n t en ds t o x 1 = %i and x2 = %i ’ , ceil ( x ( 1 ) ) ,floor ( x ( 2 ) ) )

149

Chapter 11 Eigenvalues and Eigenvectors

Scilab code Exa 11.1 Eigenvalues and Eigenvectors

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 1 . 1 / / E i g e n v a l u e s and E i g e n v e c t o r s / / P ag e n o . 3 3 3 clc ; clear ; close ; A1=[0.6;0.2];A2=[-0.2;0.6];A3=[-0.6;-0.2];A4 =[0.2;-0.6]; T=[1.1,-0.3;-0.3,1.9]; B1=T*A1;B2=T*A2;B3=T*A3;B4=T*A4; disp ( B 4 , B 3 , B 2 , B 1 , ’ The t r an s fo r me d v e c t o r s a r e : ’ ) disp ( ’ T h es e p o i n t s l i e o n t h e e l l i p s e : ’ ) printf ( ’ 2 2\ n ( x−3y)+(3x+y) \n−−−−−− −−−−−−\ n 16 4\ n\n ’ ) A 5 = [ 0 ; 2 / sqrt (10)]; disp ( ’ The v e c t o r ( 0 , 2 / 1 0 ˆ ( 1 / 2 ) ) l i e s on t h e c i r c l e : ’ ) 2\ nx + y = 4 \ n 10\ n\n printf ( ’ 2 −\n ’) B5=T*A5;

150

16 disp ( ’ A l s o l i e s o n t h e s am e e l l i p s e ’ , B 5 ) 17 printf ( ’ \n\nWe can s e e t ha t t h e r e i s a l i n e a r

r e l a t i o n s h i p b etw ee n t he f i r s t 4 v e c t o r s and t h e i r r e s p e c t i v e t ra n s fo rm en d v e c t o r s t hr ou gh t he s c a l a r s known a s e i g e n v e c t o r s and e i g e n v a l u e s respecti vely ’)

Scilab code Exa 11.2 Leverriers Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

/ / E x am pl e 1 1 . 2 / / L e v e r r i e r ’ s M et ho d / / P ag e n o . 3 3 7 clc ; close ; clear ; A=[2,2,2;2,5,5;2,5,1]; A1=A; C(1)=0; for j = 1 : 3 for k = 1 : 3 if ( j = = k ) C(1)=C(1)+A1(j,k) end end end disp (A , ’A= ’ ) disp ( A 1 , ’ A1= ’ ) printf ( ’ \nC1=’ ) disp ( C ( 1 ) ) ; for i = 2 : 3 A 2 = A * ( A 1 - C ( i - 1 ) *eye ( 3 , 3 ) ) ; printf ( ’ \n\n\nA%i=’ ,i ) disp ( A 2 ) ; C(i)=0;

151

25 for j = 1 : 3 for k = 1 : 3 26 27 if ( j = = k ) C(i)=C(i)+A2(j,k)/i 28 29 end end 30 31 end printf ( ’ \nC%i=’ ,i ) 32 33 disp ( C ( i ) ) 34 A1=A2; 35 end 36 printf ( ’ \n\n\ n Th er ef or e t h e c h a r a c t e r i s t i c

p o l y no m i a l i s : \ n 3

2\ nx − %ix − % ix %i = 0 ’ ,C

(1),C(2),C(3)) 37 38 39 40 41 42 43

// v e r i f i c a t i o n printf ( ’ \n\ n V e r i f i c a t i o n : ’ ) s = poly (0 , ” s ” ) ; p = poly (A , ’ x ’ ) ; A = A - eye ( 3 , 3 ) * % s ; disp (p , ’= ’ ,A )

Scilab code Exa 11.3 Danilevsky Method

1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 1 . 3 / / D a n i l e v s k y Me th od / / P ag e n o . 3 4 1 clc ; close ; clear ; A=[-1,0,0;1,-2,3;0,2,-3]; G = [ A ; eye ( 3 , 3 ) ] ; disp ( G ) ;

/ / t r a n s f o rm a t i o n t o f r o b e n i u s m at ri x 152

10 for k = 3 : - 1 : 2 g(k)=0; 11 12 for j = 1 : k - 1 if ( g ( k ) < G ( k , j ) ) 13 14 g(k)=G(k,j) p=j; 15 16 end end 17 18 if ( g ( k ) ~ = 0 ) 19 for j = 1 : 3 20 r(1,j)=G(k,j) 21 end for i = 1 : 6 22 23 G(i,k-1)=G(i,k-1)/g(k) end 24 25 disp ( G ) for j = 1 : 3 26 27 if ( j ~ = k - 1 ) l=G(k,j) 28 29 for i = 1 : 6 30 G(i,j)=G(i,j)-l*G(i,k-1) 31 end 32 end end 33 34 disp ( G ) end 35 36 for j = 1 : 3 for i = 1 : 3 37 38 c(i,1)=G(i,j) end 39 40 G(k-1,j)=0 41 for i = 1 : 3 42 G(k-1,j)=G(k-1,j)+r(1,i)*c(i,1) 43 end 44 end 45 disp ( G ) 46 end 47

153

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85

// p a r t i t i o n g for i = 4 : 6 for j = 1 : 3 T(i-3,j)=G(i,j) end end disp (T , ’T=’ )

/ / e i g e n v a l u e s c o m pu t a ti o n printf ( ’ \n\n\ n C h a r a t e r i s t i c p o ly n om i al : ’ ) p = poly (A , ’ x ’ ) disp ( p ) printf ( ’ \n\n\ n E i g e n v a l u e s : ’ ) a = roots ( p ) disp ( a ’ )

/ / e i g e n v e c t o r s c om pu t at io n for k = 1 : 3 m =2 for l = 1 : 3 y(l,k)=a(k,1)^(m) m=m-1; end end printf ( ’ \n\n ’ ) disp (y , ’ y= ’ )

/ / e i g e n v e c t o r c o m pu t a ti o n for k = 1 : 3 for l = 1 : 3 y1(l,1)=y(l,1) y2(l,1)=y(l,2) y3(l,1)=y(l,3) end x1=T*y3; x2=T*y2; x3=T*y1; end

154

86 printf ( ’ \n\ n E i g en v e ct o r s : \ n ’ ) 87 for i = 1 : 3 88 printf ( ’ | %.1 f | \ t \ t | %.1 f | \ t \ t | %.1 f | ’ , x 1 ( i , 1 ) , x 2 ( i ,1),x3(i,1)) 89 printf ( ’ \n ’ ) 90 end

Scilab code Exa 11.4 Power Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 1 . 4 //Power Method / / P ag e n o . 3 4 5 clc ; close ; clear ; A=[1,2;3,4]; e=0.001; q0=[1;1]; for i = 1 : 5 q1=A*q0; a = max ( q 1 ) for j = 1 : 2 q2(j)=q1(j)/a; end printf ( ’ \ nq ( %i ) = %. 4 f

q ( % i ) = %. 4 f \n

a = %. 4 f %. 4 f

Scaled %i \n\n ’ ,

i,q1(1),a,i,q2(1),q1(2),q2(2)) q1=q2; q0=q1;

16 17 18 end 19 printf ( ’ H ence t he l a r g e s t

e i g e n v a l u e i s %. 4 f w i th t he c o r r e sp o n di n g e i g e n v e c t o r a s %. 4 f \n

155

%i ’ , a , q 0 ( 1 ) , q 0 ( 2 ) )

Scilab code Exa 11.5 Inverse Power Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

/ / E x am pl e 1 1 . 5 / / I n v e r s e P ow er Me th od / / P ag e n o . 3 4 7 clc ; close ; clear ;

A=[7,6,-3;-12,-20,24;-6,-12,16]; e=10^-6; X=[1;1;1]; B=0; Y=[0;0;0] a=0;l=0; for i = 1 : 2 printf ( ’When a=%i \n ’ , a ) ; C = A - a * eye () ; disp (C , ”C=” ) C _ 1 = inv ( C ) ; disp ( C _ 1 , ” I n v e r s e o f C=” ) ; printf ( ’ \n\ n I t r lambda

X’) 19

printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 20 21 22 23 24 25

for j = 1 : 1 0 printf ( ’ \ n%i %f %f %f ’ , j - 1 , l , X ( 1 ) , X ( 2 ) , X ( 3 ) ) ; Y=C_1*X; B = max ( Y ) ; e1 = abs ( l - B ) ; X=Y/B;

156

%f

26 m=0; for k = 1 : 3 27 28 m=m+(Y(k)-X(k))^2; end 29 30 e2 = sqrt ( m ) ; er = max ( e 1 , e 2 ) ; 31 32 if ( e r < e ) break 33 34 end 35 l=B; 36 37 end a=-3; 38 39 printf ( ’ \n\n\n\n ’ ) 40 end 41 printf ( ’ \n\n\ nNo te : C om pu ta ti on o f Y i s i n t h e b oo k ’ )

wrong g i v e n

Scilab code Exa 11.6 Rayleigh Quotient

1 2 3 4 5 6 7 8 9 10 11 12 13

/ / E x am pl e 1 1 . 6 // Rayleigh Quotient / / P ag e n o . 3 4 8 clc ; close ; clear ; A=[10,7,8,7;7,5,6,5;8,6,10,9;7,5,9,10]; q0=[1;1;1;1]; for i = 0 : 4 X=(Aî)*q0; l=(X’*A*X)/(X’*X) printf ( ’ \nLambda ( %i) = %f \ n ’ , i + 1 , l ) end printf ( ’ \n\ n Do mi na nt E i g e n v a l u e = % f\ n\n\n ’ ,l )

157

14 15 16 17 18 19 20 21 22 23 24 25 26

e=0.001; for i = 1 : 5 q1=A*q0; a = max ( q 1 ) for j = 1 : 4 q2(j)=q1(j)/a; end q1=q2; q0=q1; end disp ( q 2 , ’ C o r re s p on d i ng E i g e n v e c t o r = ’ )

Scilab code Exa 11.7 Jacobi Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 1 . 7 // Jac obi ’ s Method / / P ag e n o . 3 5 5 clc ; close ; clear ; A=[1,1,1/2;1,1,1/4;1/2,1/4,2]; C=A; V =[ sqrt (2) ,0,1/2; sqrt (2) ,0,1/4;3/(4* sqrt ( 2 ) ) , - 1 / ( 4 * sqrt ( 2 ) ) , 2 ] S = eye ( 3 , 3 ) disp (A , ”A =” ) VI=0; for i = 1 : 3 for j = 1 : 3 if ( i ~ = j ) VI=VI+A(i,j)^2

// i n i t i a l

o f f d i a g norm 158

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

end end

end V I = sqrt ( V I ) ; // f i n a l t h r es h o l d VF=VI*10^-7; V1=VI/3; o = poly ( 0 , ” o ” ) ; for i=1:3 for q=2:3 for p=q-1:-1:1 if ( A ( p , q ) > V 1 ) a=-A(p,q); b=(A(p,p)-A(q,q))/2 if (b~=0) w = b * a b s ( 1 / b ) * ( a / sqrt ( a ^ 2 + b ^ 2 ) ) ; else w = ( a / sqrt ( a ^ 2 + b ^ 2 ) ) ; end s i n 0 = w / sqrt (2*(1+ sqrt ( 1 - w ^ 2 ) ) ) ; c o s 0 = sqrt ( 1 - s i n 0 ^ 2 ) end B(p,p)=A(p,p)*cos0^2+A(q,q)*sin0^2-2*A(p,q)* sin0*cos0 B(q,q)=A(p,p)*sin0^2+A(q,q)*cos0^2+2*A(p ,q)*sin0*cos0 B(p,q)=(A(p,p)-A(q,q))*sin0*cos0+A(p,q) *(cos0^2-sin0^2) S(i,i)=S(i,i) S(i,p)=S(i,p)*cos0-S(i,q)*sin0 S(i,q)=S(i,p)*sin0+S(i,q)*cos0 end

end end disp ( B , ”B =” ) disp ( S , ” S = ” ) printf ( ’ \ n \ n \ n C o mp mp u t a t i o n e r r o r p r o v i d e d b y b oo oo k ’ )

159

i n t he he s o l u t i o n

Scilab code Exa 11.8 Recursive Formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

/ / E x am am pl pl e 1 1 . 8 / / R e c u r s i v e F or o r mu m u la la / / P ag ag e n o . 3 5 7 c l c ; close ; clear ; A=[2,-1,0,0;-1,2,-1,0;0,-1,2,-1;0,0,-1,2]; l = poly ( 0 , ” l ” ) ; p0=1; p1=A(1,1)-l; for i=2:4 p2=(A(i,i)-l)*p1-A(i,i-1)^2*p0; p0=p1; p1=p2; printf ( ’ \ n \ n p % i ( l ) = ’ , i ) ; disp ( p 2 ) end

Scilab code Exa 11.9 QR Method

1 2 3 4 5 6

/ / E x am am pl pl e 1 1 . 9 //QR //Q R Method / / P ag ag e n o . 3 6 0 c l c ; close ; clear ; A=[2,-1,0;-1,2,-1;0,-1,2];

160

7 deff ( ’ y=c ( i , j ) ’ , ’ y=A( j , j ) / s q r t ( (A( i , j )ˆ2+A( )ˆ2+A( j , j ) ˆ2 ) ) ’ ) 8 deff ( ’ y=s2 ( i , j ) ’ , ’ y=A( i , j ) / s q r t ( (A( i , j ) ˆ2+ ˆ2+A( j , j ) ˆ2 ) ) ’) 9 disp ( A , ’A= ’A= ’ ) 10 l 0 = 0 ; f = 1 ; m = 0 ; s = 0 ; w = 0 ; 11 f o r n = 1 : 5 for j=1:2 12 13 for k=1:2 14 V(j,k)=A(j,k) 15 end 16 end disp ( V , ’V= 17 ’V= ’ ) 18 p = poly ( V , ’ x ’ ) ; disp ( ’ =0 ’ , p ) ; 19 20 a = roots ( p ) ; for j=1:2 21 22 printf ( ’ \ n a ( % i ) = % f ’ , j , a ( j ) ) end 23 24 if ( abs ( a ( 1 ) - V ( 1 , 1 ) ) < = abs ( a ( 2 ) - V ( 1 , 1 ) ) ) 25 a=a(1) 26 else 27 a=a(2) end 28 29 n ’ ,a) printf ( ’ \ n a = % f \ s=s+a; 30 31 A = A - a * ey e () R=A;Q= eye (3,3); 32 33 for j=1:2 34 35 for i=j+1:3 36 for k=1:3 / /C /C m a t r i x

evaluation 37 38 39 40 41

for l=1:3 if(k==l) i f ( k == = = i | k == == j ) C(k,l)=c(i,j) else

161

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

C(k,l)=1 end end if( k > l ) i f ( k == = = i & l == == j ) C(k,l)=-1*s2(i,j) else C(k,l)=0 end end if( k < l ) i f ( k == = = j & l == == i ) C(k,l)=s2(i,j) else C(k,l)=0 end end

end end

R=C*R; Q=Q*C’; end

end disp ( Q , ’Q= ’Q= ’ , R , ’R=’ ) ’ Q∗R= ’ ) disp ( Q * R , ’Q A=R*Q; ’A= ’ ) disp ( A , ’A= end l1=l0+s; for i=2:3 for j=2:3 V(i-1,j-1)=A(i,j) end end disp ( V , ’V= ’V= ’ ) p = poly ( V , ’ x ’ ) ;

162

80 disp ( ’ =0 ’ , p ) ; a = roots ( p ) ; 81 82 for j=1:2 printf ( ’ \ n a ( % i ) = % f ’ , j , a ( j ) ) 83 84 end l2=l1+a(1) 85 86 l3=l1+a(2) disp ( l 3 , ’ l 3 = ’ , l 2 , ’ l 2 = ’ , l 1 , ’ l 1 = ’ ) 87 88 printf ( ’ \ n \ n \ n N o t e : V a lu l u es es o f V v a r i e s i n e a c h s t e p

r e s u l t i n g i n d i f f e r e n t r e s u l t s due t o e r r o r i n book c a l c u l a t i o n ’ )

Scilab code Exa 11.10 LU Method

1 2 3 4 5 6

/ / E x am am pl pl e 1 1 . 1 0 //LU //L U Meth Method od / / P ag ag e n o . 3 6 3 c l c ; close ; clear ; A = [ 1 2 0 , 8 0 , 4 0 , - 1 6 ; 8 0 , 1 2 0 , 1 6 , - 4 0 ; 4 0 , 1 6 , 1 2 0 , - 8 0 ; - 1 6 , - 4 0, - 8 0 , 1 2 0 ] ;

7 8 9 10 11 12 13 14 15 16 17

disp ( A , ”A =” ) L= eye (4,4); for l=1:20 for j=1:4 for i = 1 : j k =0 for p=1:i-1 k=k-A(i,p)*A(p,j) end A(i,j)=A(i,j)+k end

163

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

for i = j + 1 : 4 k=0; for p = 1 : j - 1 k=k-A(i,p)*A(p,j) end A(i,j)=(A(i,j)+k)/A(j,j) end

end disp (A , ” M o d i f i e d A = ” ) for i = 1 : 4 for j = 1 : 4 if i > j then L(i,j)=A(i,j) else U(i,j)=A(i,j) end end end disp (U , ”U =” ,L , ”L =” ) A=U*L; printf ( ’ \n\ n Af te r %i i t e r a t i o n s , m at ri x A =\n\n ’ ,l ) for i = 1 : 4 for j = 1 : 4 printf ( ’ %. 2 f \ t ’ , A ( i , j ) ) end printf ( ’ \n ’ ) end end printf ( ’ \n\ n T h er e f o r e t he e i g e n v a l u e s a r e t he

d i a go n a l e le m e nt s f t he t ra n sf or m e d t r i a n g u l a r m a t ri x a r e : \ n\n ’ ) 47 for i = 1 : 4 48 printf ( ’ % . 2 f , ’ , A ( i , i ) ) 49 end

164

Scilab code Exa 11.11 Generalized Eigenvalue Problem

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

/ / E x am pl e 1 1 . 1 1 / / G e n e r a l i z e d E i g e n v a l u e P ro bl em / / P ag e n o . 3 6 5 clc ; close ; clear ; A=[1,1,0.5;1,1,0.25;0.5,0.25,2] B=[2,2,2;2,5,5;2,5,11] disp (B , ”B =” ,A , ”A =” ) for i = 1 : 3 G ( i , i ) = sqrt ( B ( i , i ) ) end G = [ B ; eye ( 3 , 3 ) ] ;

/ / t r a n s f o rm a t i o n t o f r o b e n i u s m at ri x for k = 3 : - 1 : 2 g(k)=0; for j = 1 : k - 1 if ( g ( k ) < G ( k , j ) ) g(k)=G(k,j) p=j; end end if ( g ( k ) ~ = 0 ) for j = 1 : 3 r(1,j)=G(k,j) end for i = 1 : 6 G(i,k-1)=G(i,k-1)/g(k) end for j = 1 : 3

165

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

if ( j ~ = k - 1 ) l=G(k,j) for i = 1 : 6 G(i,j)=G(i,j)-l*G(i,k-1) end end

end end for j = 1 : 3 for i = 1 : 3 c(i,1)=G(i,j) end G(k-1,j)=0 for i = 1 : 3 G(k-1,j)=G(k-1,j)+r(1,i)*c(i,1) end end end

// p a r t i t i o n g for i = 4 : 6 for j = 1 : 3 T(i-3,j)=G(i,j) end end

/ / e i g e n v a l u e s c o m pu t a ti o n p = poly (B , ’ x ’ ) a = roots ( p ) printf ( ’ \n\ n D i a g o n a l i z e d M at ri x B = \n\n ’ ) for i = 1 : 3 for j = 1 : 3 if i ~ = j then B(i,j)=0 else B(i,j)=a(i) end end

166

69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

end disp ( B )

