Numerical Methods for Scientific and Eng

7/26/2019 Numerical Methods for Scientific and Eng

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Scilab Textbook Companion for

Numerical Methods For Scientific And

Engineering Computation

by M. K. Jain, S. R. K. Iyengar And R. K.Jain1

Created byM Vamsi KrishanB.Tech (pursuing)

Electronics Engineering

Visvesvaraya National Institute of TechnologyCollege TeacherMr. Nagarajun

Cross-Checked bySantosh Kumar, IITB

July 4, 2014

1Funded by a grant from the National Mission on Education through ICT,http://spoken-tutorial.org/NMEICT-Intro. This Textbook Companion and Scilabcodes written in it can be downloaded from the Textbook Companion Projectsection at the website http://scilab.in


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Scilab numbering policy used in this document and the relation to theabove 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 particularExample of the above book)

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

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Contents

List of Scilab Codes 4

2 TRANSCENDENTAL AND POLINOMIAL EQUATIONS 9

3 SYSTEM OF LINEAR ALGEBRIC EQUATIONS AND

EIGENVALUE PROBLEMS 46

4 INTERPOLATION AND APPROXIMATION 59

5 DIFFERENTIATION AND INTEGRATION 77

6 ORDINARY DIFFERENTIAL EQUATIONS INNITIAL VALUE

PROBLEMS 91

7 ORDINARY DIFFERENTIAL EQUATIONS BOUNDARY

VALUE PROBLEM 102

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Exa 3.9 solution using the inverse of the matrix . . . . . . . . 50

Exa 3.10 decomposition method . . . . . . . . . . . . . . . . . . 51Exa 3.11 inverse using LU decoposition . . . . . . . . . . . . . . 51Exa 3.12 solution by decomposition method . . . . . . . . . . . 52Exa 3.13 LU decomposition . . . . . . . . . . . . . . . . . . . . 53Exa 3.14 cholesky method . . . . . . . . . . . . . . . . . . . . . 54Exa 3.15 cholesky method . . . . . . . . . . . . . . . . . . . . . 55Exa 3.21 jacobi iteration method . . . . . . . . . . . . . . . . . 56Exa 3.22 solution by gauss siedal method. . . . . . . . . . . . . 57Exa 3.27 eigen vale and eigen vector . . . . . . . . . . . . . . . 57Exa 4.3 linear interpolation polinomial . . . . . . . . . . . . . 59Exa 4.4 linear interpolation polinomial . . . . . . . . . . . . . 60

Exa 4.6 legrange linear interpolation polinomial . . . . . . . . 61Exa 4.7 polynomial of degree two . . . . . . . . . . . . . . . . 61Exa 4.8 solution by quadratic interpolation . . . . . . . . . . . 62Exa 4.9 polinomial of degree two. . . . . . . . . . . . . . . . . 63Exa 4.15 forward and backward difference polynomial . . . . . . 63Exa 4.20 hermite interpolation . . . . . . . . . . . . . . . . . . 64Exa 4.21 piecewise linear interpolating polinomial . . . . . . . . 65Exa 4.22 piecewise quadratic interpolating polinomial . . . . . . 66Exa 4.23 piecewise cubical interpolating polinomial . . . . . . . 67Exa 4.31 linear approximation . . . . . . . . . . . . . . . . . . . 68

Exa 4.32 linear polinomial approximation . . . . . . . . . . . . 69Exa 4.34 least square straight fit . . . . . . . . . . . . . . . . . 70Exa 4.35 least square approximation . . . . . . . . . . . . . . . 71Exa 4.36 least square fit . . . . . . . . . . . . . . . . . . . . . . 71Exa 4.37 least square fit . . . . . . . . . . . . . . . . . . . . . . 72Exa 4.38 gram schmidt orthogonalisation . . . . . . . . . . . . . 73Exa 4.39 gram schmidt orthogonalisation . . . . . . . . . . . . . 74Exa 4.41 chebishev polinomial . . . . . . . . . . . . . . . . . . . 76Exa 5.1 linear interpolation. . . . . . . . . . . . . . . . . . . . 77Exa 5.2 quadratic interpolation . . . . . . . . . . . . . . . . . 78Exa 5.10 jacobian matrix of the given system . . . . . . . . . . 78

Exa 5.11 solution by trapizoidal and simpsons . . . . . . . . . . 79Exa 5.12 integral approximation by mid point and two point . . 79Exa 5.13 integral approximation by simpson three eight rule . . 80Exa 5.15 quadrature formula. . . . . . . . . . . . . . . . . . . . 81Exa 5.16 gauss legendary three point method . . . . . . . . . . 82

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Exa 5.17 gauss legendary method . . . . . . . . . . . . . . . . . 82

Exa 5.18 integral approximation by gauss chebishev . . . . . . . 83Exa 5.20 integral approximation by gauss legurre method. . . . 84Exa 5.21 integral approximation by gauss legurre method. . . . 85Exa 5.22 integral approximation by gauss legurre method. . . . 85Exa 5.26 composite trapizoidal and composite simpson . . . . . 86Exa 5.27 integral approximation by gauss legurre method. . . . 87Exa 5.29 double integral using simpson rule . . . . . . . . . . . 88Exa 5.30 double integral using simpson rule . . . . . . . . . . . 89Exa 6.3 solution to the system of equations . . . . . . . . . . . 91Exa 6.4 solution ti the IVP . . . . . . . . . . . . . . . . . . . . 92Exa 6.9 euler method to solve the IVP . . . . . . . . . . . . . 93

Exa 6.12 solution ti IVP by back euler method . . . . . . . . . 93Exa 6.13 solution ti IVP by euler mid point method. . . . . . . 94Exa 6.15 solution ti IVP by taylor expansion. . . . . . . . . . . 94Exa 6.17 solution ti IVP by modified euler cauchy and heun . . 95Exa 6.18 solution ti IVP by fourth order range kutta method. . 96Exa 6.20 solution to the IVP systems . . . . . . . . . . . . . . . 96Exa 6.21 solution ti IVP by second order range kutta method . 97Exa 6.25 solution ti IVP by third order adamsbashfort meth . . 98Exa 6.27 solution ti IVP by third order adams moult method . 99Exa 6.32 solution by numerov method . . . . . . . . . . . . . . 100

Exa 7.1 solution to the BVP by shooting method. . . . . . . . 102Exa 7.3 solution to the BVP by shooting method. . . . . . . . 103Exa 7.4 solution to the BVP by shooting method. . . . . . . . 104Exa 7.5 solution to the BVP . . . . . . . . . . . . . . . . . . . 105Exa 7.6 solution to the BVP by finite differences . . . . . . . . 107Exa 7.11 solution to the BVP by finite differences . . . . . . . . 108AP 1 shooting method for solving BVP. . . . . . . . . . . . 110AP 2 euler method . . . . . . . . . . . . . . . . . . . . . . . 111AP 3 eigen vectors and eigen values. . . . . . . . . . . . . . 111AP 4 adams bashforth third order method . . . . . . . . . . 113AP 5 newton raphson method . . . . . . . . . . . . . . . . . 113

AP 6 euler cauchy solution to the simultanioys equations . . 114AP 7 simultaneous fourth order range kutta . . . . . . . . . 115AP 8 fourth order range kutta method . . . . . . . . . . . . 116AP 9 modified euler method . . . . . . . . . . . . . . . . . . 117AP 10 euler cauchy or heun . . . . . . . . . . . . . . . . . . . 118

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AP 11 taylor series. . . . . . . . . . . . . . . . . . . . . . . . 118

AP 12 factorial . . . . . . . . . . . . . . . . . . . . . . . . . . 119AP 13 mid point nmethod. . . . . . . . . . . . . . . . . . . . 119AP 14 back euler method . . . . . . . . . . . . . . . . . . . . 120AP 15 composite trapizoidal rule . . . . . . . . . . . . . . . . 121AP 16 simpson rule . . . . . . . . . . . . . . . . . . . . . . . 122AP 17 simpson rule . . . . . . . . . . . . . . . . . . . . . . . 123AP 18 approximation to the integral by simpson method. . . 124AP 19 integration by trapizoidal method. . . . . . . . . . . . 124AP 20 jacobian of a given matrix . . . . . . . . . . . . . . . . 124AP 21 linear interpolating polinomial . . . . . . . . . . . . . 125AP 22 iterated interpolation . . . . . . . . . . . . . . . . . . 125