/ / e i g e n v e c t o r s c om pu t at io n for k = 1 : 3 m =2 for l = 1 : 3 y(l,k)=a(k)^(m) m=m-1; end end printf ( ’ \n\n ’ )

for k = 1 : 3 for l = 1 : 3 y1(l,1)=y(l,1) y2(l,1)=y(l,2) y3(l,1)=y(l,3) end x1=T*y3; x2=T*y2; x3=T*y1; end printf ( ’ \n\ n E i g e n v e c t o r s o f B a re : \ n\n ’ ) for i = 1 : 3 printf ( ’ | %.5 f | \ t \ t | %.5 f | \ t \ t | %.5 f | ’ , x 3 ( i , 1 ) , x 2 ( i ,1),x1(i,1)) printf ( ’ \n ’ ) end x=[x3,x2,x1]

B=[2,2,2;2,5,5;2,5,11] G =0 for i = 1 : 3

167

106 107 108 109 110 111 112 113 114 115 116 117 118

for j = 1 : 3 if i = = j then G ( i , j ) = sqrt ( B ( i , j ) ) else G(i,j)=0; end end end B = inv ( G ) * x ’ * A * x * inv ( G ) disp (B , ” E i g e n v e c t o r s o f A =” ) printf ( ’ \n\n\ nNo te : C om pu ta ti on E r r or i n bo ok i n c ac u la ti o n o f e i ge n ve ct or o f B thus f o r A ’ )

168

Chapter 12 Interpolation and Extrapolation

Scilab code Exa 12.1 Linear Interpolation Technique

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 1 2 . 1 // L i n ea r I n t e r p o l a t i o n T ec hn iq ue / / P ag e n o . 3 7 2 clc ; close ; clear ;

’) printf ( ’ x : f=[1,4,9,16,25]; for i = 1 : 5 printf ( ’ %i \ t ’ ,i ) end printf ( ’ \ n f ( x ) : ’) for i = 1 : 5 printf ( ’ %i \ t ’ , f ( i ) ) end x=2.5; x 1 = 2 ; x 2 = 3 ; printf ( ’ \n\ n f o r ( 2 , 4 ) and ( 3 , 9 ) ’ ) f(2.5)=f(x1)+(f(x2)-f(x1))*(x-x1)/(x2-x1) 169

18 19 20 21 22 23 24 25 26 27 28 29 30

printf ( ’ \ n f ( 2 . 5 ) = %. 1 f ’ , f ( 2 . 5 ) )

x=2.5; x 1 = 2 ; x 2 = 4 ; printf ( ’ \n\ n f o r ( 2 , 4 ) and ( 4 , 1 6 ) ’ ) f(2.5)=f(x1)+(f(x2)-f(x1))*(x-x1)/(x2-x1) printf ( ’ \ n f ( 2 . 5 ) = %. 1 f ’ , f ( 2 . 5 ) )

x=2.5; x 1 = 1 ; x 2 = 3 ; printf ( ’ \n\ n f o r ( 1 , 1 ) and ( 3 , 9 ) ’ ) f(2.5)=f(x1)+(f(x2)-f(x1))*(x-x1)/(x2-x1) printf ( ’ \ n f ( 2 . 5 ) = %. 1 f ’ , f ( 2 . 5 ) )

printf ( ’ \n\ n E xa ct v a l u e = %. 2 f ’ ,2.5^2)

Scilab code Exa 12.2 Lagarangian Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 1 2 . 2 / / L a g a r a n g i a n M et ho d / / P ag e n o . 3 7 3 clc ; close ; clear ; xk=[-1,0,2,5]; yk=[10,7,7,22]; P=0; x = poly (0 , ” x ” ) ; for k = 0 : 3 p=yk(k+1) for j = 0 : 3 if ( j ~ = k ) p=p*((x-xk(j+1))/(xk(k+1)-xk(j+1))) end end

170

18 P=P+p; 19 end 20 disp (P , ’P=’ )

Scilab code Exa 12.3 Aitken Nevilles Method

1 2 3 4 5 6

/ / E x am pl e 1 2 . 3 / / A i t k e n −N e v i l l e ’ s M et ho d / / P ag e n o . 3 7 8 clc ; close ; clear ; function [ x , y , z ] = t r a n ( a , b )


e x ch a n gi n g v a l u e s 7 z=a;y=b;x=z; 8 endfunction 9 deff ( ’ y=P( a , b , c , d , e ) ’ , ’ y =( c ( d ) ∗b ( d + 1 )−c ( d+e ) ∗b ( d ) ) / ( a ( d+e )−a ( d ) ) ’ ) // f u n c t i o n f o r f i n d i n g

polynomials 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

xi=[0.8,1,1.2,1.4,1.6]; yi=[2.2255,2.7183,3.3201,4.0552,4.9530]; x=1.23 [xi(5),xi(1),a]=tran(xi(1),xi(5)) [xi(4),xi(1),a]=tran(xi(1),xi(4)) [xi(3),xi(2),a]=tran(xi(2),xi(3)) [xi(2),xi(1),a]=tran(xi(1),xi(2)) [yi(5),yi(1),a]=tran(yi(1),yi(5)) [yi(4),yi(1),a]=tran(yi(1),yi(4)) [yi(3),yi(2),a]=tran(yi(2),yi(3)) [yi(2),yi(1),a]=tran(yi(1),yi(2)) for i = 1 : 5 x_xi(i)=x-xi(i); end printf ( ’ x i x−x i y i \n ’ )

171

25 printf ( ’−−−−−−−−−−−−−−−−−−−−−−−−\n ’ ) 26 for i = 1 : 5 27 %. 2 f \ t%f \n ’ , x i ( i ) , x _ x i ( i ) , y i ( i ) printf ( ’%. 1 f ) 28 end 29 printf ( ’ \n\ n P o l y n o m i a l s \n ’ ) 30 printf ( ’−−−−−−−−−−−\n ’ ) 31 for i = 1 : 4 32 for j = 1 : 5 - i 33 printf ( ’ %f \n ’ , P ( x i , y i , x _ x i , j , i ) ) 34 yi(j)=P(xi,yi,x_xi,j,i) 35 end printf ( ’ \n\n\n ’ ) 36 37 end

Scilab code Exa 12.4 Newtons Divided Difference Interpolation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 2 . 4 / /Newton ’ s D i vi d ed D i f f e r e n c e I n t e r p o l a t i o n / / P ag e n o . 3 8 1 clc ; close ; clear ;

x=[0,1,2,3,4,5] y=[1,2,5,10,17,26]; y1=y; deff ( ’ y i=P( a , b , d , e ) ’ , ’ y i =( b ( d + 1 )−b (d ) ) /( a ( d+e )−a ( d ) ) ’ ) // f u n c t i o n f o r f i n d i n g p o ly n om i al s for i = 1 : 3 for j = 1 : 6 - i z(j,i)=P(x,y,j,i) y(j)=z(j,i) end end

172

16 z ( 6 , 1 ) = 0 ; 17 printf ( ’ x

y f ( x0 , x1 ) f ( x0 , x1 , x3 ) f ( x0 , x1 , x2 , x3 ) \n ’ ) 18 printf ( ’ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 19 20 21 22 23 24

for j = 1 : 6 printf ( ’ %i %i \ t%i \ t \ t% i \ t \ t%i \n ’ , x ( 1 , j ) ,y1(1,j),z(j,1),z(j,2),z(j,3)) end x1=2.6; f=y1(4)+(x1-x(4))*(z(4,1))+(x1-x(4))*(x1-x(5))*z (4,2) printf ( ’ \n\ n f ( 2 . 6 ) =% . 2 f ’ ,f )

Scilab code Exa 12.5 Interpolation Methods

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

/ / E x am pl e 1 2 . 5 / / I n t e r p o l a t i o n Metho ds / / P ag e n o . 4 0 3 clc ; close ; clear ; x=[0,1,2,3,4]; y=[0,1,8,27,64];

/ / I n v e r s e l a g r a n g e Method P=0; y1=20; for k = 0 : 4 p=x(k+1) for j = 0 : 4 if ( j ~ = k ) p=p*((y1-y(j+1))/(y(k+1)-y(j+1)))

173

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

end end P=P+p;

end disp (P , ’ I n v e r s e L ag ra ng e i n t e r p o l a t i o n x= ’ )

// Newton ’ s d i v i d e d i f f e r e n c e i n t e r p o l a t i o n x1=x; deff ( ’ xi=P( a , b , d , y ) ’ , ’ x i =( b ( d + 1 )−b ( d ) ) / ( a ( d+y )−a ( d ) ) ’ ) // f u n c t i o n f o r f i n d i n g p o ly n om i al s for i = 1 : 2 for j = 1 : 5 - i z(j,i)=P(y,x,j,i) x(j)=z(j,i) end end z(5,1)=0; printf ( ’ \n\n y \ t x f ( y0 , y1 ) f ( y0 , y1 , y3 ) \n ’) printf ( ’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’) for j = 1 : 5 printf ( ’ %i \ t%i \ t%i \ t \ t% i \ t \n ’ , y ( 1 , j ) , x 1 ( 1 , j),z(j,1),z(j,2)) end y1=20; f=x1(4)+(y1-y(4))*(z(4,1))+(y1-y(4))*(y1-y(5))*z (4,2) printf ( ’ \n\ nNewton D i v i d e D i f f e r e n c e x ( 2 0 ) =%. 2 f ’ ,f )

41 42 43 x = x 1 ; 44 // I t e r a t e d L i n e a r I n t e r p o l a t i o n 45 function [ x , y , z ] = t r a n ( a , b )


e x ch a n gi n g v a l u e s 46 z=a;y=b;x=z; 47 e n d f u n c t i o n 48 deff ( ’ y=P( a , b , c , d , e ) ’ , ’ y =( c ( d ) ∗b ( d + 1 )−c ( d+e ) ∗b ( d ) ) / (

174

a ( d+e )−a ( d ) ) ’ ) polynomials 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83

// f u n c t i o n f o r f i n d i n g

y1=20

[y(4),y(1),a]=tran(y(1),y(4)) [y(3),y(2),a]=tran(y(2),y(3)) [x(4),x(1),a]=tran(x(1),x(4)) [x(3),x(2),a]=tran(x(2),x(3)) for i = 1 : 5 y1_y(i)=y1-y(i); end printf ( ’ y \ ty1 −y \ t x \n ’ ) printf ( ’−−−−−−−−−−−−−−−−−−\n ’ ) for i = 1 : 5 printf ( ’%. 1 f \ t%i \ t% i \n ’ , y ( i ) , y 1 _ y ( i ) , x ( i ) ) end printf ( ’ \n\ n P o l y n o m i a l s \n ’ ) printf ( ’−−−−−−−−−−−\n ’ ) for i = 1 : 4 for j = 1 : 5 - i printf ( ’ %f \n ’ , P ( y , x , y 1 _ y , j , i ) ) x(j)=P(y,x,y1_y,j,i) end printf ( ’ \n\n ’ ) end printf ( ’ I t e r a t e d L i n e a r I n t e r p o l a t i o n x ( 2 0 ) = %f ’ ,x ( j))

x=[0,1,2,3,4]; y=[0,1,8,27,64]; y1=y;

// S u gg e st ed I n t e r p o l a t i o n for i = 1 : 4 for j = 1 : 5 - i z(j,i)=y(j+1)-y(j); y(j)=z(j,i) end

175

84 e n d 85 printf ( ’ \ n \ n \ n x \ t y \ t d y \ t d 2 y \ t d 3 y \ t d 4 y \ n ’ ) 86 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’ ) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n 87 f o r i = 1 : 5 printf ( ’ %i \ t % i \ t % i \ t % i \ t % i \ t % i \ n ’ , x ( i ) , y 1 ( i ) , z ( 88 i,1),z(i,2),z(i,3),z(i,4)) 89 e n d 90 s = poly ( 0 , ’ s ’ ) 91 p = y 1 ( 4 ) ; k = 3 ; 92 f o r i = 1 : 3 93 r=1; for j = 1 : i 94 95 r=r*(s+(j-1)) end 96 97 r=r*z(k,i)/factorial(j); k=k-1; 98 99 p=p+r; printf ( ’ \ n \ n St 100 S t a ge ge %i : ’ , i ) 101 disp ( p ) 102 e n d 103 s 0 = - 7 / 1 9 ; 104 disp ( s 0 , ’ s0= s0= ’ ) ; 105 s 1 = ( - 7 - s 0 * ( s 0 + 1 ) * 6 ) / 1 9 106 disp ( s 1 , ’ s1= s1= ’ ) 107 disp ( 3 + s 1 , ’ x 1= 1= ’ ) 108 s 2 = ( - 7 - s 1 * ( s 1 + 1 ) * 6 - s 1 * ( s 1 + 1 ) * ( s 1 + 2 ) ) / 1 9 109 disp ( s 2 , ’ s2= s2= ’ ) 110 x 2 = 3 + s 2 ; 111 disp ( x 2 , ’ S u gg g g e st s t ed e d I n t e r p o l a t i o n x ( 2 0 )= )= ’ ) ;

Scilab code Exa 12.6 Chebyshev Interpolating Polynomial

176

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

/ / E x am am pl pl e 1 2 . 6 / / Ch C h eb e b ys ys he h e v I n t e r p o l a t i n g P o ly l y n om o m i al al / / P ag ag e n o . 4 0 7 c l c ; close ; clear ; deff ( ’y=f ( x ) ’ , ’y =1/(1 =1/(1+ex +ex p(− p( − x ) ) ’ ) ; a=-2;b=2;n=3; D=%pi/(2*n+2) for k=0:n t(k+1)=- cos (D*(2*k+1)) x(k+1)=((a+b)/2)+(b-a)*t(k+1)/2 y(k+1)=f(x(k+1)) C(k+1)=0 end for j=0:n for k = 0 : n L=(2*k+1)*D C ( j + 1 ) = C ( j + 1 ) + y ( k + 1 ) *c *cos ( j * L ) end end C(1)=C(1)/(n+1); for j=1:n C(j+1)=2*C(j+1)/(n+1) end

x = poly ( 0 , ” x ” ) T(1)=1;T(2)=x; for j=1:n-1 T(j+2)=2*x*T(j+1)-T(j) end P=C(1)*T(1) for j=1:n P=P+C(j+1)*T(j+1) end P3 ( x ) = ’ ) disp ( P , ’ P3 printf ( ’ \ n \ n \ n N o t e : B o o k h a s C a l c u l a t i o n ca l c u la t i o n of c o e f f i c i e n t s ’ )

177

errors in

Scilab code Exa 12.7 Double Interpolation

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

/ / E x am am pl pl e 1 2 . 7 / / Do D o ub ub l e I n t e r p o l a t i o n / / P ag ag e n o . 4 0 9 c l c ; close ; clear ; x=[0,1,2,3,4]; y=[0,1,2,3,4]; z = [ 0 , 1 , 8 , 2 7 , 6 4 ; 1 , 3 , 1 1 , 3 1 , 6 9 ; 4 , 7 , 1 6 , 3 7 , 7 6 ; 9 , 1 3 , 2 3 , 4 5 ,8 5 ; 1 6 , 2 1 , 3 2 , 5 5 printf ( ’ y / x ’ ) for i=1:5 printf ( ’ \ t%i ’ , x ( i ) ) end for i=1:5 printf ( ’ \ n % i ’ , y ( i ) ) for j=1:5 printf ( ’ \ t%i ’ , z ( j , i ) ) end end printf ( ’ \ n \ n \ n ’ ) for i=1:5 x=2.5; x1=2;x2=3; z1(1,i)=z(i,x1+1)+(z(i,x2+1)-z(i,x1+1))*(x-x1)/( x2-x1) end printf ( ’ V al al u uee s o f z a t x = 2 . 55:: \ n \ n y ’ ) for i=1:5 printf ( ’ \ t%i ’ , y ( i ) )

178

28 29 30 31 32 33 34 35 36

end printf ( ’ \ n z ’ ) for i=1:5 printf ( ’ \ t%g ’ , z 1 ( i ) ) end y=1.5; y1=1;y2=2; z2=z1(y1+1)+(z1(y2+1)-z1(y1+1))*(y-y1)/(y2-y1) = 2 . 5 a n d y =1 = 1 . 5 : %g ’ , z 2 ) printf ( ’ \ n \ n V a l u e o f z a t x =2

Scilab code Exa 12.8 Spline Interpolation

1 2 3 4 5 6 7

/ / E x am am pl pl e 1 2 . 8 // S p l i ne I n t e r p o l a t i o n / / P ag ag e n o . 4 1 4 c l c ; close ; clear ; xi=[0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17]; yi = [ 0 . 1 1 1 0 , 0 . 1 2 3 4 , 0 . 1 3 6 1 , 0 . 1 4 9 1 , 0 . 1 6 2 3 , 0 . 1 7 5 9 , 0 . 1 8 9 7 ,0 . 2 0 3 8 ] ;

8 h=0.01; 9 10 p i ( 1 ) = 0 ; q i ( 1 ) = 0 ; p i ( 8 ) = 0 ; q i ( 8 ) = 0 ; 11 f o r i = 2 : 7 pi(i)=-1/(4+pi(i-1)) 12 13 qi(i)=((6/h^2)*(yi(i+1)-2*yi(i)+yi(i-1))-qi(i-1) )/(4+pi(i-1)) 14 e n d 15 s i 2 ( 8 ) = 0 ; 16 s i 2 ( 1 ) = 0 ; s i 1 ( 8 ) = 0 ; 17 s i 1 ( 1 ) = 0 ; 18 f o r i = 7 : - 1 : 2

179

si2(i)=pi(i)*si2(i+1)+qi(i) end for i=2:8 si1(i)=si1(i-1)+h*(si2(i)+si2(i-1))/2 end printf ( ’ \ n i \ t x i \ t f i \ i \ t pi \t\t q i \ t \ t si1 ’) 25 printf ( ’ \ n 19 20 21 22 23 24

s i 2 \ t \ t

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 26 f o r i = 1 : 8 27 t% g \ t%g t% g \ t % f \ f \ t % f \ f \ t % f \ f \ t%f ’ , i , x i ( i ) , printf ( ’ \ n %i \ t%g yi(i),pi(i),qi(i),si2(i),si1(i)) 28 e n d 29 x = 0 . 1 3 2 5 ; 30 i = 4 ; 31 s = y i ( i ) + ( x - x i ( i ) ) * s i 1 ( i ) + ( s i 2 ( i ) * ( x - x i ( i ) ) ^ 2 ) / 2 + ( ( si2(i+1)-si2(i))/(xi(i+1)-xi(i)))*((x-xi(i))^3)/6 32 printf ( ’ \ n \ n S pl pl i ne I n t e r p o l a te d Value o f s ( 0 . 1 3 2 5 ) i s : %f ’ , s )

180

Chapter 13 Numerical Differentiation

Scilab code Exa 13.1 Differentiation

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 3 . 1 // D i f f e r e n t i a t i o n / / P ag e n o . 4 2 0 clc ; close ; clear ;

deff ( ’y=f ( x ) ’ , ’y=xˆ2+5 ’ ) deff ( ’ y=f1 ( x , h ) ’ , ’ y=(f ( x+h )− f ( x ) ) /h ’ ) h=0.01;x=2.4 d=f1(x,h) d1=(f1(x+h,h)-f1(x))/h printf ( ’ dy \n −− = %g\n d x ’ ,d ) printf ( ’ \n\n\n d 2 y \n −−− = %g\n d x2 ’ , d 1 )

Scilab code Exa 13.2 Calculation of x coordinate of Minimum Point

181

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 3 . 2 // C a l c ul a t i o n o f x−c o o r d i n a t e o f Minimum P o i n t / / P ag e n o . 4 2 2 clc ; close ; clear ; for i = 1 : 7 for j = 1 : 6 z(i,j)=0 end end h=0.2 x printf ( ’

d3

y d4 \n ’ )

d

d2

13 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

for i = 1 : 7 z(i,1)=i/5; end z(1,2)=2.10022 z(2,2)=1.98730 z(3,2)=1.90940 z(4,2)=1.86672 z(5,2)=1.85937 z(6,2)=1.88755 z(7,2)=1.95147 for i = 3 : 6 for j = 1 : 9 - i z(j,i)=z(j+1,i-1)-z(j,i-1) end end disp ( z )

s = poly (0 , ’ s ’ ) p=z(5,2);k=4; for i = 3 : 5 r=1; for j = 1 : i - 2

182

36 37 38 39 40 41 42 43 44 45 46 47

r=r*(s+(j-1)) end r=r*z(k,i)/factorial(j); k=k-1; p=p+r;

end disp ( p ) s=(-z(4,3)+z(3,4)/2)/z(3,4) disp (s , ’ s= ’ ) x=z(5,1)+s*h disp (x , ’ x= ’ )