AP 23 newton divided differences interpolation order two . . 125AP 24 legrange fundamental polynomial . . . . . . . . . . . . 125AP 25 legrange interpolation . . . . . . . . . . . . . . . . . . 126AP 26 quadratic approximation. . . . . . . . . . . . . . . . . 127AP 27 straight line approximation . . . . . . . . . . . . . . . 127AP 28 aitken interpolation . . . . . . . . . . . . . . . . . . . 128AP 29 newton divided differences interpolation . . . . . . . . 128AP 30 hermite interpolation . . . . . . . . . . . . . . . . . . 128AP 31 newton backward differences polinomial . . . . . . . . 129AP 32 gauss jorden . . . . . . . . . . . . . . . . . . . . . . . 131

AP 33 gauss elimination with pivoting . . . . . . . . . . . . . 131AP 34 gauss elimination . . . . . . . . . . . . . . . . . . . . . 132AP 35 eigen vector and eigen value. . . . . . . . . . . . . . . 134AP 36 gauss siedel method . . . . . . . . . . . . . . . . . . . 135AP 37 jacobi iteration method . . . . . . . . . . . . . . . . . 136AP 38 back substitution . . . . . . . . . . . . . . . . . . . . . 136AP 39 cholesky method . . . . . . . . . . . . . . . . . . . . . 137AP 40 forward substitution . . . . . . . . . . . . . . . . . . . 137AP 41 L and U matrices. . . . . . . . . . . . . . . . . . . . . 138AP 42 newton raphson method . . . . . . . . . . . . . . . . . 138AP 43 four itterations of newton raphson method. . . . . . . 139

AP 44 secant method . . . . . . . . . . . . . . . . . . . . . . 139AP 45 regula falsi method. . . . . . . . . . . . . . . . . . . . 140AP 46 four iterations of regula falsi method . . . . . . . . . . 140AP 47 four iterations of secant method . . . . . . . . . . . . 140AP 48 five itterations by bisection method. . . . . . . . . . . 141

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AP 49 bisection method . . . . . . . . . . . . . . . . . . . . . 142

AP 50 solution by newton method given in equation 2.63 . . 143AP 51 solution by secant method given in equation 2.64 . . . 144AP 52 solution by secant method given in equation 2.65 . . . 144AP 53 solution to the equation having multiple roots . . . . . 145AP 54 solution by two iterations of general iteration . . . . . 145AP 55 solution by aitken method . . . . . . . . . . . . . . . . 146AP 56 solution by general iteration. . . . . . . . . . . . . . . 146AP 57 solution by multipoint iteration given in equation 33 . 147AP 58 solution by multipoint iteration given in equation 31 . 148AP 59 solution by chebeshev method. . . . . . . . . . . . . . 148AP 60 solution by five iterations of muller method . . . . . . 149

AP 61 solution by three iterations of muller method . . . . . 150

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Chapter 2

TRANSCENDENTAL AND

POLINOMIAL EQUATIONS

Scilab code Exa 2.1 intervals containing the roots of the equation

1 / / The e q u a t i on8x 3 12x

22x+3==0ha s t h re e r e a l

r o o t s .2 // t he gr a ph o f

t h i s f u n c t io nc an b eo b s e r v e d h e r e .

3 xset ( window ,0);4 x = - 1 : . 0 1 : 2 . 5 ;

//

d e f i n i n g t h e r an ge o f x .5 deff ( [ y]= f (x ) , y=8x 312x22x+3 ) ;// d e f i n i n g t he c u n ct i o n

6 y = feval ( x , f ) ;

7

9


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8 a = gca () ;

910 a . y _ l o c at i o n = o r i g i n ;11

12 a . x _ l o c at i o n = o r i g i n ;13 plot ( x , y )

// i n s t r u c t i o n t o p l o t t he gr ap h14

15 t i t l e ( y = 8x 3 12x 22x+3 )16

17 // from t he abov e p l o t we c an i n f r e t ha t t he

f u n c t i o n h as r o o t s b et we en18 / / t h e i n t e r v a l s ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 2 ) .