Scilab code Exa 13.3 Newton Forward Difference Formula

1 2 3 4 5

/ / E x am pl e 1 3 . 3 / / Newton ’ s F or wa rd D i f f e r e n c e F or mu la / / P ag e n o . 4 2 3

clc ; close ; clear ; printf ( ’ x\ t \ t d4 \n ’ ) \t 6 printf ( ’

y\ t \ t

d\ t \ t

d2 \ t \ t

d3 \ t

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 7 h=0.05; 8 z = [ 1 . 0 0 , 1 . 0 0 0 0 0 ; 1 . 0 5 , 1 . 0 2 4 7 0 ; 1 . 1 0 , 1 . 0 4 8 8 1 ; 1 . 1 5 , 1 . 0 7 23 8 ; 1 . 2 0 , 1 . 0 9 5 4 9 deff ( ’ y=f 1 (x , s ) ’ , ’ y=(z (x , 3 ) +(s −1/2) ∗ z (x ,4 )+z (x ,5 ) ∗ ( 3 ∗ s ˆ2 −6∗ s + 2) / 6 ) / h ’ ) 10 deff ( ’ y=f 2 (x , s ) ’ , ’ y=(z (x ,4 )+z (x ,5 ) ∗ ( s −1) ) / h ˆ 2 ’ ) 11 deff ( ’ y=f 3 (x , s ) ’ , ’ y=z ( x , 5 ) /hˆ 3 ’ ) 12 for i = 3 : 6

183

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

for j = 1 : 9 - i z(j,i)=z(j+1,i-1)-z(j,i-1) end end printf ( ’ \n ’ ) for i = 1 : 7 for j = 1 : 6 if z ( i , j ) = = 0 then printf ( ’ \ t ’ ) else printf ( ’%. 7 f \ t ’ , z ( i , j ) ) end end printf ( ’ \n ’ ) end s = poly (0 , ’ s ’ ) p=z(5,2);k=4; for i = 3 : 5 r=1; for j = 1 : i - 2 r=r*(s+(j-1)) end r=r*z(k,i)/factorial(j); k=k-1; p=p+r;

end disp (p , ’ y ( s ) = ’ ) printf ( ’ \n\ ny1 (1 ) = %g ’ , f 1 ( 1 , 0 ) ) printf ( ’ \n\ ny2 (1 ) = %g ’ , f 2 ( 1 , 0 ) ) printf ( ’ \n\ ny3 (1 ) = %g ’ , f 3 ( 1 , 0 ) ) printf ( ’ \n\ n y1 ( 1 . 0 2 5 ) = %g ’ , f 1 ( 1 , 0 . 5 ) )

184

Scilab code Exa 13.4 Newton Backward Difference Formula

1 2 3 4 5

/ / E x am pl e 1 3 . 4 / / Newto n ’ s B ac kw ar d D i f f e r e n c e F or m ul a / / P ag e n o . 4 2 5

clc ; close ; clear ; x\ t \ t printf ( ’ d4 \n ’ ) \t 6 printf ( ’

y\ t \ t

d\ t \ t

d2 \ t \ t

d3 \ t

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 7 h=0.02; 8 z = [ 0 . 9 6 , 1 . 8 0 2 5 ; 0 . 9 8 , 1 . 7 9 3 9 ; 1 . 0 0 , 1 . 7 8 5 1 ; 1 . 0 2 , 1 . 7 7 6 3 ; 1. 0 4 , 1 . 7 6 7 3 ] ; 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

deff ( ’ y=f 1 (x , s ) ’ , ’ y=(z (x ,3 ) +(s +1/2) ∗ z (x ,4 ) )/h ’ ) for i = 3 : 6 for j = 1 : 7 - i z(j,i)=z(j+1,i-1)-z(j,i-1) end end printf ( ’ \n ’ ) for i = 1 : 5 for j = 1 : 6 if z ( i , j ) = = 0 then printf ( ’ \ t ’ ) else printf ( ’%. 7 f \ t ’ , z ( i , j ) ) end end printf ( ’ \n ’ ) end printf ( ’ \n\ ny1 (1 ) = %g ’ , f 1 ( 2 , 0 ) ) printf ( ’ \n\ n y1 ( 1 . 0 3 ) = %g ’ , f 1 ( 4 , 0 . 5 ) )

185

Scilab code Exa 13.5 Stirlings Central Difference Derivatives

1 2 3 4 5

/ / E x am pl e 1 3 . 5 // S t i r l i n g s C en tr al D i f f e r e n c e D e r i v a t i v e s / / P ag e n o . 4 2 6

clc ; close ; clear ; x\ t \ t printf ( ’ ’) 6 printf ( ’

y\ t \ t

d\ t \ t

d2 \ t \ t

d3 \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 7 h=0.01;s=0.5; 8 deff ( ’ y=f 1 (x , s ) ’ , ’ y = ( ( z ( x , 3 ) +z ( x −1 ,3) )/2+s ∗ z ( x − 1 , 4 ) +(z ( x −1 ,5)+z (x −2 , 5 ) ) ∗ ( 3 ∗ s ˆ2 −1 ) / 1 2 ) / h ’ ) 9 deff ( ’ y=f 2 (x , s ) ’ , ’ y=(z (x −1 , 4 ) ) / h ˆ 2 ’ ) 10 deff ( ’ y=f 3 (x , s ) ’ , ’ y=(z (x −1 ,5)+z ( x −2 ,5) ) /(2 ∗ h ˆ 3 ) ’ ) 11 z = [ 1 . 0 0 , 1 . 0 0 0 0 0 ; 1 . 0 1 , 1 . 0 0 4 9 9 ; 1 . 0 2 , 1 . 0 0 9 9 5 ; 1 . 0 3 , 1 . 0 1 48 9 ; 1 . 0 4 , 1 . 0 1 9 8 12 13 14 15 16 17 18 19 20 21 22 23 24

for i = 3 : 5 for j = 1 : 1 9 - i z(j,i)=z(j+1,i-1)-z(j,i-1) end end printf ( ’ \n ’ ) for i = 1 : 1 7 for j = 1 : 5 if z ( i , j ) = = 0 then printf ( ’ \ t ’ ) else printf ( ’%. 7 f \ t ’ , z ( i , j ) ) end

186

25 end printf ( ’ \n ’ ) 26 27 end 28 printf ( ’ \n\ ny1 ( 1 . 1 2 5 ) = %g 0 . 4 7 7 1 4 0 4 ) ’ ,f1(13,0.5)) 29 printf ( ’ \n\ ny2 ( 1 . 1 2 5 ) = %g −0.20951) ’ ,f2(13,0.5)) 30 printf ( ’ \n\ ny3 ( 1 . 1 2 5 ) = %g ’ ,f3(13,0.5))

( e xa c t v a l u e = ( e xa c t v a l u e = ( e xa c t v a l u e = 0 . 2 7 9 3 5 )

Scilab code Exa 13.6 Extrapolation

1 2 3 4 5 6

/ / E x am pl e 1 3 . 6 // Extrap olati on / / P ag e n o . 4 3 0 clc ; close ; clear ; x=[-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8]; y = [ 0 . 2 0 1 9 , 0 . 3 0 1 1 9 , 0 . 4 4 9 3 3 , 0 . 6 7 0 3 2 , 1 , 1 . 4 9 1 8 2 , 2 . 2 2 5 5 4 ,3 . 3 2 0 1 2 , 4 . 9 5 3 0

7 for i = 1 : 4 8 printf ( ’ \ nh = %g\n ’ , x ( 1 0 - i ) ) 9 y1=(y(10-i)-y(i))/(2*x(10-i)) 10 printf ( ’ f 1 ( 0 ) = %g\n\n ’ , y 1 ) 11 end

Scilab code Exa 13.7 Richardson Extrapolation

1 / / E x am pl e 1 3 . 7

187

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

/ / R i c h ar d s o n E x t r a p o l a t i o n / / P ag e n o . 4 3 1 clc ; close ; clear ; deff ( ’y=f ( x ) ’ , ’ y=exp (2 ∗ x ) ’ ) e=10^-4;h=0.8; D1=0; for i = 1 : 4 printf ( ’ \n ’ ) for j = 1 : i if j = = 1 then D(i,j)=(f(h)-f(-h))/(2*h) else D(i,j)=D(i,j-1)+(D(i,j-1)-D(i-1,j-1)) /(2^(2*(j-1))-1) end printf ( ’ %g\ t \ t ’ , D ( i , j ) ) end h=h/2 end printf ( ’ \n\n\ t \ t \ t \ t \ t \ t 2 x \ nHence , t h e d e r i v a t i v e

o f the f un ct io n y = f ( x ) = e %g ’ , D ( i , j ) )

Scilab code Exa 13.8 Application

1 2 3 4 5 6 7

/ / E x am pl e 1 3 . 8 // Appli cation / / P ag e n o . 4 3 3 clc ; close ; clear ; deff ( ’y=f ( x ) ’ , ’ y=2/xˆ2 ’ ) a=1;b=2;a1=1;b1=0;

188

a t x=0 i s D (3 , 3 ) =

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

N=4; h=(b-a)/(N+1); for j = 1 : N s(j)=f(a+j*h) end for i = 1 : N for j = 1 : N if a bs ( i - j ) = = 1 then A(i,j)=-1 end if i = = j then A(i,j)=2+s(i)*h^2 end end if i = = 1 then k(i,1)=s(i)+a1/h^2 elseif i = = N k(i,1)=s(i)+b1/h^2 else k(i,1)=s(i) end end disp (A , ’A = ’ ) disp (k , ’ k = ’ )

189

Chapter 14 Numerical Integration

Scilab code Exa 14.2 Simpsons 1 3rd Rule

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17

/ / E x am pl e 1 4 . 2 / / S i m ps o ns 1 /3 r d R ul e / / Pa ge no 4 42 clc ; clear ; close ; x(1,1)=0 for i = 2 : 9 x(1,i)=x(1,1)+(i-1)*10 end y = [ 3 0 , 3 1 . 6 3 , 3 3 . 4 4 , 3 5 . 4 7 , 3 7 . 7 5 , 4 0 . 3 3 , 4 3 . 2 9 , 4 6 . 6 9 , 5 0 . 67 ]

// t r a p e z o i d a l r u l e S=0; h=(x(9)-x(1))/8 for j = 1 : 9 S=0; for i = 1 : j

190

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

end

if ( i == 1 | i == j ) S=S+y(i) else S=S+2*y(i) end

end S=S*h/2 printf ( ’ \n V e l o c i t y a t t ( %i ) = %. 2 f ’ , x ( j ) , S ) y1(j)=S

y1(1)=0;

/ / S i m ps o ns 1 /3 r d R ul e S=0; h=(x(9)-x(1))/8 for i = 1 : 9 if ( i == 1 | i = =9 ) S=S+y1(i) elseif ( ( ( i ) / 2 ) - fix ( ( i ) / 2 ) = = 0 ) S=S+4*y1(i) else S=S+2*y1(i) end end S=S*h/3; S=S/1000 printf ( ’ \n\ n S i m p so n s 1 / 3 r d R u l e Sum = %g km ’ ,S )

Scilab code Exa 14.3 Trapezoidal Rule and Simpsons Rule

1 / / E x am pl e 1 4 . 3 2 / / T r a p e z o i d a l R ul e and S i mp s on s R ul e 3 / / P ag e n o . 4 4 2

191

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

clc ; close ; clear ; n=2;a=0;b=1; h=(b-a)/n deff ( ’y=f ( x ) ’ , ’y =1/(1+x ) ’ ) for i = 0 : 2 x(i+1)=i/2; y(i+1)=f(x(i+1)) end printf ( ’ x i \ t ’ ) for i = 1 : 3 printf ( ’ %g\ t ’ , x ( i ) ) end printf ( ’ \n y i \ t ’ ) for i = 1 : 3 printf ( ’ 1/%g\ t ’ ,1+(i-1) /2) end

// t r a p e z o i d a l r u l e S=0; for i = 1 : 3 if ( i == 1 | i = =3 ) S=S+y(i) else S=S+2*y(i) end end S=S*h/2 printf ( ’ \n\ n T r a p e z o i d a l R u le Sum = %g ’ ,S )

/ / S i m ps o ns 1 /3 r d R ul e S=0; for i = 1 : 3 if ( i == 1 | i = =3 ) S=S+y(i) elseif ( ( ( i ) / 2 ) - fix ( ( i ) / 2 ) = = 0 ) S=S+4*y(i) else S=S+2*y(i)

192

42 end 43 end 44 S = S * h / 3 45 printf ( ’ \n\ n S i m p so n s 1 / 3 r d R u l e Sum = %g ’ ,S )

Scilab code Exa 14.5 Romberg Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

/ / E x am pl e 1 4 . 5 / /Romberg Method / / P ag e n o . 4 5 7 clc ; close ; clear ; deff ( ’y=f ( x ) ’ , ’y =1/(1+x ) ’ ) h=[0.5,0.25,0.125] for k = 1 : 3 for i = 0 : h ( k ) : 1 x(i/h(k)+1)=i; y(i/h(k)+1)=f(x(i/h(k)+1)) end n=1+(1/h(k))

// t r a p e z o i d a l r u l e

S=0; for i = 1 : n if ( i == 1 | i == n ) S=S+y(i) else S=S+2*y(i) end end S=S*h(k)/2 printf ( ’ \n\ nI (%g) = %g ’ , h ( k ) , S ) z(2*k-1,1)=S

193

27 end 28 for i = 2 : 3 29 for k = 1 : 4 - i z(k*2+i-2,i)=z(2*k-1+i,i-1)+(z(2*k-1+i,i-1)-z(2* 30 k-3+i,i-1))/3 31 end 32 end 33 34 printf ( ’ \n\n ’ ) 35 disp (z , ’ The T ab le o f v a l u e s : ’ )

Scilab code Exa 14.7 Gaussian Quadrature Formula

1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 4 . 7 / / G a u s s i a n Q u a d ra t u r e F or mu la / / P ag e n o . 4 6 3 clc ; close ; clear ;

deff ( ’y=f ( x ) ’ , ’ y= c o s ( x ) ∗ l o g ( x ) ’ ) s=0; for i = 0 : 2 : 2 0 0 0 s = s + i n t e g r a t e( ’ ( ( −1) ˆ( i /2 ) ) ∗ ( x ˆ i ) / f a c t o r i a l ( i ) ∗ l o g ( x ) ’ , ’ x ’ ,0,1) 10 end 11 disp (s , ’ T i l l 1 0 0 0 t e rm s . . . . I = ’ )

Scilab code Exa 14.8 Gauss Legendre Two Point Rule

1 / / E x am pl e 1 4 . 8

194

2 3 4 5 6 7 8 9 10

/ / G a us s L e g e n d re Two P o i n t R u le / / P ag e n o . 4 7 2 clc ; close ; clear ; deff ( ’y=f ( x ) ’ , ’ y=1/(x+3) ’ ) s = i n t e g r a t e( ’ f ( x ) ’ , ’ x ’ ,-1,1) printf ( ’ By D i r e c t Method , I = %g ’ ,s ) s = f ( - 1 / sqrt ( 3 ) ) + f ( 1 / sqrt ( 3 ) ) printf ( ’ \n\n By G au ss−L eg en dr e 2 p o i n t r u l e , I = %g ’ ,s )

Scilab code Exa 14.9 Gauss Legendre Three Point Rule

1 2 3 4 5 6 7 8 9 10

/ / E x am pl e 1 4 . 9 / / G au ss L e g en d r e T hr ee P o i n t R ul e / / P ag e n o . 4 7 3 clc ; close ; clear ; deff ( ’y=f ( x ) ’ , ’ y=1/(x+3) ’ ) s = i n t e g r a t e( ’ f ( x ) ’ , ’ x ’ ,-1,1) printf ( ’ By D i r e c t Method , I = %g ’ ,s ) s = 5 / 9 * f ( - sqrt ( 3 / 5 ) ) + 8 / 9 * f ( 0 ) + 5 / 9 * f ( sqrt ( 3 / 5 ) ) printf ( ’ \n\n By G au ss−L eg en dr e 3 p o i n t r u l e , I = %g ’ ,s )

Scilab code Exa 14.10 Spline Integration Method

1 / / E x am pl e 1 4 . 1 0 2 / / S p l i n e I n t e g r a t i o n Method

195

3 4 5 6 7 8 9 10

/ / P ag e n o . 4 7 8 clc ; close ; clear ;

deff ( ’y=f ( x ) ’ , ’ y= s i n d ( % p i∗x ) ’ ) deff ( ’ y=f1 ( x , h ) ’ , ’ y=(f ( x+h )− f ( x ) ) /h ’ ) h=0.01; n=2;h=0.5;a=0;b=1; disp ( i n t e g r a t e( ’ f ( x ) ’ , ’ x ’ ,0,1), ’ I = ’ )

Scilab code Exa 14.11 Trapezoidal Rule

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

/ / E x am pl e 1 4 . 1 / / T r a p e z o i d a l R ul e / / Pa ge no 4 40 clc ; clear ; close ; x1=1.46 for i = 1 : 6 x(1,i)=x1+i/100 end y=[3.86,3.90,3.96,4.02,4.06,4.12]

// t r a p e z o i d a l r u l e S=0; h=(x(6)-x1)/6 for i = 1 : 6 if ( i == 1 | i = =6 ) S=S+y(i) else S=S+2*y(i) end end S=S*h/2 printf ( ’ \n I = %g ’ ,S )

196

Scilab code Exa 14.14 Trapezoidal and Simpsons Rule

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

/ / E x am pl e 1 4 . 1 4 / / T r a p e z o i d a l and S i mp s on s R ul e / / P ag e n o . 4 8 6 clc ; close ; clear ; x(1)=0.5;y(1)=0.5;h=0.25 for i = 2 : 3 x(i)=x(i-1)+h y(i)=y(i-1)+h end printf ( ’ y /x \ t | \ t%g\t%g\t%g ’ , x ( 1 ) , x ( 2 ) , x ( 3 ) ) printf ( ’ \n−−−−−−−−|−−−−−−−−−−−−−−−−−−−−−−−−−−−’ ) for i = 1 : 3 printf ( ’ \n%g\ t | \ t ’ , y ( i ) ) for j = 1 : 3 z(i,j)=x(j)*y(i) printf ( ’ %g\ t ’ , z ( i , j ) ) end end

// t r a p e z o i d a l r u l e s=0; for i = 1 : 3 for j = 1 : 3 if i ==1 & j ==1 then s=s+z(i,j) elseif i = =3 & j ==3 s=s+z(i,j) else s=s+2*z(i,j)

197

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

end end

end s=(s*(h^2))/4 printf ( ’ \n\n ’ ) disp (s , ’ T r a p e z o i d a l R ul e Sum = ’ ) printf ( ’ \n\n ’ )

/ / s i m p s on s r u l e s=0; for i = 1 : 3 for j = 1 : 3 if i / 2 - int ( i /2 ) == 0 & j /2 -int ( j / 2 ) = = 0 then s=s+16*z(i,j) elseif i / 2 - int ( i /2 ) ~= 0 & j /2 -int ( j / 2 ) ~ = 0 s=s+z(i,j) else s=s+4*z(i,j) end end end s=(s*(h^2))/9 disp (s , ’ S i m p s o n s R u l e Sum = ’ )

Scilab code Exa 14.15 Trapezoidal and Simpsons Rule

1 2 3 4 5 6 7 8

/ / E x am pl e 1 4 . 1 5 / / T r a p e z o i d a l and S i mp s on s R ul e / / P ag e n o . 4 8 7 clc ; close ; clear ; x(1)=0;y(1)=0;h=0.25 for i = 2 : 5 x(i)=x(i-1)+h

198

9 y(i)=y(i-1)+h 10 e n d 11 printf ( ’ y /x / x \ t | \ t%g t% g \ t%g t% g \ t%g t% g \ t%g t% g \ t%g ’ , x ( 1 ) , x ( 2 ) , x ( 3 ) , x(4),x(5)) 12 printf ( ’ \ n

−−−−−−−−|−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

for i=1:5 n%g \ t | \ t ’ , y ( i ) ) printf ( ’ \ n%g\ for j=1:5 z(i,j)=x(j)*y(i) %g\ t ’ , z ( i , j ) ) printf ( ’ %g\ end end

// t r a p e z o i d a l r u l e s=0; for i=1:5 for j=1:5 i f i == = = 1 & j == = = 1 then s=s+z(i,j) elseif i = =5 =5 & j == ==5 s=s+z(i,j) else s=s+2*z(i,j) end end end s=(s*(h^2))/4 printf ( ’ \ n \ n ’ ) u l e Sum = ’ ) disp ( s , ’ T r a p e z o i d a l R ul printf ( ’ \ n \ n ’ )

/ / s i m p s on on s r u l e s=0; for i=1:5 for j=1:5 i f i / 2 - i n t ( i /2 / 2 ) == = = 0 & j /2 / 2 -i - i n t ( j / 2 ) = = 0 then

199

44 i f i = = j then s=s+16*z(i,j) 45 46 else s=s+4*z(i,j) 47 48 end 49 50 elseif i / 2 - i n t ( i /2 / 2 ) ~= ~ = 0 & j /2 / 2 -i -int (j/2)~=0 s=s+z(i,j) 51 52 else 53 s=s+4*z(i,j) 54 end 55 end 56 e n d 57 s = ( s * ( h ^ 2 ) ) / 9 58 disp ( s , ’ S i m p s o n s R u l e Sum = ’ )

Scilab code Exa 14.16 Multiple Integration with Variable Limits

1 2 3 4 5 6 7 8 9 10 11

/ / E x am am pl pl e 1 4 . 1 6 / / M u l ti t i p l e I n t e g r a t i o n w it i t h V a r i ab a b l e L i m it it s / / P ag ag e n o . 4 9 1 c l c ; close ; clear ;

deff ( ’ z=f (x ) ’ , ’ z=x+1 ’ ) y+3)ˆ2 ’ ) deff ( ’ z=f1 ( y ) ’ , ’ z=(y+1)ˆ 3 ∗ ( y+3)ˆ2 s = 5 / 9 * f ( - sqrt ( 3 / 5 ) ) + 8 / 9 * f ( 0 ) + 5 / 9 * f ( sqrt ( 3 / 5 ) ) s = s * 5 / 9 * f 1 ( - sqrt ( 3 / 5 ) ) + 8 / 9 * f 1 ( 0 ) + 5 / 9 * f 1 ( sqrt ( 3 / 5 ) ) s=s/256; disp ( s , ’ I = ’ )

200


1 2 3 4 5 6 7

/ / E x am am pl pl e 1 4 . 1 8 // Int egra tio n / / P ag ag e n o . 4 9 4 c l c ; close ; clear ; s = i n t e g r a t e( e ( ’ x ˆ 2∗ 2 ∗ s i n ( x ˆ 2 ) ’ , ’ x ’ ,0,1) disp ( s , ’ I = ’ )


1 2 3 4 5 6 7

/ / E x am am pl pl e 1 4 . 1 9 // Int egra tio n / / P ag ag e n o . 4 9 4 c l c ; close ; clear ; s = i n t e g r a t e( e ( ’ si n ( t ) / t ’ , ’ t ’ ,1,999) disp ( s , ’ I = ’ )

201

Chapter 15 Numerical Solutions of Ordinary Differential Equations Initial Value Problem

Scilab code Exa 15.1 Ordinary Differential Equation

1 2 3 4 5 6

/ / E x am am pl pl e 1 5 . 1 // / / O r di d i na n a ry r y D i f f e r e n t i a l E qu q u at a t io io n / / P ag ag e n o . 5 0 3 c l c ; clear ; close ; s = l o g ( 2 ) / l o g (1.02) disp ( s , ’ T im im e T ak ak en en = ’ )

Scilab code Exa 15.6 Taylor Method

1 / / E x am am pl pl e 1 5 . 6

202

2 3 4 5 6 7 8 9 10 11 12 13

// Tay lor Me Meth thod od / / P ag ag e n o . 5 1 0 c l c ; clear ; close ;

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

deff ( ’ y=f1 ( x , y ) ’ , ’ y=xˆ2+yˆ 2 ’ ) deff ( ’ y=f2 ( x , y ) ’ , ’ y=2 y=2∗ x +2∗ +2 ∗ y ∗ f1 ( x , y ) ’ ) y=2+2∗∗ f1 ( x , y ) ˆ2+2 ˆ2+2 ∗ y ∗ f2 ( x , y ) ’ ) deff ( ’ y=f3 ( x , y ) ’ , ’ y=2+2 deff ( ’ y=f4 ( x , y ) ’ , ’ y=6 y=6∗ f1 ( x , y ) ∗ f2 ( x , y ) +2∗ +2∗ y ∗ f3 ( x , y ) ’ ) h=0.2; for l=1:2 a=0;y=0;x=0; n h = % g\ g\ n printf ( ’ \ n−−−−−−−−−−−−−−−\ −−−−−−−−−−−−−−−\n n ’ ,h) −−−−−−−−−−−−−−−\n −−−−−−−−−−−−−−−\ for i=1:4 x=a+(i-1)*h k=0; for j=1:4 i f j = = 1 then k=k+(h^j)*f1(x,y)/factorial(j) elseif j = = 2 k=k+(h^j)*f2(x,y)/factorial(j) elseif j = = 3 k=k+(h^j)*f3(x,y)/factorial(j) else k=k+(h^j)*f4(x,y)/factorial(j) end end y=y+k; g \ n \ ny(%g) ny(%g) = %g %g \ n \ n ’ , x , x + 0 . 2 , y ) printf ( ’ \ n x = % g\ end h=h+0.2; end

203

Scilab code Exa 15.7 Picard Method

1 2 3 4 5 6 7 8 9 10

/ / E x am pl e 1 5 . 7 / / P i c a r d M et ho d / / P ag e n o . 5 1 1 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=xˆ2+y ˆ2 ’ ) y(1)=0; for i = 1 : 2 y ( i + 1 ) = y ( 1 ) + i n t e g r a t e( ’ f ( x , y ( i ) ) ’ , ’ x ’ ,0,i /10) printf ( ’ \n y ( %g ) = %g\n ’ , i / 1 0 , y ( i + 1 ) ) end

Scilab code Exa 15.8 Euler Method

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 5 . 8 / / E u l e r M et ho d / / P ag e n o . 5 1 3 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x+y ’ ) y(1)=1; h=0.1; for i = 1 : 6 printf ( ’ \ ny(%g) = %g\n ’ , (i - 1 ) / 1 0 , y ( i ) ) y(i+1)=y(i)+h*f((i-1)/10,y(i)) end

Scilab code Exa 15.9 Trapezium Method

204

1 2 3 4 5 6 7 8 9 10 11 12 13 14

/ / E x am pl e 1 5 . 9 / / T r a p e z i u m M et ho d / / P ag e n o . 5 1 6 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x∗yˆ2 ’ ) y=1; h=0.2; y2 = poly (0 , ’ y 2 ’ ) for i = 1 : 2 x=(i-1)*h; x1=x+h y1 = roots ( - y 2 + y + h * ( f ( x , y ) + f ( x 1 , y 2 ) ) / 2 ) printf ( ’ \n Y( %i ) = %g o r %g\n ’ , i , y 1 ( 1 ) , y 1 ( 2 ) ) end

Scilab code Exa 15.10 Heun Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14

/ / E x am pl e 1 5 . 1 0 / /Heun Method / / P ag e n o . 5 1 7 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=y∗2/x ’ ) y=2; h=0.25; for i = 1 : 4 x=1+(i-1)*h x1=x+h ye=y+h*f(x,y) y=y+h*(f(x,y)+f(x1,ye))/2 printf ( ’ \n y ( %g ) = %g\n ’ , x 1 , y ) end

205

Scilab code Exa 15.11 Midpoint Method

1 2 3 4 5 6 7 8

/ / E x am pl e 1 5 . 1 1 / / M i d p o i n t M et ho d / / P ag e n o . 5 1 8

clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=y+x ’ ) y=1; h=0.2; printf ( ’ i \ t x i \ t y i \ t s l o p e 1 \ t s l o p e 2 \ ty ( i +1) \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’) 9 for i = 1 : 3 10 x=(i-1)*h s1=f(x,y); 11 12 s2=f(x+h/2,y+s1*h/2); printf ( ’ %i \t%g\t%g\t%g\t%g ’ , i - 1 , x , y , s 1 , s 2 ) 13 14 y=y+s2*h; printf ( ’ \t%g\n ’ ,y ) 15 16 end

Scilab code Exa 15.12 Modified Midpoint Method

1 2 3 4 5

/ / E x am pl e 1 5 . 1 2 / / M o d i f i e d M i d p o i n t Met ho d / / P ag e n o . 5 1 9 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=y+x ’ )

206

y=1; h=0.2; Z(1)=y; Z(2)=Z(1)+h*f(0,Z(1)) printf ( ’ Z ( % i ) = %g ’ , 1 , Z ( 2 ) ) for i = 2 : 5 x=(i-1)*h; Y(i-1)=(Z(i)+Z(i-1)+h*f(x,Z(i)))/2 Z(i+1)=Z(i-1)+2*h*f(x,Z(i)) printf ( ’ \n Y ( % i ) = % g\n\n\n Z ( % i ) = %g ’ , i - 1 , Y ( i -1),i,Z(i+1)) 16 end 17 printf ( ’ \n\n\n y 4 = %g ’ , ( 4 * Y ( 4 ) - Y ( 2 ) ) / 3 ) 6 7 8 9 10 11 12 13 14 15

Scilab code Exa 15.13 Single Step Method

1 2 3 4 5 6 7 8 9

/ / E x am pl e 1 5 . 1 3 / / S i n g l e S t e p Method / / P ag e n o . 5 2 1 clc ; clear ; close ;

deff ( ’y=f ( x ) ’ , ’ y=xˆ 2 ’ ) deff ( ’ y= f 1 ( x ) ’ , ’ y=1/(1−x ) ’ ) y=1;h=0.2; printf ( ’ n\ tXn \tYn ( by s i n g l e −s t e p m et ho d ) \ tYn (

computed) \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 for i = 1 : 6 11 x=(i-1)*h if i <6 then 12 13 n ’ ,i-1,x printf ( ’ %i \t% . 2 f \t% . 5 f \ t \ t \ t \t% . 5 f \ ,y,f1(x))

207

14 else printf ( ’ %i \t% . 2 f \t% . 5 f \ t \ t \ t \ t \n ’ , i - 1 , x , y ) 15 16 end y=y+h*f(y); 17 18 end

Scilab code Exa 15.14 Second Order Runge Kutta Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

/ / E x am pl e 1 5 . 1 4 / / S e c o n d O r d er Ru ng e K u tt a M et ho d / / P ag e n o . 5 2 5 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y=1;x=1;h=0.1;

/ / s i m p l e r u ng e k u t t a method

K1=h*f(x,y); K2=h*f(x+h,y+K1); y1=y+(K1+K2)/2 printf ( ’ \ ny ( 1 . 1 ) by s i m p l e r un g e k u tt a method = %g\n \n ’ , y 1 )

/ / e u l e r c a uc h y m et ho d

K1=h*f(x,y); K2=h*f(x+h/2,y+K1/2); y1=y+(K1+K2) printf ( ’ y ( 1 . 1 ) by e u l e r c au ch y method = %g\n\n ’ , y 1 )

/ / o p t i m a l m et ho d

K1=h*f(x,y); K2=h*f(x+2*h/3,y+2*K1/3); y1=y+(K1+3*K2)/4 printf ( ’ y ( 1 . 1 ) by o p t i m a l method = %g ’ , y 1 )

208

Scilab code Exa 15.15 Third Order Runge Kutta Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

/ / E x am pl e 1 5 . 1 5 / / T h i r d O r d er R un ge K u tt a Me th od / / P ag e n o . 5 2 6 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y=1;x=1;h=0.1;

/ / s c h em e 1

K1=h*f(x,y); K2=h*f(x+h/2,y+K1/2); K3=h*f(x+h/2,y-K1+2*K2); y1=y+(K1+4*K2+K3)/6 printf ( ’ \ ny ( 1 . 1 ) by s ch em e 1 = %g\n\n ’ , y 1 )

/ / s c h em e 2

K1=h*f(x,y); K2=h*f(x+h/3,y+K1/3); K3=h*f(x+2*h/3,y+2*K2/3); y1=y+(K1+3*K3)/4 ’ ,y1) printf ( ’ \ ny ( 1 . 1 ) by s ch em e 2 = %. 7 f \n\n

Scilab code Exa 15.16 Fourth Order Runge Kutta Method

1 / / E x am pl e 1 5 . 1 6 2 / / F o u r t h O r d er Ru ng e K u tt a Met ho d 3 / / P ag e n o . 5 2 8

209

4 5 6 7 8 9 10 11 12 13

clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y=1;x=1;h=0.1; K1=h*f(x,y); K2=h*f(x+h/2,y+K1/2); K3=h*f(x+h/2,y+K2/2); K4=h*f(x+h,y+K3); disp ( K 4 , ’ K4 = ’ , K 3 , ’ K3 = ’ , K 2 , ’ K2 = ’ , K 1 , ’ K1 = ’ ) y1=y+(K1+2*K2+2*K3+K4)/6 printf ( ’ \ ny ( 1 . 1 ) = %. 8 f \n\n ’ ,y1)

Scilab code Exa 15.17 New Variant of Runge Kutta Method

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 5 . 1 7 / / New V a r i a n t o f R un ge K u tt a M et ho d / / P ag e n o . 5 3 0

13 14

clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y=1;x=1;h=0.1; K1=h*f(x,y); K2=h*f(x+h/2,y+K1/2); K3=h*f(x+h/2,y+K2/2); K4=h*f(x+h,y+K3); K5=h*f(x+3*h/4,y+(5*K1+7*K2+13*K3-K4)/32) disp ( K 5 , ’ K5 = ’ , K 4 , ’ K4 = ’ , K 3 , ’ K3 = ’ , K 2 , ’ K2 = ’ , K 1 , ’ K1 =’) y1=y+(K1+2*K2+2*K3+K5)/6 printf ( ’ \ ny ( 1 . 1 ) = %. 8 f \n\n ’ ,y1)

210

Scilab code Exa 15.18 Runge Kutta Merson Method

1 2 3 4 5 6 7

/ / E x am pl e 1 5 . 1 8 / / R un ge K u t t a M e rs o n M et ho d / / P ag e n o . 5 3 2 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x+y ’ ) y=1;x=0;h=0.1; printf ( ’ n\ t Xn\ t Yn\ t K1\ t K2\ t K3\ t K4\ t K5\tY(n+1)

\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 8 9 10 11 12 13 14 15

16 17 18

for i = 0 : 1 4 K1=h*f(x,y); K2=h*f(x+h/3,y+K1/3); K3=h*f(x+h/3,y+(K1+K2)/6); K4=h*f(x+h/2,y+(K1+3*K3)/8); K5=h*f(x+h,y+(K1-3*K3+4*K4)/2) y1=y+(K1+4*K4+K5)/6 . 3 f \t% printf ( ’ \n %i \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f ’ , i , x , y , K 1 , K 2 , K 3 , K 4 , K 5 , y 1 ) y=y1; x=x+h; end

Scilab code Exa 15.19 Runge Kutta Fehlberg Method

1 2 3 4 5 6

/ / E x am pl e 1 5 . 1 9 / / R un ge K u tt a F e h l b e r g Me th od / / P ag e n o . 5 3 5 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y=1;x=1;h=0.1;

211

7 8 9 10

11 12 13 14 15 16 17

K1=h*f(x,y); K2=h*f(x+h/4,y+K1/4); K3=h*f(x+3*h/8,y+3*(K1+3*K2)/32); K 4 = h * f ( x + 1 2 * h / 1 3 , y + 1 9 3 2 * K 1 / 2 1 9 7 - 7 2 0 0 * K 2 / 2 1 9 7 + 7 2 9 6 *K 3 /2197); K5=h*f(x+h,y+439*K1/216-8*K2+3680*K3/513-845*K4 /4104) K6=h*f(x+h/2,y-8*K1/27+2*K2-3544*K3/2565+1859*K4 /4104-11*K5/40) disp ( K 6 , ’ K6 = ’ , K 5 , ’ K5 = ’ , K 4 , ’ K4 = ’ , K 3 , ’ K3 = ’ , K 2 , ’ K2 = ’ , K 1 , ’ K1 = ’ ) y1=y+(25*K1/216+1408*K3/2565+2197*K4/4104-K5/5) y11=y+(16*K1/135+6656*K3/12825+28561*K4/56430-9*K5 /50+2*K6/55) printf ( ’ \ ny ( 1 . 1 ) = %. 9 f \n\n ’ ,y1) ’ ,y11) printf ( ’ \ ny ˜ ( 1 . 1 ) = % . 9 f \n\n

Scilab code Exa 15.20 Carp Karp Runge Kutta Method

1 2 3 4 5 6 7 8

/ / E x am pl e 1 5 . 2 0 / / C a rp K ar p R un ge K u t ta M et ho d / / P ag e n o . 5 3 7 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y = 1 ; x = 1 ; h = 0 . 1 ; printf ( ’ \n ’ ) U=[0,1/5,3/10,3/5,1,7/8]; v =[0,0,0,0,0;1/5,0,0,0,0;3/40,9/40,0,0,0;3/10,-9/10,6/5,0,0;-11/5

9 a=[37/378,0,250/621,125/594,0,512/1771]; 10 a1 = [ 2 8 2 5 / 2 7 6 4 8 , 0 , 1 8 5 7 5 / 4 8 3 8 4 , 1 3 5 2 5 / 5 5 2 9 6 , 2 7 7 / 1 4 3 3 6 , 1 /4 ] ;

212

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

for l = 1 : 5 K(1)=h*f(x,y); for i = 2 : 6 k=0; for j = 1 : i - 1 k=k+v(i,j)*K(j) end K(i)=h*f(x+U(i)*h,y+k) end k=0; for i = 1 : 6 k=k+a(i)*K(i) end y1=y+k; k=0; for i = 1 : 6 k=k+a1(i)*K(i) end y11=y+k; for i = 1 : 6 printf ( ’K%i = %.9 f \n ’ , i , K ( i ) ) end printf ( ’ \ n y ( 1 . 1 ) = Y%i = %. 9 f \n ’ ,l,y1) printf ( ’ y ˜ ( 1 . 1 ) = Y%i ˜ = %. 9 f \n ’ ,l,y11) y=y1; printf ( ’ \n\n\n ’ ) end

Scilab code Exa 15.21 Implicit Runge Kutta Method

1 / / E x am pl e 1 5 . 2 1 2 / / I m p l i c i t Runge K ut ta Meth od 3 / / P ag e n o . 5 3 9

213

4 5 6 7 8 9 10 11 12 13

clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x−y ’ ) y = 1 ; x = 1 ; h = 0 . 1 ; printf ( ’ \n ’ ) U=[0,1/5]; v=[0,0;1/2,1/2]; a2=1; K(1)=h*f(x,y); K(2)=(x+h/2-y-K(1)/2)/(1/h-1/2) y1=y+(K(1)+a2*K(2)) printf ( ’ \ ny ( 1 . 1 ) = %. 9 f \n\n ’ ,y1)

Scilab code Exa 15.22 Linear Multi Step Method

1 2 3 4 5 6 7

/ / E x am pl e 1 5 . 2 2 / / L i n e a r M u l t i S t e p Metho d / / P ag e n o . 5 4 0 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x+y ’ ) y(1)=1;y(2)=1;x(1)=0;h=0.1; printf ( ’ n\ tXn \ t \tYn \ t \ t f n \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n 0\ t%g\ t \t% . 3 f \ t \t% . 3 f \n ’ , x ( 1 ) , y ( 1 ) , f ( x ( 1 ) , y ( 1 ) ) ) ; 8 for i = 2 : 1 1 9 x(i)=(i-1)*h; y(i+1)=(-y(i)-y(i-1)+h*(f(x(i),y(i))+f(x(i-1),y( 10 i-1))))/2; 11 printf ( ’ %i \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \n ’ , i - 1 , x ( i ) , y ( i),f(x(i),y(i))) 12 end

214

Scilab code Exa 15.23 Milne Simpson Predictor Corrector Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

/ / E x am pl e 1 5 . 2 3 / / M i ln e S im ps on P r e d i c t o r C o r r e c t o r Method / / P ag e n o . 5 4 4

clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=y+exp ( x ) ’ ) h=0.5; y=[1,1.824,3.718,7.722] for i = 1 : 4 x=(i-1)*h; f1(i)=f(x,y(i)); printf ( ’ \ n f % i = %g ’ , i - 1 , f 1 ( i ) ) end y41=y(1)+4*h*(2*f1(4)-f1(3)+2*f1(2))/3 f4=f(x+h,y41); y4=y(3)+h*(f4+4*f1(4)+f1(3))/3 ’ ,y41) printf ( ’ \n\n\ n P r e d i c t o r = %. 9 f \n\n printf ( ’ E v a l u a t o r = % . 9 f \n\n ’ ,f4) printf ( ’ C o r r e c t o r = % . 9 f ’ , y 4 )

Scilab code Exa 15.24 Improved Milne Simpson Predictor Corrector Method

1 2 3 4 5

/ / E x am pl e 1 5 . 2 4 / / I mp ro ve d M il n e S im ps on P r e d i c t o r C o r r e c t o r Method / / P ag e n o . 5 4 6 clc ; clear ; close ;

215

6 deff ( ’ y=f (x , y ) ’ , ’ y=y−xˆ2 ’ ) 7 y(1)=1;h=0.25;x=0; 8 printf ( ’ n \ tXn tX n \ tYn tY n \ t f n \tY tY n \ tY ‘ n+1\ n+1\ tm(n+1) \ tv (n \ tY ‘ n \ tYn

+1 ) \ n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 f1(1)=f(x,y(1)); 10 f o r i = 1 : 3 11 K1=h*f(x,y(i)); 12 K2=h*f(x+2*h/3,y(i)+2*K1/3); 13 y(i+1)=y(i)+(K1+3*K2)/4 14 n ’ ,i-1,x,y(i),f1(i printf ( ’ %i \ t% . 3 f \t \ t% . 3 f \t \ t% . 3 f \ )) 15 x=x+h f1(i+1)=f(x,y(i+1)) 16 17 e n d 18 Y 3 1 = 0 19 f o r i = 3 : 1 0 Y41=y(i-2)+4*h*(2*f1(4)-f1(3)+2*f1(2))/3 20 //

predictor 21 22 23 24

25 26 27 28 29 30 31 e n d

m4=Y41+28*(y(i+1)-Y31)/29 // mod ifie r // eval uato r v4=f(x+h,m4) Y4=y(i)+h*(v4+4*f1(4)+f1(3))/3 // cor rec tor printf ( ’ %i \ t% . 3 f \t t% . 3 f \t \ t% . 3 f \t \ t% . 3 f \t \ \ t% . 3 f \t \ t% . 3 f \ \ t% . 3 f \t \ t% . 3 f \ \ n ’ , i , x , y ( i + 1 ) , f 1 ( 4 ) , Y 3 1 , y ( i + 1 ) , Y41,m4,v4) y(i+2)=Y4 Y31=Y41; f1(2)=f1(3); f1(3)=f1(4); f1(4)=f(x+h,y(i+2)) x=x+h

216

Scilab code Exa 15.25 Hamming Predictor Corrector Method

1 2 3 4 5 6 7 8

/ / E x am am pl pl e 1 5 . 2 5 / / Ha Ha m m i n g P r e d i c t o r C o r r e c t o r M e t h o d / / P ag ag e n o . 5 4 8 c l c ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=y−xˆ2 ’ ) y(1)=1;h=0.25;x=0; printf ( ’ n \ tXn tX n \ tYn tY n \ t f n \tY n+1 \tm(n+1) \ t v ( \ tY ‘ n \ t Y c ( n ) \ tY ‘ n+1\

n+1) \ t Y c ( n + 1 )\ )\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 9 f1(1)=f(x,y(1)); 10 f o r i = 1 : 3 11 K1=h*f(x,y(i)); K2=h*f(x+2*h/3,y(i)+2*K1/3); 12 13 y(i+1)=y(i)+(K1+3*K2)/4 printf ( ’ %i \ t% . 3 f \t 14 n ’ ,i-1,x,y(i),f1(i \ t% . 3 f \t \ t% . 3 f \ )) 15 x=x+h 16 f1(i+1)=f(x,y(i+1)) 17 e n d 18 Y 3 1 = y ( 4 ) ; Y c = 0 19 f o r i = 3 : 1 0 Y41=y(i-2)+4*h*(2*f1(4)-f1(3)+2*f1(2))/3 // 20

predictor 21 22 23

m4=Y41+112*(Y31-Yc)/121 // mod ifie r // eval uato r v4=f(x+h,m4) Y4c=(9*y(i+1)-y(i-1))/8+3*h*(v4+2*f1(4)-f1(3))/8

// cor rec tor 24 25

26 27 28

Y4=Y4c+9*(Y41-Y4c)/121 // f i n a l v al ue t% . 3 f \t printf ( ’ %i \ t% . 3 f \t \ t% . 3 f \t \ t% . 3 f \t \ \ t% . 3 f \t \ t% . 3 f \ \ \ t% . 3 f \t \ t% . 3 f \ t% . 3 f \t n ’ ,i,x,y(i+1),f1(4),Y31,Yc, Y41,m4,v4,Y4c) y(i+2)=Y4 Y31=Y41; f1(2)=f1(3);

217

29 30 31 32 33 e n d

f1(3)=f1(4); f1(4)=f(x+h,y(i+2)) Yc=Y4c x=x+h

Scilab code Exa 15.26 Multi Valued Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

/ / E x am am pl pl e 1 5 . 2 6 / / M u l t i V a l u e d M et et ho ho d / / P ag ag e n o . 5 5 3 c l c ; clear ; close ;

y=2∗ xˆ2 xˆ 2 − y ’ ) deff ( ’ y=f1 ( x , y ) ’ , ’ y=2 h=0.1;x=0;y=-1; y=4∗ x− f 1 (x , y ) ’ ) deff ( ’ y=f2 ( x , y ) ’ , ’ y=4 deff ( ’ y=f3 ( x , y ) ’ , ’ y=4 y=4− f 2 (x , y ) ’ ) B=[1,1,1,1;0,1,2,3;0,0,1,3;0,0,0,1]; y0=[y;h*f1(x,y);h^2*f2(x,y)/2;h^3*f3(x,y)/6] y01=y0; r=[0;1;3/4;1/6]

disp ( r , ’ I f r = ’ ) printf ( ’ \ n \ n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n x = 0 \ t \ t \ t x = 0 . 1 \ t \ t \ t \ t x = 0 . 2 \ n \ t ’ ) 17 f o r i = 1 : 2 18 y11=B*y01 19 s(i)=h*(f1(x+h,y11(1)))-y11(2) 20 y1=y11+s(i)*r i f i = = 2 then 21 22 break end 23

218

24 y2=y1; y22=y11; 25 26 y01=y1 27 e n d 28 printf ( ’ \ t ( s = %. 5 g ) \ t \ t

( s = %. 9 f ) \ n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n Y0\ Y0 \ t \ t Y ‘ i \ t \ t Y1\ Y1 \ t \ t Y ‘ 2 \ t \ t Y2\ Y2 \ n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’ ,s(1),s(2))

29 f o r i = 1 : 4 30 ’%. 5 f \ t% . 5 f \ t% . 5 f \ t% . 5 f \ t% . 5 f \ printf ( ’%. \ n ’ , y 0 ( i ),y22(i),y2(i),y11(i),y1(i)) 31 e n d 32 y 0 = [ y ; h * f 1 ( x , y ) ; h ^ 2 * f 2 ( x , y ) / 2 ; h ^ 3 * f 3 ( x , y ) / 6 ] 33 y 0 1 = y 0 ; 34 r = [ 5 / 1 2 ; 1 ; 3 / 4 ; 1 / 6 ] 35 disp ( r , ’ I f r = ’ ) 36 printf ( ’ \ n \ n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n x = 0 \ t \ t \ t x = 0 . 1 \ t \ t \ t \ t x = 0 . 2 \ n \ t ’ ) 37 f o r i = 1 : 2 38 y11=B*y01 39 s(i)=h*(f1(x+h,y11(1)))-y11(2) y1=y11+s(i)*r 40 41 i f i = = 2 then break 42 43 end y2=y1; 44 45 y22=y11; y01=y1 46 47 e n d 48 printf ( ’ \ t ( s = %. 5 g ) \ t \ t ( s = %. 9 f ) \ n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n Y0\ Y0 \ t \ t Y ‘ i \ t \ t Y1\ Y1 \ t \ t Y ‘ 2 \ t \ t Y2\ Y2 \ n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’ ,s(1),s(2)) 49 f o r i = 1 : 4 50 ’%. 5 f \ t% . 5 f \ t% . 5 f \ t% . 5 f \ t% . 5 f \ printf ( ’%. \ n ’ , y 0 ( i

219

),y22(i),y2(i),y11(i),y1(i)) 51 e n d

Scilab code Exa 15.27 First order ODE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

/ / E x am am pl pl e 1 5 . 2 7 / / F i r s t o r d e r ODE / / P ag ag e n o . 5 5 8 c l c ; clear ; close ; deff ( ’ y=f 1 (x , y1 , y2 ) ’ , ’ y=y1 ∗ y2+x ’ ) =y1−x ’ ) deff ( ’ y=f 2 (x , y1 , y2 ) ’ , ’ y=y1− h=0.2;x=0;y1=0;y2=1;

/ / h e un u n m et et ho ho d printf ( ’ Heun Heun Method Method : \ n \ n x \ t y 1 \ t y 2 \ n n’) −−−−−−−−−−−−−−−−−−−−−−−−−\n −−−−−−−−−−−−−−−−−−−−−−−−−\ Y=[y1;y2] for i=1:8

F=[f1(x,Y(1),Y(2));f2(x,Y(1),Y(2))] Y1=Y+h*F x=x+h; F1=[f1(x,Y1(1),Y1(2));f2(x,Y1(1),Y1(2))] Y=Y+(h/2)*(F+F1) %g\ t% . 3 f \t printf ( ’ %g\ \ t% . 3 f \ \ n ’ , x - h , Y ( 1 ) , Y ( 2 ) )

end

// c l a s s i c a l run ge ku tt a met method hod printf ( ’ \ n \ n \ n C l a s s i c a l R u n g e K ut u t ta ta Met hod : \ n \ n n \ t x tY n \ tK 1 \ tK 2 \ tK 3 \ tK 4 \ tY(n+1) \ n \ tYn −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 220

25 Y = [ y 1 ; y 2 ] ; x = 0 ; 26 for i = 1 : 6 27 K1=h*[f1(x,Y(1),Y(2));f2(x,Y(1),Y(2))] K2=h*[f1(x+h/2,Y(1)+K1(1)/2,Y(2)+K1(2)/2);f2(x+h 28 /2,Y(1)+K1(1)/2,Y(2)+K1(2)/2)] K3=h*[f1(x+h/2,Y(1)+K2(1)/2,Y(2)+K2(2)/2);f2(x+h 29 /2,Y(1)+K2(1)/2,Y(2)+K2(2)/2)] K4=h*[f1(x+h,Y(1)+K3(1),Y(2)+K3(2));f2(x+h,Y(1)+ 30 K3(1),Y(2)+K3(2))] 31 Y1=Y+(K1+2*K2+2*K3+K4)/6 32 . 3 f \t% . 3 f \t% . 3 f \ printf ( ’ %i \t% . 2 f \t% . 3 f \t% . 3 f \t%

t% . 3 f \n\ t \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’ ,i-1,x,Y(1),K1(1),K2(1),K3(1),K4(1),Y1(1),Y 33 34 35 end

(2),K1(2),K2(2),K3(2),K4(2),Y1(2)) Y=Y1; x=x+h

Scilab code Exa 15.28 Differential Equation

1 2 3 4 5 6 7 8 9 10 11

/ / E x am pl e 1 5 . 2 8 // D i f f e r e n t i a l E qu at io n / / P ag e n o . 5 6 2 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=2∗y ˆ 2 / ( 1 + x ) ’ ) h=0.1;z(1)=-1; for i = 1 : 1 1 printf ( ’ \nZ(%g) = %g\n ’ , (i - 1 ) / 1 0 , z ( i ) ) z(i+1)=z(i)+h*f((i-1)/10,z(i)) end

221

222

Chapter 16 Numerical Solutions of Ordinary Differential Equations Boundary Value Problems

Scilab code Exa 16.1 Outline of Linear Shooting Method

1 2 3 4 5 6 7 8 9 10 11 12 13

/ / E x am pl e 1 6 . 1 / / O u t l i ne o f L i n e a r S h o o ti n g Method / / P ag e n o . 5 7 2 clc ; close ; clear ;

deff ( ’y=f ( x ) ’ , ’ y=xˆ 2 ’ ) ; h=0.5;X0=0;Y0=1;Z1=[-1,-1.5,-1.1771];i=1;Y1=Y0; for j = 1 : 3 Z0=Z1(i); i=i+1 Y0=1; for n = 1 : 2 printf ( ’ \ n Fo r n = %i \n −−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’ ,n-1) 14 K1(1)=h*Z0;

223

15 printf ( ’ \n K11 = %g ’ , K 1 ( 1 ) ) ; K1(2)=h*f(Y0); 16 17 printf ( ’ \n K12 = %g ’ , K 1 ( 2 ) ) ; K2=h*f(Y0+K1(2)) 18 19 printf ( ’ \n K22 = %g ’ , K 2 ) ; Z0=Z0+(K1(2)+K2)/2 20 21 printf ( ’ \n Z%i = %g ’ , n , Z 0 ) ; K2=h*Z0; 22 23 printf ( ’ \n K21 = %g ’ , K 2 ) ; 24 Y0=Y0+(K1(1)+K2)/2 25 printf ( ’ \n Y%i = %g ’ , n , Y 0 ) ; 26 printf ( ’ \n\n\n ’ ) if n = = 1 then 27 28 Y2=Y0 end 29 30 end printf ( ’ \n\n\n ’ ) 31 32 end 33 printf ( ’ H ence t he s o l u t i o n i s y (%g ) = %i , y (%g) = % .4 f and y (%g ) = %. 1 f ’ , X 0 , Y 1 , X 0 + h , Y 2 , X 0 + 2 * h , Y 0 )

Scilab code Exa 16.2 Linear Shooting Method

1 2 3 4 5 6 7 8 9 10

/ / E x am pl e 1 6 . 2 / / L i n e a r S h o o t i n g Meth od / / P ag e n o . 5 7 6 clc ; close ; clear ;

deff ( ’ y=f 1 (x , y , y1 ) ’ , ’ y=−x ∗ y1+xˆ2∗ y+2∗xˆ3 ’ ) deff ( ’ y=F1( x , y , y1 ) ’ , ’ y=−x ∗ y1+xˆ2∗ y+2∗xˆ3 ’ ) deff ( ’ y=F2( x , y , y1 ) ’ , ’ y=−x ∗ y1+xˆ2∗ y ’ ) a=0;b=1; y0=1;y1=-1;n=5;

224

11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

h=(b-a)/n y=y0;y01=0;x=a; for i = 0 : 5 yi1(1,i+1)=y K1=h*y01; R1=h*F1(x,y,y01); K2=h*(y+R1/2); R2=h*F1(x+h/2,y+K1/2,y01+R1/2) K3=h*(y01+R2/2) R3=h*F1(x+h/2,y+K2/2,y01+R2/2) K4=h*(y+R3) R4=h*F1(x+h,y+K3,y01+R3) y=y+(K1+2*K2+2*K3+K4)/6 y01=y01+(R1+2*R2+2*R3+R4)/6 x=x+h end y=0;y01=1;x=a; for i = 0 : 5 yi2(1,i+1)=y K1=h*y01; R1=h*F2(x,y,y01); K2=h*(y+R1/2); R2=h*F2(x+h/2,y+K1/2,y01+R1/2) K3=h*(y01+R2/2) R3=h*F2(x+h/2,y+K2/2,y01+R2/2) K4=h*(y+R3) R4=h*F2(x+h,y+K3,y01+R3) y=y+(K1+2*K2+2*K3+K4)/6 y01=y01+(R1+2*R2+2*R3+R4)/6 x=x+h end for i = 1 : 6 yi(i)=yi1(1,i)+((y1-yi1(6))/yi2(6))*yi2(i) end y=1;x=a;y01=y1 for i = 0 : 5 yir(1,i+1)=y; K1=h*y01;

225

49 R1=h*f1(x,y,y01); K2=h*(y+R1/2); 50 51 R2=h*f1(x+h/2,y+K1/2,y01+R1/2) K3=h*(y01+R2/2) 52 53 R3=h*f1(x+h/2,y+K2/2,y01+R2/2) K4=h*(y+R3) 54 55 R4=h*f1(x+h,y+K3,y01+R3) y=y+(K1+2*K2+2*K3+K4)/6 56 57 y01=y01+(R1+2*R2+2*R3+R4)/6 58 x=x+h 59 end 60 x = a ; 61 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n\ t x ’ ) 62 63 64 65 66 67 68 69 70 71 72 73 74

for i = 1 : 6 printf ( ’ \t% . 1 f \ t ’ ,x ) x=x+h end printf ( ’ \n\ t y ’ ) for i = 1 : 6 printf ( ’ \t% . 4 f \ t ’ , y i ( i ) ) end printf ( ’ \n by RK ’ ) for i = 1 : 6 printf ( ’ \t% . 4 f \ t ’ , y i r ( i ) ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 75 printf ( ’ \n\n\ nNote : Co mpu tatio n e r r o r i n c a l c u l a t i o n o f v a l u e s by RK method p e r fo r m e d i n b oo k ’ )

226

Scilab code Exa 16.3 Multiple Shooting Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

/ / E x am pl e 1 6 . 3 / / M u l t i p l e S h o o t i n g Metho d / / P ag e n o . 5 7 7 clc ; close ; clear ; h=0.25;x=0;y1=0; deff ( ’y=f ( x ) ’ , ’ y=−(4∗h ˆ 2 ) / (1 + x ) ˆ 2 ’ ) deff ( ’ y= f 1 ( x ) ’ , ’ y=−2∗(1+(h ˆ2 ) /(1+x ) ˆ2 ) ’ ) for i = 1 : 4 x=x+h B(i)=f(x); for j = 1 : 4 if i ==4 & i == j A(i,j)=f1(x)+1/4 A(i,j-1)=2 elseif j = = i then A(i,j)=f1(x) A(i,j+1)=1 if j - 1 ~ = 0 then A(i,j-1)=1 end end end end y = inv ( A ) * B disp (B , ”B =” ,A , ’A = ’ ) printf ( ’ \n\n\n x : ’ ) for i = 1 : 5 printf ( ’ \t% . 2 f ’ ,x ) x=x+h end x = 0 ; printf ( ’ \n y : \ t% . 2 f ’ , y 1 ) ; for i = 1 : 4 printf ( ’ \t% . 4 f ’ , y ( i ) ) end

227

Scilab code Exa 16.4 Finite Difference Method

1 2 3 4 5 6 7

/ / E x am pl e 1 6 . 4 / / F i n i t e D i f f e r e n c e Method / / P ag e n o . 5 8 2 clc ; close ; clear ; x=0;h=0.25;q=-1;Y(1)=-2;Y(5)=1; printf ( ’ \n i \ t x i \ t Y i \ t p i \ t q i \ t r i \ n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’) 8 for i = 1 : 5 r(i)=-x^2 9 10 if i >1 & i <5 then printf ( ’ %i \t%g\ t% s \t%g\ t%i \t%g\n ’ , i - 1 , x , ” ? ” 11 ,x,q,r(i)) 12 else 13 printf ( ’ %i \t%g\t%g\t%g\ t%i \t%g\n ’ , i - 1 , x , Y ( i ),x,q,r(i)) end 14 15 x=x+h 16 end 17 x = 0 ; 18 printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\n ’) 19 for i = 1 : 3 20 x=x+h 21 for j = 1 : 3 22 if i = = j then A(i,j)=2+h^2*q 23 24 elseif i < j & abs ( i - j ) ~ = 2

228

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

A(i,j)=-1+h*x/2 elseif i > j & abs ( i - j ) ~ = 2 A(i,j)=-1-h*x/2 end

end if i = = 3 then B(i)=-h^2*r(i+1)+(-h*x/2+1)*Y(1+2*(i-1)) else B(i)=-h^2*r(i+1)+(h*x/2+1)*Y(1+2*(i-1)) end B(i)=(-1)^(i+1)*B(i)

end disp (B , ”B =” ,A , ’A = ’ ) y = inv ( A ) * B for i = 1 : 3 Y(i+1)=y(i) end x=0; disp ( ” The S o l u ti o n i s : ” ,B , ”B =” ,A , ’A = ’ ) printf ( ’ x : ’ ) for i = 1 : 5 printf ( ’ \ t % . 2 f ’ ,x ) x=x+h end x = 0 ; printf ( ’ \n y : ’ ) ; for i = 1 : 5 printf ( ’ \t% . 3 f ’ , Y ( i ) ) end

Scilab code Exa 16.5 Non Linear Problem

1 / / E x am pl e 1 6 . 5 2 / / Non L i n e a r P ro bl em

229

3 4 5 6 7 8 9 10

/ / P ag e n o . 5 8 4 clc ; close ; clear ;

deff ( ’y=f ( x ) ’ , ’y =2/(1+x ) ’ ) Y=[1,0.75,0.75,0.75,0.5];h=0.25 A = [ - 2 , 1 , 0 ; 1 , - 2 , 1 ; 0 , 1 , - 2 ] ; A _ 1 =inv ( A ) disp ( A _ 1 , ” I n v e r s e o f A =” ,A , ”A =” ) printf ( ’ \nThe S o l u t i o n o f t he s ys te m i s : \n\n

I t e r a t i o n \ t Y0\ t \ t Y1\ t \ t Y2\ t \ t Y3\ t \ t Y4\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 11 for i = 0 : 6 printf ( ’ \n 12 %i ’ ,i ) 13 for j = 1 : 5 if j <4 & i ~=0 then 14 15 Y(j+1)=y(j) end 16 17 printf ( ’ \ t \t% . 4 f ’ , Y ( j ) ) end 18 19 x=0; 20 for j = 1 : 3 21 x=x+h 22 if j ~ = 2 then B(j)=h^2*f(x)*Y(j+1)^2-Y(1+2*(j-1)) 23 24 else B(j)=h^2*f(x)*Y(j+1)^2 25 26 end end 27 28 y=A_1*B 29 end

Scilab code Exa 16.6 Collocation Method

230

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 6 . 6 / / C o l l o c a t i o n Method / / P ag e n o . 5 8 9 clc ; close ; clear ;

h1=0.000001;h=0.25;x=0; Y(1)=0;Y(5)=0; deff ( ’y=p ( x ) ’ , ’ y=1 ’ ) deff ( ’y=q ( x ) ’ , ’ y=−2/(1+x)ˆ2 ’ ) deff ( ’y=f ( x ) ’ , ’ y = (2∗x −4 ) / (1 + x ) ˆ 4 ’ ) deff ( ’ y= f i ( x , j ) ’ , ’ y=(1−x ) ∗xˆ j ’ ) // f u n c t i o n f o r deff ( ’ y=f1 ( x , y ) ’ , ’ y=(−x+y ) /h1 ’ )

differentiation 13 for i = 1 : 4 x=x+h 14 15 for j = 1 : 4 A(i,j)=p(x)*f1(f1(fi(x,j),fi(x+h1,j)),f1(fi( 16 x+h1,j),fi(x+2*h1,j)))+f1(p(x),p(x+h1))* f1(fi(x,j),fi(x+h1,j))+q(x)*fi(x,j) 17 end 18 end 19 x = 0 ; 20 for i = 1 : 4 x=x+h 21 22 B(i)=f(x) 23 end 24 disp (B , ’B = ’ ,A , ”A =” ) 25 C = inv ( A ) * B 26 x = 0 ; 27 for i = 2 : 4 28 x=x+h; 29 for j = 1 : 4 30 Y(i)=Y(i)+C(j)*fi(x,j) 31 end 32 end 33 disp (Y , ” S o l u t i o n M at ri x Y = ” )

231

Chapter 18 Numerical Solutions of Parabolic Partial Differential Equations

Scilab code Exa 18.4 Forward Difference Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 8 . 4 / / F or wa rd D i f f e r e n c e Method / / P ag e n o . 6 2 4 clc ; clear ; close ; h=0.2;k=0.02; r=k/h^2; printf ( ’ \n j \ t t \ t | \ t i −−>\t ’ ) for i = 0 : 5 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t x −−>\t ’ ) for i = 0 : 5 printf ( ’%. 3 f \ t ’ ,(i)/5) end

232

16 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 for j = 1 : 6 18 printf ( ’ \n %i \t% . 3 f \ t | \ t \ t ’ , j - 1 , ( j - 1 ) / 5 0 ) for i = 1 : 6 19 20 if i ==1 | i ==6 then u(j,i)=0; 21 22 elseif j = = 1 then 23 u ( j , i ) = sin ( % p i * ( i - 1 ) / 5 ) 24 else 25 u(j,i)=(u(j-1,i-1)+u(j-1,i+1))/2 end 26 27 printf ( ’%. 3 f \ t ’ , u ( j , i ) ) end 28 29 end

Scilab code Exa 18.5 Bender Schmidt Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14

/ / E x am pl e 1 8 . 5 / / B e n d e r S c h mi d t Met ho d / / P ag e n o . 6 2 5 clc ; clear ; close ; h=0.1;k=0.005; r=k/h^2; printf ( ’ \n j |\ t i for i = 0 : 1 0 %i \ t printf ( ’ end printf ( ’ \n | | \ t x for i = 0 : 1 0 printf ( ’%. 3 f \ t

−−>\t ’ ) ’ ,i ) −−>\t ’ ) ’ ,(i) /10) 233

15 end 16 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 for j = 1 : 9 printf ( ’ \n %i 18 | \ t \ t ’ ,j-1) 19 for i = 1 : 1 1 if i == 1 | i = =1 1 then 20 21 u(j,i)=0; 22 elseif j = = 1 then 23 u ( j , i ) = sin ( % p i * ( i - 1 ) / 1 0 ) 24 else u(j,i)=u(j-1,i)/2+(u(j-1,i-1)+u(j-1,i+1) 25 ) /4 end 26 27 printf ( ’%. 3 f \ t ’ , u ( j , i ) ) end 28 29 end

Scilab code Exa 18.6 Crank Nicolson Method

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 8 . 6 / / C ra nk N i c o l s o n Metho d / / P ag e n o . 6 3 1 clc ; clear ; close ; h=1/2;k=1/8; r=k/h^2; for i = 1 : 2 : 3 for j = 1 : 9 if i ==1 | j ==1 then u(i,j)=0; end if i = = 3 then

234

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

u(i,j)=(j-1)/8 end

end

end a=[3,-1,0;-1,3,-1;0,-1,3]; a = inv ( a ) ; for j = 2 : 9 u(2,j)=(u(1,j-1)+2*u(2,j-1)+u(3,j-1)+u(1,j)+ u(3,j))/6 end u=u’ printf ( ’ \ n f o r h = 1 / 2\ n\n ’ ) printf ( ’ i \ \ j −−> ’ ) for i = 1 : 3 printf ( ’ \ t%i \ t ’ ,i ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

for i = 1 : 9 printf ( ’ \n %i ’ ,i ) for j = 1 : 3 printf ( ’ \ t %. 9 f ’ , u ( i , j ) ) end end

h=1/4;k=1/8; r=k/h^2; for i = 1 : 4 : 5 for j = 1 : 9 if i ==1 | j ==1 then u(i,j)=0; end if i = = 5 then u(i,j)=(j-1)/8

235

48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63 64

end end

end a=[3,-1,0;-1,3,-1;0,-1,3]; a = inv ( a ) ; for j = 2 : 9 b=[u(1,j-1)-u(2,j-1)+u(3,j-1)+u(1,j);u(2,j -1)-u(3,j-1)+u(4,j-1);u(3,j-1)-u(4,j-1)+u (5,j-1)+u(5,j)] x=a*b u(2,j)=x(1);u(3,j)=x(2);u(4,j)=x(3); end u=u’ printf ( ’ \n\n\n\n\ n f o r h = 1 / 4\ n\n ’ ) printf ( ’ i \ \ j −−> ’ ) for i = 1 : 5 printf ( ’ \ t%i \ t ’ ,i ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 65 for i = 1 : 9 66 printf ( ’ \n %i ’ ,i ) for j = 1 : 5 67 68 %. 9 f ’ , u ( i , j ) ) printf ( ’ \ t end 69 70 end

Scilab code Exa 18.7 Gauss Seidel Method

1 / / E x am pl e 1 8 . 7 2 / / G au ss S e i d e l Method 3 / / P ag e n o . 6 3 7

236

4 5 6 7 8 9 10 11 12 13 14 15 16

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=4∗x−4∗xˆ2 ’ ) h=0.2;k=0.04; r=k/h^2; printf ( ’ \n k \ t | \ t i −−>\t ’ ) for i = 0 : 5 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t x −−>\t ’ ) for i = 0 : 5 printf ( ’%. 2 f \ t ’ ,(i)/5) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 for k = 1 : 7 printf ( ’ \n %i \ t | \ t \ t ’ ,k-1) 18 19 for i = 1 : 6 if i ==1 | i ==6 then 20 21 u(k,i)=0; 22 elseif k = = 1 then 23 u(k,i)=f((i-1)/5) 24 else u(k,i)=(u(k-1,i-1)+u(k-1,i+1))/2 25 26 end printf ( ’%. 2 f \ t ’ , u ( k , i ) ) 27 28 end 29 end

Scilab code Exa 18.8 ADI Method

1 / / E x am pl e 1 8 . 8 2 / /ADI Method

237

3 4 5 6 7 8 9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

/ / P ag e n o . 6 4 2 clc ; clear ; close ; for i = 1 : 4 for j = 1 : 5 P(i,j)=20 end end r=1;k=0; for i = 1 : 6 v1(i)=20 u1(i)=20 end P1 = [ 2 5 , 3 0 , 3 5 , 5 0 , 6 0 ; 3 5 , 0 , 0 , 0 , 7 0 ; 4 5 , 0 , 0 , 0 , 8 0 ; 6 0 , 7 0 , 8 0 , 10 0 , 9 0 ] for i = 1 : 4 printf ( ’ \n ’ ) for j = 1 : 5 printf ( ’ %i \ t ’ , P ( i , j ) ) end if i = = 2 then printf ( ’−−> ’ ) end printf ( ’ \ t ’ ) for j = 1 : 5 printf ( ’ %i \ t ’ , P 1 ( i , j ) ) if i >1 & i <4 & j >1 & j <5 then P1(i,j)=P(i,j) end end end P1v=P1;P1h=P1; for i = 1 : 6 for j = 1 : 6 if i = = j then Av(i,j)=1+2*r e l s ei f a b s( i -j ) == 1 & i +j ~ =5 & i + j ~= 9

238

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71

Av(i,j)=-r end

end end for i = 1 : 6 for j = 1 : 6 if i = = j then Ah(i,j)=1+2*r e l s ei f a b s( i - j) = =1 & i + j ~= 7 Ah(i,j)=-r end end end n =8 for l = 1 : n k=0; for j = 0 : 2 for i = 1 : 2 if i = = 1 then Bv(i+j+k)=r*P1h(i+1,j+1)+(1-2*r)*P1h(i +1,j+2)+r*P1h(i+1,j+3)+r*P1h(i,j+1) else Bv(i+j+k)=r*P1h(i+1,j+1)+(1-2*r)*P1h(i +1,j+2)+r*P1h(i+1,j+3)+r*P1h(i+2,j+1) end end k=k+1; end k=0; B h = [ r * 3 0 + ( 1 - 2 * r ) * v 1 ( 1 ) + r * v 1 ( 4 ) + r * 3 5 ; r * 3 5 + ( 1 - 2 * r ) * v1 (3)+r*v1(5);r*v1(1)+(1-2*r)*v1(2)+r*v1(3)+r*(70); r*v1(1)+(1-2*r)*v1(2)+r*(70+45);r*v1(3)+(1-2*r)* v1(4)+r*80;r*v1(5)+(1-2*r)*v1(6)+r*(100+80)] for i = 1 : 6 v(i,l)=v1(i) end for i = 1 : 6 u(i,l)=u1(i)

239

72 end v1 = inv ( A v ) * B v 73 74 u1 = inv ( A h ) * B h k=1; 75 76 for i = 2 : 3 for j = 2 : 4 77 78 P1h(i,j)=u1(i+j+k-4) end 79 80 k=k+2 81 end 82 k=0; 83 for j = 2 : 4 for i = 2 : 3 84 85 P1v(i,j)=v1(i+j+k-3) end 86 87 k=k+1 end 88 89 end 90 printf ( ’ \n\n\n\ n R e s u l ts f o r V e r t i c a l T r a n s v e rs e i n C el s iu s : \ n ’ ) 91 for i = 1 : 7 92 printf ( ’ \n ’ ) 93 if i = = 1 then printf ( ’ I t r −−> ’ ) 94 95 for j = 1 : n printf ( ’ \ t %i ’ , j - 1 ) 96 97 end printf ( ’ \n 98

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 99 else 100 printf ( ’ v %i ’ , i - 1 ) 101 for j = 1 : n 102 printf ( ’ \t% . 2 f ’ , v ( i - 1 , j ) ) 103 end 104 end 105 end 106 printf ( ’ \n\n\n\ n R e s u l ts f o r H o r i z o n ta l T ra n sv er s e i n

240

C el s iu s : \ n ’ ) 107 for i = 1 : 7 108 printf ( ’ \n ’ ) if i = = 1 then 109 110 printf ( ’ I t r −−> ’ ) for j = 1 : n 111 112 printf ( ’ \ t %i ’ , j - 1 ) end 113 114 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 115 else printf ( ’ u %i ’ , i - 1 ) 116 117 for j = 1 : n printf ( ’ \t% . 2 f ’ , u ( i - 1 , j ) ) 118 119 end end 120 121 end

241

Chapter 19 Numerical Solutions of Hyperbolic Partial Differential Equations

Scilab code Exa 19.3 Simple Explicit Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 1 9 . 3 / / S i mp l e E x p l i c i t Method / / P ag e n o . 6 5 8 clc ; clear ; close ; c=-2;dt=0.07;dx=0.2; r = abs ( c ) * d t / d x ; printf ( ’ \n x \ t i \ t | \ t j −−>\t ’ ) for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t ) end

242

16 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 for j = 1 : 6 18 t%i \ t | \ t \ t ’ , ( j - 1 ) * d x , j - 1 ) printf ( ’ \n % . 1 f \ for i = 1 : 7 19 20 if i = = 1 then u(j,i)=0; 21 22 elseif j = = 1 then 23 u(j,i)=1 24 else 25 u(j,i)=(1-r)*u(j,i-1)+r*u(j-1,i-1) end 26 27 printf ( ’%. 3 f \ t ’ , u ( j , i ) ) end 28 29 end

Scilab code Exa 19.4 Simple Implicit Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14

/ / E x am pl e 1 9 . 4 / / S i mp l e I m p l i c i t Method / / P ag e n o . 6 5 9 clc ; clear ; close ; c=-2;dt=0.07;dx=0.2; r = abs ( c ) * d t / d x ; printf ( ’ \n x \ t i \ t | \ t j −−>\t ’ ) for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t )

243

15 end 16 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 for j = 1 : 6 printf ( ’ \n % . 1 f \ 18 t%i \ t | \ t \ t ’ , ( j - 1 ) * d x , j - 1 ) 19 for i = 1 : 7 if i = = 1 then 20 21 u(j,i)=0; 22 elseif j = = 1 then 23 u(j,i)=1 24 else u(j,i)=(1/(1+r))*u(j,i-1)+r*u(j-1,i)/(1+ 25 r) end 26 27 printf ( ’%. 3 f \ t ’ , u ( j , i ) ) end 28 29 end

Scilab code Exa 19.5 Lax Wendroff Method

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 1 9 . 5 / / L a x W e n d r o f f M et ho d / / P ag e n o . 6 6 0 clc ; clear ; close ; c=-2;dt=0.07;dx=0.2; r = abs ( c ) * d t / d x ; printf ( ’ \n x \ t i \ t | \ t j −−>\t ’ ) for i = 0 : 6 printf ( ’ %i \ t ’ ,i ) end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ )

244

13 14 15 16 17

for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t ) end i=1; printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 18 for j = 1 : 7 19 for i = 1 : 6 20 if j = = 1 then 21 u(i,j)=0; 22 u(i+1,j)=0; elseif i = = 1 then 23 24 u(i,j)=1 else 25 26 u(i,j)=r*(r-1)*u(i+1,j-1)/2+(1-r^2)*u(i, j-1)+r*(1+r)*u(i-1,j-1)/2 27 end end 28 29 end 30 for i = 1 : 6 31 t%i \ t | \ t \ t ’ , ( i - 1 ) * d x , i - 1 ) printf ( ’ \n % . 1 f \ 32 for j = 1 : 7 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) 33 34 end 35 end

Scilab code Exa 19.6 Wendroff Method

1 2 3 4

/ / E x am pl e 1 9 . 6 / / W e n d r o f f M et ho d / / P ag e n o . 6 6 1 clc ; clear ; close ;

245

5 6 7 8 9 10 11 12 13 14 15 16

c=2;k=0.07;h=0.2; a=(h+k*c)/(h-k*c) printf ( ’ \n x \ t i \ t | \ t j −−>\t ’ ) for i = 0 : 6 printf ( ’ %i \ t ’ ,i ) end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * k ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 for i = 1 : 6 18 t%i \ t | \ t \ t ’ , ( i - 1 ) * h , i - 1 ) printf ( ’ \n % . 1 f \ for j = 1 : 7 19 20 if j = = 1 then u(i,j)=0; 21 22 elseif i = = 1 then 23 u(i,j)=1 24 else 25 u(i,j)=u(i-1,j-1)+(u(i,j-1)-u(i-1,j))/a end 26 27 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) end 28 29 end

Scilab code Exa 19.7 Leapfrog Method

1 / / E x am pl e 1 9 . 7 2 / / L e a p f r o g M et ho d 3 / / P ag e n o . 6 6 2

246

4 5 6 7 8 9 10 11 12 13 14 15 16

clc ; clear ; close ; c=2;k=0.07;h=0.2; r=c*k/h printf ( ’ \n x \ t i \ t | \ t j −−>\t ’ ) for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * k ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 17 18 for j = 1 : 7 19 for i = 1 : 6 if j ==1 | j ==2 & i ~=1 then 20 21 u(i,j)=0; 22 u(i+1,j)=0; 23 elseif i = = 1 then 24 u(i,j)=1 else 25 26 u(i,j)=u(i,j-2)-r*(u(i+1,j-1)-u(i-1,j-1) ) 27 end end 28 29 end 30 for i = 1 : 6 31 t%i \ t | \ t \ t ’ , ( i - 1 ) * h , i - 1 ) printf ( ’ \n % . 1 f \ 32 for j = 1 : 7 33 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) 34 end 35 end

247

Scilab code Exa 19.8 Variable Coefficients

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

/ / E x am pl e 1 9 . 8 // V a ri a b l e C o e f f i c i e n t s / / P ag e n o . 6 6 3 clc ; clear ; close ;

// s i mp l e e x p l i c i t method printf ( ’ \n\nBy S i m pl e E x p l i c i t Method : \ n\n ’ ) dt=0.05;dx=0.2; x=0; printf ( ’ \n i \ t x \ t r \ t | \ t j −−>\t ’ ) for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n \ t \ t \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 19 for j = 1 : 6 20 r = sqrt ( 1 + 2 * x ) * d t / d x ; 21 printf ( ’ \n %i \t% . 3 f \t% . 3 f \ t | \ t \ t ’ , ( j - 1 ) , x , r ) for i = 1 : 7 22 23 if i = = 1 then u(j,i)=0; 24 25 elseif j = = 1 then u(j,i)=1 26 27 else u(j,i)=(1-r)*u(j,i-1)+r*u(j-1,i-1) 28

248

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

end

end printf ( ’%. 3 f \ t ’ , u ( j , i ) ) end x=x+dx

/ / s i m p l e i m p l i c i t method printf ( ’ \n\n\nBy S i mp l e I m p l i c i t Method : \ n ’ ) c=-2;dt=0.05;dx=0.2;x=0 r \ t | \ t j −−>\t ’ ) printf ( ’ \n i \ t x \ t for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n \ t \ t \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 49 for j = 1 : 6 r = sqrt ( 1 + 2 * x ) * d t / d x ; 50 51 printf ( ’ \n %i \t% . 3 f \t% . 3 f \ t | \ t \ t ’ , ( j - 1 ) , x , r ) for i = 1 : 7 52 53 if i = = 1 then u(j,i)=0; 54 55 elseif j = = 1 then u(j,i)=1 56 57 else 58 u(j,i)=(1/(1+r))*u(j,i-1)+r*u(j-1,i)/(1+ r) 59 end 60 printf ( ’%. 3 f \ t ’ , u ( j , i ) ) 61 end x=x+dx 62 63 end

249

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

/ / w e n d r o f f m et ho d printf ( ’ \n\n\nBy Wendro ff Method : \ n ’ ) k=0.05;h=0.2; x=0.1; c \ t a \ t | \ t j −−>\t ’ ) printf ( ’ \n i \ t x \ t for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n \ t \ t \ t \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * k ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 79 for i = 1 : 6 c = sqrt ( 1 + 2 * x ) ; 80 81 a=(h+k*c)/(h-k*c) 82 printf ( ’ \n %i \t% . 3 f \t% . 3 f \t% . 3 f \ t | \ t \ t ’ , ( i - 1 ) , x h/2,c,a) 83 for j = 1 : 7 if j = = 1 then 84 85 u(i,j)=0; u(i+1,j)=0; 86 87 elseif i = = 1 then u(i,j)=1 88 89 else u(i,j)=u(i-1,j-1)+(u(i,j-1)-u(i-1,j))/a 90 91 end 92 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) 93 end 94 x=x+h 95 end

250

Scilab code Exa 19.9 Inhomogeneous 1st Order Hyperboolic Differential

Equation 1 / / E x am pl e 1 9 . 9 2 // I nh om og en eo us 1 s t O rder H y pe r bo o l ic

Di ff er en ti al

Equation 3 / / P ag e n o . 6 6 5 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18


// s i mp l e e x p l i c i t method printf ( ’ \n\nBy S i m pl e E x p l i c i t Method : \ n ’ ) c=-2;dt=0.07;dx=0.2; r = abs ( c ) * d t / d x ; i \ t x \ t | \ t j −−>\t ’ ) printf ( ’ \n for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 19 x = 0 ; 20 for j = 1 : 6 21 printf ( ’ \n %i \t% . 1 f \ t | \ t \ t ’ , j - 1 , x ) for i = 1 : 7 22 23 if i = = 1 then u ( j , i ) = exp ( - x ) ; 24 25 elseif j = = 1 then u(j,i)=1 26

251

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

else

end

u(j,i)=(1-r)*u(j,i-1)+r*u(j-1,i-1)+dt*2* x end printf ( ’%. 3 f \ t ’ , u ( j , i ) )

end x=x+dx

/ / s i m p l e i m p l i c i t method printf ( ’ \n\n\nBy S i mp l e I m p l i c i t Method : \ n ’ ) c=-2;dt=0.07;dx=0.2; r = abs ( c ) * d t / d x ; printf ( ’ \n i \ t x \ t | \ t j −−>\t ’ ) for i = 0 : 6 printf ( ’ %i \ t ’ ,i ) end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * d t ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 49 x = 0 ; 50 for j = 1 : 6 printf ( ’ \n %i \t% . 1 f \ t | \ t \ t ’ , j - 1 , x ) 51 52 for i = 1 : 7 if i = = 1 then 53 54 u ( j , i ) = exp ( - x ) ; 55 elseif j = = 1 then 56 u(j,i)=1 57 else 58 u(j,i)=(1/(1+r))*u(j,i-1)+r*u(j-1,i)/(1+ r)+dt*2*x end 59 60 printf ( ’%. 3 f \ t ’ , u ( j , i ) )

252

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

end

end x=x+dx

/ / w e n d r o f f m et ho d printf ( ’ \n\n\nBy Wendro ff Method : \ n ’ ) c=2;k=0.07;h=0.2; a=(h+k*c)/(h-k*c) printf ( ’ \n x \ t i \ t | \ t j −−>\t ’ ) for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * k ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 79 x = 0 ; 80 for i = 1 : 6 81 printf ( ’ \n % . 1 f \ t%i \ t | \ t \ t ’ , x , i - 1 ) for j = 1 : 7 82 83 if j = = 1 then u ( i , j ) = exp ( - x ) ; 84 85 elseif i = = 1 then u(i,j)=1 86 87 else u(i,j)=u(i-1,j-1)+(u(i,j-1)-u(i-1,j))/a 88 +(2*h*k)*(x+h/2)/(a*(h+c*k)) 89 end 90 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) 91 end 92 x=x+h 93 end

253

Scilab code Exa 19.10 Non Linear 1st Order Hyperboolic Differential Equa-

tion 1 / / E x am pl e 1 9 . 1 0 2 / /Non L i n ea r 1 s t O rder H y pe r bo o l ic

Di ff ere nt ia l

Equation 3 / / P ag e n o . 6 6 7 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

clc ; clear ; close ; c=-2;k=0.05;h=0.2; r = abs ( c ) * k / h ; printf ( ’ \n i \ t x \ t | \ t j −−>\t ’ ) for i = 0 : 6 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t t −−>\t ’ ) for i = 0 : 6 printf ( ’%. 3 f \ t ’ , i * k ) end i=1; x=0; printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 19 for j = 1 : 7 for i = 1 : 6 20 21 if j = = 1 then u ( i , j ) = exp ( - x ) ; 22 23 u ( i + 1 , j ) = exp ( - ( x + h ) ) ; elseif i = = 1 then 24 25 u(i,j)=1 else 26

254

27

28 29 30 31 32 33 34 35 36 37 38 39

u(i,j)=u(i,j-1)-k*(u(i+1,j-1)^2-u(i-1,j -1)^2)/(4*h)+k^2*((u(i+1,j-1)+u(i,j -1))*(u(i+1,j-1)^2-u(i,j-1)^2)-(u(i,j -1)+u(i-1,j-1))*(u(i,j-1)^2-u(i-1,j -1)^2))/(8*h^2)

end x=x+h end

end x=0; for i = 1 : 6 printf ( ’ \n %i \t% . 1 f \ t | \ t \ t ’ , i - 1 , x ) for j = 1 : 7 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) end x=x+h end


1 2 3 4 5 6 7 8 9 10 11 12 13

/ / E x am pl e 1 9 . 1 1 / / F i n i t e D i f f e r e n c e Method / / P ag e n o . 6 7 0

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y= s i n ( % p i∗ x ) ’ ) deff ( ’ y=g ( x ) ’ , ’ y=0 ’ ) a=1;b=1;c=1;n=5;m=10; h=a/n;k=b/m;r=c*k/h; r1=r^2;r2=r1/2;s1=1-r1;s2=2*(1-r2) printf ( ’ \n i ’ ) for i = 1 : n printf ( ’ \ t %i ’ ,i ) end

255

14 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ nfi ’) 15 16 17 18 19 20 21 22 23 24 25 26 27 28

for i = 1 : n f1(i)=f(h*(i-1)) printf ( ’ \t% . 3 f ’ , f 1 ( i ) ) end printf ( ’ \ n g i ’ ) for i = 1 : n g1(i)=g(h*(i-1)) printf ( ’ \ t %g ’ , g 1 ( i ) ) end printf ( ’ \n\n\n i / j −−> ’ ) for i = 1 : m printf ( ’ \ t %i ’ ,i ) end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 29 for j = 1 : m 30 for i = 1 : n 31 if i ==1 | i == n then 32 u(i,j)=0; elseif j = = 1 33 34 u(i,j)=f1(i) elseif j = = 2 35 36 u(i,j)=s1*f1(i)+k*g1(i)+r2*(f1(i+1)+f1(i -1)) 37 else u(i,j)=s2*u(i,j-1)+r1*u(i-1,j-1)+u(i+1,j 38 -1)-u(i,j-2) 39 end 40 41 end 42 end 43 for i = 1 : n printf ( ’ \n %i \ t ’ ,i ) 44 45 for j = 1 : m

256

46 printf ( ’ \t% . 3 f ’ , u ( i , j ) ) end 47 48 end

Scilab code Exa 19.12 Hyperbolic Partial Differential Equations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

/ / E x am pl e 1 9 . 1 2 / / H y p e r b o l ic P a r t i a l D i f f e r e n t i a l E qu at io ns / / P ag e n o . 6 7 3

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=12∗x ’ ) Ua(1)=0.25; Ua(2)=0.75 A=[1,-2;1,2]; x1 = inv ( A ) * U a ; and Tb = %g ’ , x 1 ( 1 ) , x 1 ( 2 ) ) printf ( ’Xb = %g A=[2,-1;2,1]; B=[-7.5;-8.5]; x2 = inv ( A ) * B ; and Qb = %g ’ , x 2 ( 1 ) , x 2 ( 2 ) ) printf ( ’ \n\n Pb = %g x1(1)=x1(1)-Ua(1) du=x1’*x2 printf ( ’ \n\n dU = %g ’ , d u ) Ub=f(Ua(1))+du; printf ( ’ \n\n M o d i f i e d Ub = %g ’ , U b )

Scilab code Exa 19.13 Hyperbolic Differential Equations in 2D or 3D

1 / / E x am pl e 1 9 . 1 3

257

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

// H y p e r b o l ic D i f f e r e n t i a l E qu at io ns i n 2D o r 3D / / P ag e n o . 6 7 5 clc ; clear ; close ; deff ( ’ y=f (x , y ) ’ , ’ y=x∗(2 − x ) ∗ y∗(2 − y ) ’ ) c2=3;k=0.4;h=0.4;c2=3;s2=0.5 for l = 0 : 1 1 if l = = 0 then printf ( ’ \n t = %i \n\n i \ t x \ t | \ t j −−>\t ’ ,l ) for i = 0 : 5 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t y −−>\t ’ ) for i = 0 : 5 printf ( ’%. 3 f \ t ’ , i * k ) end x=0; printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 20 for i = 1 : 6 21 y=0; 22 printf ( ’ \n %i \t% . 3 f \ t | \ t \ t ’ , i - 1 , x ) for j = 1 : 6 23 24 if i ==1 | i ==6 then u(i,j)=0; 25 26 elseif j = =1 | j ==6 then u(i,j)=0 27 28 else u(i,j)=f(x,y) 29 30 end 31 printf ( ’%. 3 f \ t ’ , u ( i , j ) ) 32 y=y+k; 33 end 34 x=x+h 35 end 36 u 2 = u ; 37 else

258

38 39 40 41 42 43 44 45 46 47

printf ( ’ \n\n\n t = %i \n\n i \ t x \ t | \ t j −−>\t ’ ,l ) for i = 0 : 5 %i \ t ’ ,i ) printf ( ’ end printf ( ’ \n | \ t | \ t | \ t y −−>\t ’ ) for i = 0 : 5 printf ( ’%. 3 f \ t ’ , i * k ) end x=0; printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 48 49 50 51 52 53 54 55 56 57

58 59

60 61 62 63 64 65 66 67 68 69

for i = 1 : 6 y=0; printf ( ’ \n %i \t% . 3 f \ t | \ t \ t ’ , i - 1 , x ) for j = 1 : 6 if i ==1 | i ==6 then u(i,j)=0; elseif j = =1 | j ==6 then u(i,j)=0 elseif l = = 1 u(i,j)=s2*(u1(i+1,j)+u1(i-1,j)+u1(i, j+1)+u1(i,j-1)-4*u1(i,j))+2*u1(i, j) else u(i,j)=s2*(u1(i+1,j)+u1(i-1,j)+u1(i, j+1)+u1(i,j-1)-4*u1(i,j))+2*u1(i, j)-u2(i,j) end printf ( ’%. 4 f \ t ’ , u ( i , j ) ) y=y+k; end x=x+h end end if l >1 then u2=u1 end

259

70 u 1 = u ; 71 end

260

Chapter 20 Numerical Solutions of Elliptical Partial Differential Equations

Scilab code Exa 20.1 Direct Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 2 0 . 1 / / D i r e c t M et ho d / / P ag e n o . 6 8 2 clc ; clear ; close ; h=1/3; A=[-4,1,1,0;1,-4,0,1;1,0,-4,1;0,1,1,-4] x=0; for i = 1 : 4 x=x+h if i = = 4 then B(i,1)=0 else B ( i , 1 ) = - 1 * sin ( x * % p i ) ^ 2 end end

261

16 17 18 19

disp (A , ’A = ’ ) disp (B , ’B = ’ ) U = inv ( A ) * B disp (U , ’U = ’ )

Scilab code Exa 20.2 Five Point Formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

/ / E x am pl e 2 0 . 2 / / F i v e P o i n t F or mu la / / P ag e n o . 6 8 3 clc ; clear ; close ; A=[-4,1,1,0;1,0,-4,1;1,-4,0,1;0,1,1,-4]; B=[-25;-150;-25;-150]; u1 = inv ( A ) * B ; j=0;k=1 for i = 1 : 4 j=j+1; printf ( ’ \ nu%i%i = %g\n ’ , k , j , u 1 ( i ) ) if i = = 2 then j=0;k=2 end end printf ( ’ \n\n U = \n ’ ) for i = 1 : 4 printf ( ’ \n ’ ) for j = 1 : 4 if j = = 1 then u(i,j)=0 elseif j = = 4 u(i,j)=100 elseif ( i ==1 | i == 4) & j ==2 u(i,j)=25

262

27 28 29 30 31 32 33 end 34 end

elseif i = =1 | i ==4 u(i,j)=u(i,j-1)*2 else u(i,j)=u1((j-1)*2-i+2) end printf ( ’ \t%g\ t ’ , u ( i , j ) )


1 2 3 4 5 6

/ / E x am pl e 2 0 . 3 / / F i n i t e D i f f e r e n c e Method / / P ag e n o . 6 8 5 clc ; clear ; close ; printf ( ’ I t r \ t \ t U11\ t \ t U21\ t \ t U12\ t \ t U22\n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 7 for i = 1 : 4 8 for j = 1 : 4 if j = = 1 then 9 10 u(i,j)=0 elseif j = = 4 11 12 u(i,j)=100 elseif ( i ==1 | i == 4) & j ==2 13 14 u(i,j)=25 15 elseif i = =1 | i ==4 16 u(i,j)=u(i,j-1)*2 17 else u(i,j)=0 18 19 end end 20

263

21 end 22 for k = 0 : 1 7 23 %i \ t \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \n ’, printf ( ’ k,u(3,2),u(3,3),u(2,2),u(2,3)) 24 for i = 3 : - 1 : 2 for j = 2 : 3 25 26 u1(i,j)=(u(i,j+1)+u(i,j-1)+u(i-1,j)+u(i +1,j))/4 27 end 28 end 29 for i = 3 : - 1 : 2 30 for j = 2 : 3 u(i,j)=u1(i,j) 31 32 end end 33 34 end 35 disp (u , ’U = ’ )

Scilab code Exa 20.4 Seven Point Formula

1 2 3 4 5

/ / E x am pl e 2 0 . 4 / / S e v en P o i n t F or mu la / / P ag e n o . 6 8 6 clc ; clear ; close ; printf ( ’ I t r \ t \ t U1 11 \ t \ t U 21 1\ t \ t U1 21 \ t \ t U 22 1\n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 6 for i = 1 : 4 7 for j = 1 : 4 8 for k = 3 : - 1 : 1 if k = = 3 then 9 10 u(i,j,k)=100 elseif ( i ==1 | i ==4 | j ==1 | j ==4) & k 11

264

12 13 14 15

==2 u(i,j,k)=300 elseif k = = 2 u(i,j,k)=0 elseif ( i==1 | i ==4 | j ==1 | j ==4) & k ==1 u(i,j,k)=500 else u(i,j,k)=700 end

16 17 18 19 20 end 21 end 22 end 23 k =2 24 for l = 0 : 1 4 25 %i \ t \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \n ’, printf ( ’ l,u(3,2,2),u(3,3,2),u(2,2,2),u(2,3,2)) 26 for i = 3 : - 1 : 2 for j = 2 : 3 27 28 u1(i,j)=(u(i,j+1,k)+u(i,j-1,k)+u(i-1,j,k )+u(i+1,j,k)+u(i,j,k+1)+u(i,j,k-1))/6 29 end 30 end for i = 3 : - 1 : 2 31 32 for j = 2 : 3 u(i,j,2)=u1(i,j) 33 34 end end 35 36 end

Scilab code Exa 20.5 Nine Point Formula

1 / / E x am pl e 2 0 . 5

265

2 3 4 5 6

/ / N i n e P o i n t F or mu la / / P ag e n o . 6 8 8 clc ; clear ; close ; printf ( ’ I t r \ t \ t U11\ t \ t U12\ t \ t U21\ t \ t U22\n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 7 for i = 1 : 4 8 for j = 1 : 4 9 if j = = 1 then 10 u(i,j)=0 11 elseif j = = 4 u(i,j)=100 12 13 elseif ( i ==1 | i == 4) & j ==2 u(i,j)=25 14 15 elseif i = =1 | i ==4 u(i,j)=u(i,j-1)*2 16 17 else u(i,j)=0 18 19 end 20 end 21 end 22 for k = 0 : 1 7 printf ( ’ 23 %i \ t \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \ t \t% . 3 f \n ’, k,u(3,2),u(2,2),u(3,3),u(2,3)) for i = 3 : - 1 : 2 24 25 for j = 2 : 3 u1(i,j)=(u(i+1,j-1)+u(i-1,j-1)+u(i+1,j 26 +1)+u(i-1,j+1)+4*(u(i,j+1)+u(i,j-1)+u (i-1,j)+u(i+1,j)))/20 27 end 28 end 29 for i = 3 : - 1 : 2 30 for j = 2 : 3 31 u(i,j)=u1(i,j) 32 end end 33 34 end

266

35 disp (u , ’ The S o l u ti o n o f t he System i s = ’ )

Scilab code Exa 20.6 Five Point Formula

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

/ / E x am pl e 2 0 . 6 / / F i v e P o i n t F or mu la / / P ag e n o . 6 8 9 clc ; clear ; close ; h=0.25;k=0.25;y=1;x=0; deff ( ’x=f ( y ) ’ , ’ x=yˆ 3 ’ ) for i = 1 : 5 x=0; printf ( ’ \n%g\ t | ’ ,y ) for j = 1 : 5 if ( i == 1 | i = =5 ) u(i,j)=f(x) elseif j = = 5 u(i,j)=f(x) else u(i,j)=0 end x=x+k; printf ( ’ %f \ t ’ , u ( i , j ) ) end y=y-h end printf ( ’ \n\ t

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 26 x = 0 ; for j = 1 : 5 27

267

28 %g\ t ’ ,x ) printf ( ’ \ t x=x+k 29 30 end 31 printf ( ’ \n\n\n I t r \ t U11\ t U12\ t U13\ t U21\ t U22\ t

U23\ t U31\ t U32\ t U33\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 32 33 for l = 0 : 2 0 34 y=0; 35 printf ( ’

%i \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \n ’ , l , u ( 4 , 2 ) , u ( 4 , 3 ) , u ( 4 , 4 ) ,

36 37 38 39 40 41 42 43 44 45 46 47 end

u(3,2),u(3,3),u(3,4),u(2,2),u(2,3),u(2,4)) for i = 4 : - 1 : 2 y=y+k for j = 2 : 4 u1(i,j)=(u(i,j+1)+u(i,j-1)+u(i-1,j)+u(i +1,j)-h^2*y)/4 end end for i = 4 : - 1 : 2 for j = 2 : 4 u(i,j)=u1(i,j) end end

Scilab code Exa 20.7 Laplace Distribution

1 2 3 4

/ / E x am pl e 2 0 . 7 / / L a p la c e D i s t r i b u t i o n / / P ag e n o . 6 9 4 clc ; clear ; close ;

268

5 6 dr=3;r0=4;dth=%pi/4; 7 deff ( ’ y=f ( u1 , u2 , u3 , u4 ) ’ , ’y=(u1+u3+(dr ∗ ( u3−u1 ) ) /(2 ∗ r 0

)+(u2+u4) ∗ ( dr / ( r0 ∗ dth ) ) ˆ2 ) /(2 ∗ (1 +( dr /( r0 ∗ dth ) ) ˆ2 ) ) ’) // l a p l a c e d i s t r i b u t i o n 8 for i = 1 : 8 9 U(i)=0; 10 end 11 printf ( ’ I t r \ t

U1\ t

U2\ t

U3\ t

U4\ t

U5\ t

U6\ t

U7\ t U8\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 12 for l = 0 : 1 5 13 printf ( ’ \n %i ’ ,l ) for i = 1 : 8 14 15 if i = = 1 then u1(i)=f(100,U(8),40,U(i+1)) 16 17 elseif i = = 8 u1(i)=f(100,U(i-1),40,U(1)) 18 19 else 20 u1(i)=f(100,U(i-1),40,U(i+1)) 21 end 22 end 23 24 for i = 1 : 8 U(i)=u1(i) 25 26 printf ( ’ \t% . 3 f ’ , U ( i ) ) end 27 28 end

Scilab code Exa 20.8 Spherical Coordinate System

1 / / E x am pl e 2 0 . 8

269

2 3 4 5 6

/ / S p h e r i c a l C o o r d i n a te S ys te m / / P ag e n o . 6 9 7

clc ; clear ; close ; deff ( ’ y= c o t ( x ) ’ , ’y=1/t an ( x ) ’ ) dr=5;r0=50;dth=%pi/4;dfi=%pi/4;N=-10;Z=60;Nb=0;Zt =70; 7 deff ( ’ y=f ( u1 , u2 , u3 , u4 , u5 , u6 , t h0 ) ’ , ’ y = ( ( u 1+u 3 ) / d r ˆ 2+ (

u3−u1 ) /( r0 ∗ dr )+(u2+u4 ) /( r0 ∗ dth )ˆ2+(u2 ∗ co t ( th 0 ) /( r0 ˆ2∗ dth )+(u5+u6 ) /( r0 ∗ s i n ( th0 ) ∗ d f i ) ˆ2 ) ) /(2 / dr ˆ2+2/( r0 ∗ dth )ˆ2+ co t ( th 0 ) /( r0 ˆ2 ∗ d t h ) + 2 /( r 0 ∗ s i n ( t h 0 ) ∗ d f i ) ˆ 2 ) ’ ) // l a p l a c e d i s t r i b u t i o n i n s p h e r i c a l c o o r d in a t e 8 T1=40;T2=20;H1=35;H2=10;B1=5;B2=0;t1=240;t2=180;b1 =100;b2=80;h1=210;h2=150 9 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 s =[ ”T” , ”H” , ”B” , ” t ” , ” h” , ” b ” ]; 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

for i = 1 : 8 if i <4 | i >6 then T(1,i)=T1; H(1,i)=H1; B(1,i)=B1; b(1,i)=b1; t(1,i)=t1; h(1,i)=h1; else T(1,i)=T2; H(1,i)=H2; b(1,i)=b2; B(1,i)=B2; t(1,i)=t2; h(1,i)=h2; end end h(1)=0;h(2)=0; Al=[T;H;B;t;h;b] for i = 1 : 6

270

if i = = 1 then printf ( ’ T em pe ra tu re D i s t r i b u t i o n i n O ut er S p h e r e \n ’ ) printf ( ’

31 32 33

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n’) end if i = = 4 then printf ( ’ \ n Te mp er at ur e D i s t r i b u t i o n S p h e r e \n ’ ) printf ( ’

34 35 36 37

i n I n n er

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n’) 38 39 40 41 42 43 44 45 46 47 48 49 50 51

end printf ( ’ \ n P o i n t : ’ ) for j = 1 : 8 printf ( ’ \ t%s%i ’ , s ( i ) , j ) end printf ( ’ \ n Te mp er at ur e : ’ ) for j = 1 : 8 if ( j ==1 | j == 2) & i ==5 then printf ( ’ \ t%s ’ , ” ? ” ) else printf ( ’ \ t%i ’ , A l ( i , j ) ) end end printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− ’) 52 end 53 t h 0 = 1 0 ^ - 3 0 54 U h 1 = f ( 1 0 0 0 , A l ( 5 , 8 ) , A l ( 2 , 1 ) , A l ( 5 , 2 ) , A l ( 6 , 1 ) , A l ( 4 , 1 ), th0) 55 disp ( U h 1 , ’ Uh1 = ’ ) 56 t h 0 = % p i / 4 ; 57 U h 2 = f ( 1 0 0 0 , U h 1 , A l ( 2 , 2 ) , A l ( 5 , 3 ) , A l ( 6 , 2 ) , A l ( 4 , 2 ) , t h 0) 58 disp ( U h 2 , ’ Uh2 = ’ )

271

Chapter 21 Advances in Numerical Methods Using Parallel Computing Paradigm

Scilab code Exa 21.1 Parallel Bisection Method

1 2 3 4 5 6 7 8

/ / E x am pl e 2 1 . 1 // P a r a l l e l B i s e c t i o n Method / / P ag e n o . 7 2 1 clc ; close ; clear ; deff ( ’y=f ( x ) ’ , ’ y=xˆ2− c o s ( x ) ’ ) a=0;b=1;e=0.0001;i=1; printf ( ’ I t r \ t a \ tb \ th \ t \ t x 0 \ t \ t x 1 \ t \ t x 2 \ t \ t x 3 \ t \ t x 4 \ t

\ t x 5 \n\ t \ t \ t \ t \ t y 0 \ t \ t y 1 \ t \ t y 2 \ t \ t y 3 \ t \ t y 4 \ t \ t y 5 ’ ) 9 printf ( ’ \n

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 10 while ( abs ( a - b ) > = e ) 11

272

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 end

h=(b-a)/5; y(1)=f(a); x(1)=a; printf ( ’ %i \t%g\t%g\ t%f \ t%f ’ , i , a , b , h , x ( 1 ) ) for j = 2 : 6 x(j)=x(j-1)+h; y(j)=f(x(j)); if ( y ( j - 1 ) * y ( j ) < 0 ) a=x(j-1); b=x(j); end printf ( ’ \ t%f ’ , x ( j ) ) end printf ( ’ \n\ t \ t \ t \ t \ t ’ ) for j = 1 : 6 printf ( ’ %f \ t ’ , y ( j ) ) end printf ( ’ \n ’ ) i=i+1;

Scilab code Exa 21.2 Lagrange Interpolation in Parallel Computing

1 2 3 4 5 6 7 8

/ / E x am pl e 2 1 . 2 // L a gr an ge I n t e r p o l a t i o n i n P a r a l l e l Computing / / P ag e n o . 7 2 3 clc ; close ; clear ; xi=[-1,0,2,5]; yi=[9,5,3,15]; s =[ ”x=1” , ”n=4” , ” Data : ” , ” ( − 1 , 9 ) ” , ” (0 , 5 ) ” , ” ( 2 ,3 ) ” , ”

273

(5 ,1 5) ”] 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

for i = 1 : 4 printf ( ’ \ t P r o c e s s o r \ t ’ ) end printf ( ’ \n ’ ) for i = 1 : 4 N%i\ t \ t ’ ,i ) printf ( ’ \ t end printf ( ’ \n ’ ) for i = 1 : 7 for j = 1 : 4 %s\ t \ t ’ , s ( i ) ) printf ( ’ end printf ( ’ \n ’ ) end x=1;T=0; for k = 0 : 3 p=yi(k+1) for j = 0 : 3 if ( j ~ = k ) p=p*((x-xi(j+1))/(xi(k+1)-xi(j+1))) end end T=T+p; printf ( ’ \nT( %i) = %g ’ , k + 1 , p ) end printf ( ’ \n\nT = %g ’ ,T )

Scilab code Exa 21.3 Trapezoidal Rule and Simpsons Rule in Parallel Com-

puting 1 / / E x am pl e 2 1 . 3

274

2 // T r a pe z o id a l Ru le and S im ps on s Ru le i n P a r a l l e l

Computing 3 / / P ag e n o . 7 2 6 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

clc ; close ; clear ; n=8;a=0;b=8; h=(b-a)/n deff ( ’y=f ( x ) ’ , ’y =1/(1+x ) ’ ) for i = 0 : 8 x(i+1)=i; y(i+1)=f(x(i+1)) end printf ( ’ x i \ t ’ ) for i = 1 : 9 printf ( ’ %i \ t ’ , x ( i ) ) end printf ( ’ \n y i \ t ’ ) for i = 1 : 9 printf ( ’ 1 / %i\ t ’ ,i ) end

// t r a p e z o i d a l r u l e S=0; for i = 1 : 9 if ( i == 1 | i = =9 ) S=S+y(i) else S=S+2*y(i) end end S=S*h/2 printf ( ’ \n\ n T r a p e z o i d a l R u le Sum = %g ’ ,S )

/ / S i m ps o ns 1 /3 r d R ul e S=0; for i = 1 : 9 if ( i == 1 | i = =9 ) S=S+y(i) elseif ( ( ( i ) / 2 ) - fix ( ( i ) / 2 ) = = 0 )

275

39 S=S+4*y(i) else 40 41 S=S+2*y(i) end 42 43 end 44 S = S * h / 3 45 printf ( ’ \n\ n S i m p so n s 1 / 3 r d R u l e Sum = %g ’ ,S )

Scilab code Exa 21.4 Parallel Gauss Seidel Method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

/ / E x am pl e 2 1 . 4 / / P a r a l l e l Gauss−S e i d e l Method / / P ag e n o . 7 3 0 clc ; close ; clear ;

16 17 18 19 20 21

A=[3,2;6,2]; B=[2;3]; x(1)=1/4; x(2)=1/5; e=0.002; old(1)=x(1); old(2)=x(2); new(1)=old(1); new(2)=old(2); printf ( ’ \ t \ t P r oc e s s 1\ t \ t P r o c e s s 2\ n I t r \ t \ t o l d \ tnew1 \ t \ t o l d 2 \ tnew2 \n\n ’ ) %i \ t \t%g\t%g\ t \t%g\t%g\n ’ ,0 , o l d ( 1 ) , n e w ( 1 ) , printf ( ’ old(2),new(2)) for i = 1 : 4 printf ( ’ %i ’ ,i ) for j = 1 : 2 k=0; for l = 1 : j - 1

276

22 23 24 25 26 27 28 29 30 31 32 33 34 end

k=k-(A(j,l)*old(l)); end m=0; for l = j + 1 : 2 m=m-(A(j,l)*old(l)); end new(j)=(B(j)+k+m)/A(j,j) printf ( ’ \ t \t% . 5 g \t% . 5 g ’ , o l d ( j ) , n e w ( j ) )

end printf ( ’ \n ’ ) old(1)=new(1) old(2)=new(2)

Scilab code Exa 21.5 Poissons Partial Differential Equation

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 2 1 . 5 / / P o i s s o n s P a r t i a l D i f f e r e n t i a l E qu at io n / / P ag e n o . 7 3 3 clc ; clear ; close ; s =[ ” s t ” , ”nd” , ” r d ” ] for i = 4 : 2 0 s(i)=” t h ” end h = 0 . 2 5 ; deff ( ’ y=f ( x ) ’ , ’ y=xˆ 3 ’ ) ; y = 1 ; x = 0 ; for i = 1 : 6

if i ~ = 6 then printf ( ’ %g\ t | ’ ,y ) y=y-h; x=0;

277

18 19 20 21 22 23 24 25 26 27 28

for j = 1 : 5 if i ==1 | i ==5 | j ==5 then P(i,j)=f(x) else P(i,j)=0 end printf ( ’ %f \ t ’ , P ( i , j ) ) x=x+h; end else printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n\ t ’ ) 29 x=0; for j = 1 : 5 30 31 %g\ t \ t ’ ,x ) printf ( ’ x=x+h 32 33 end end 34 35 printf ( ’ \n ’ ) 36 end 37 38 printf ( ’ \n\n\n ’ ) 39 40 for l = 0 : 1 7 y=1; 41 42 if l ~ = 0 then printf ( ’ A f te r t he %i%s I t e r a t i o n 43

: \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n %i ’ , l , s ( l ) , l )

44 45 46 47 48 49 50 51

for i = 1 : 6 if i ~ = 6 then printf ( ’ \t%g ’ ,y ) y=y-h for j = 1 : 5 printf ( ’ \t% .3 f ’ , P ( i , j ) ) end else

278

52 53 54 55 56 57 58 59 60 61

x=0; printf ( ’ \ t ’ ) for j = 1 : 5 printf ( ’ \t%g ’ ,x ) x=x+h end

end printf ( ’ \n ’ ) end printf ( ’

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 62 end 63 y = 0 ; for i = 4 : - 1 : 2 64 65 y=y+h for j = 2 : 4 66 67 P1(i,j)=(P(i,j+1)+P(i,j-1)+P(i-1,j)+P(i +1,j)-h^2*y)/4 68 end 69 end 70 for i = 4 : - 1 : 2 71 for j = 2 : 4 P(i,j)=P1(i,j) 72 73 end end 74 75 end

279

Chapter 22 Numerical Methods Using Neural Networks

Scilab code Exa 22.1 MLP Algorithm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

/ / E x am pl e 2 2 . 1 //MLP A lg or it hm / / P ag e n o . 7 4 8

clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’y =1/(1+ex p(−x ) ) ’ ) Wih=[0.1,-0.3;0.3,0.4]; Who=[0.4;0.5] i=[0.2,0.6]; t=0.7; a=10; for k = 1 : 3 printf ( ’ \n\n\ n Af te r I t e r a t i o n %i : \ n\n ’ ,k ) disp ( W i h , ’ Wih = ’ ) disp ( W h o , ’Who = ’ ) a1=i*Wih; disp ( a 1 , ’ a = ’ ) h=[f(a1(1)),f(a1(2))]

280

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

disp (h , ’ h = ’ ) b1=h*Who disp ( b 1 , ’ b1 = ’ ) o=f(b1) disp (o , ’ o = ’ ) d=o*(1-o)*(t-o) disp (d , ’d = ’ ) for j = 1 : 2 e(1,j)=h(j)*(1-h(j))*d*Who(j) end disp (e , ’ e = ’ ) dWho=a*h’*d; disp ( d W h o , ’dWho =’ ) Who=Who+dWho; dWih=a*i’*e; disp ( d W i h , ’dWih = ’ ) Wih=Wih+dWih; end

Scilab code Exa 22.2 MLP

1 2 3 4 5 6 7

/ / E x am pl e 2 2 . 2 //MLP / / P ag e n o . 7 5 8 clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ3−xˆ2+x−1 ’ ) printf ( ’ I n p u t \ t \ t D e s i r e d \ t \ t N e t w o r k \ t \ t E r r o r \n\ t \ t

Output \ t \ t O ut pu t\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 8 in

= [ 0 . 7 5 7 2 , 0 . 7 6 0 1 , 0 . 7 6 2 0 , 1 . 4 8 3 1 , 1 . 4 8 7 4 , 1 . 4 9 0 0 , 2 . 0 9 1 3 ,2 . 0 9 3 4 , 2 . 1 0 0 6 ,

281

9 n

= [ - 0 . 3 9 41 , - 0 . 38 9 6 , - 0 .3 8 6 7 , 1 . 60 5 4 , 1 . 6 25 9 , 1 . 6 3 91 , 5 . 87 6 2 , 5 . 8 96 9 , 5 . 9 6 10 for i = 1 : 1 8 printf ( ’ %. 4 f \ 11 t \t% . 4 f \ t \t% . 4 f \ t \t% . 4 f \ n ’ , i n ( i ) , f (in(i)),n(i),n(i)-f(in(i))) 12 end


1 2 3 4 5 6 7

/ / E x am pl e 2 2 . 3 / / B i s e c t i o n Metho d / / P ag e n o . 7 6 4 clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=xˆ3−xˆ2+x−1 ’ ) printf ( ’ N01\ tN02 \ tN11 \ tN12 \ tN21 \ t N e t 3 1 \ tO31 \ tN41 \

tN42 \n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− n ’) 8 N01 = [ 0 , 1 , 0 . 5 , 0 . 7 5 , 0 . 8 7 5 , 0 . 9 3 8 , 0 . 9 6 9 , 0 . 9 8 4 , 0 . 9 9 2 , 0 . 9 9 6 ,0 . 9 9 8 , 0 . 9 9 9 , 1 , 9 10 11 12 13 14 15 16 17

N02(1)=2 for i = 2 : 1 3 N02(i)=1; end for i = 1 : 1 3 net31(i)=f(N01(i+1))*f(N01(i)) if n e t 3 1 ( i ) > 0 then O31(i)=1; else

282

18 19 20 21 22 23

24 25

O31(i)=0;

end N41(i)=(1-O31(i))*(N01(i))+O31(i)*N01(i+1) N42(i)=(1-O31(i))*N01(i+1)+O31(i)*N02(i) if i = = 2 then printf ( ’%. 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \t% . 3 f \ t% . 3 f \t% . 3 f \t% . 3 f \ n ’ ,0 , N 0 2 ( i ) , f ( N 0 1 ( i ) ) , N01(i+1),f(N01(i+1)),net31(i),O31(i),N41( i),N42(i)) else . 3 f \t% . 3 f \t% . 3 f \t% . 3 f printf ( ’%. 3 f \t% . 3 f \t% . 3 f \t% \t% . 3 f \t% . 3 f \n ’ , N 0 1 ( i ) , N 0 2 ( i ) , f ( N 0 1 ( i ) ) , N 0 1 ( i +1),f(N01(i+1)),net31(i),O31(i),N41(i),N42(i) )

26 end 27 28 end 29 printf ( ’ \n\ n T h er e f o r e t he s o l u t i o n

i s %. 3 f ’ , N 4 2 ( 1 3 ) )

Scilab code Exa 22.4 Hopfield Neural Network

1 2 3 4 5 6 7 8 9 10 11 12

/ / E x am pl e 2 2 . 4 / / H o p f i e l d N e u r a l N et wo rk / / P ag e n o . 7 6 6 clc ; clear ; close ; A=[1,2,1;-1,1,1;1,0,-1]; disp ( inv ( A ) , ’ I n v e rs e o f A = ’ ,A , ’A = ’ ) for i = 1 : 3 for j = 1 : 3 k=0; for l = 1 : 3 k=k+A(i,l)*A(j,l)

283

13 end T(i,j)=k; 14 15 end 16 end 17 disp (T , ’T = ’ )

Scilab code Exa 22.5 RBF Network

1 2 3 4 5 6 7

/ / E x am pl e 2 2 . 5 //RBF Netwo rk / / P ag e n o . 7 7 3 clc ; clear ; close ; deff ( ’y=f ( x ) ’ , ’ y=10∗ s i n ( x ) ’ ) printf ( ’ I n p u t \ t \ t D e s i r e d \ t \ t N e t w o r k \ t \ t E r r o r \n\ t \ t

Output \ t \ t O ut pu t\n −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−\ n ’) 8 in

= [ 0 . 7 0 5 3 , 0 . 7 0 6 0 , 0 . 7 0 9 7 , 1 . 5 0 5 6 , 1 . 5 1 0 3 , 1 . 5 3 7 7 , 2 . 2 4 8 1 ,2 . 2 5 1 4 , 2 . 2 5 9 9 , 9 n

= [ 6 . 4 8 2 8 , 6 . 4 8 8 3 , 6 . 5 1 6 4 , 9 . 9 7 8 6 , 9 . 9 8 1 6 , 9 . 9 9 4 4 , 7 . 7 9 2 6 ,7 . 7 7 1 8 , 7 . 7 1 8 0 , 10 for i = 1 : 1 8 printf ( ’ %. 4 f \ 11 t \t% . 4 f \ t \t% . 4 f \ t \t% . 4 f \ n ’ , i n ( i ) , f (in(i)),n(i),f(in(i))-n(i)) 12 end

284

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