Scilab code Exa 2.2 interval containing the roots

1 / / The e q u a t i on

c o s ( x )x%ex==0 h as r e a lr o o t s .

2 // t he gr a ph o f t h i s f u n c t io nc an b eo b s e r v e d h e r e .

3 xset ( window ,1);4 x = 0 : . 0 1 : 2 ;

//

d e f i n i n g t h e r an ge o f x .5 deff ( [ y]= f (x ) , y= c os ( x )x%ex ) ;// d e f i n i n g t he c u n ct i o n .

6 y = feval ( x , f ) ;

7

10


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8 a = gca () ;



// i n s t r u c t i o n t o p l o t t he gr ap h14 t i t l e ( y = c os ( x )x%ex )15

16 // from t he abov e p l o t we c an i n f r e t ha t t hef u n c t i o n h as r o o t b etw een

17 / / t he i n t e r v a l ( 0 , 1 )

check AppendixAP 49for dependency:

Vbisection.sce


Vbisection5.sce

Scilab code Exa 2.3 solution to the eq by bisection method

1 // The e q ua t i on x35x+1==0ha s r e a lr o o t s .

2 // t he gr a ph o f t h i s f u n c t io nc an b e

o b s e r v e d h e r e .3 xset ( window ,2);4 x = - 2 : . 0 1 : 4 ;

//d e f i n i n g t h e r an ge o f x .

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5 deff ( [ y]= f (x ) ,y=x35x+1 ) ; //

d e f i n i n g t he c u nc t io n .6 y = feval ( x , f ) ;7

8 a = gca () ;

9



// i n s t r u c t i o n t o p l o t t he gr ap h

14 t i t l e ( y = x35x+1 )15

16 // from t he abov e p l o t we c an i n f r e t ha t t hef u n c t i o n h as r o o t s b et we en

17 // t he i n t e r v a l s ( 0 , 1 ) , ( 2 , 3 ) .18 // s i n c e we h ave be en a sk ed f o r t he s m a l l e s t

p o s i t i v e r oo t o f t h e e q ua ti on ,19 / / we a re i n t r e s t e d on t h e i n t e r v a l ( 0 , 1 )20 / / a = 0; b =1 ,21

22 // we c a l l a u se r

d e fi n e d f u nc t i o n b i s e c t i o n s o a st o f i n d t he a pp ro xi ma te23 / / r o o t o f t h e e qu at i o n w i t h a d e f i ne d p e r m i s s i b l e

e r r o r .24

25 b i s e c t i o n ( 0 , 1 , f )

26

27 / / s i n c e i n t he ex ampl e 2 . 3 we h ave bee n a sk ed t op er f o rm 5 i t t e r a t i o n s

28 // t h e a p p r o x i m a t e r oo t a f t e r 5 i t e r a t i o n s can b eo b s e r v e d .

2930

31

32 b i s e c t i o n 5 ( 0 , 1 , f )

33

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34

35 // h e n ce t h e a p p ro xi m a te r o o t a f t e r 5 i t e r a t i o n s i s0 . 2 03 12 5 w i t i n t he p e r m i s s i b l e e r r o r o f 104 ,


Vbisection.sce


Vbisection5.sce

Scilab code Exa 2.4 solution to the eq by bisection method

1 / / The e q u a t i onc o s ( x )x%ex==0 h as r e a lr o o t s .

2 // t he gr a ph o f


3 xset ( window ,3);4 x = 0 : . 0 1 : 2 ;


5 deff ( [ y]= f (x ) , y= c os ( x )x%ex ) ;// d e f i n i n g t he c u n ct i o n .

6 y = feval ( x , f ) ;

7

8 a = gca () ;9


12 a . x _ l o c at i o n = o r i g i n ;

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13 plot ( x , y )



17 / / t he i n t e r v a l ( 0 , 1 )18

19

20 / / a = 0; b =1 ,21

22 // we c a l l a u se rd e fi n e d f u nc t i o n b i s e c t i o n s o a st o f i n d t he a pp ro xi ma te

23 / / r o o t o f t h e e qu at i o n w i t h a d e f i ne d p e r m i s s i b l ee r r o r .

24

25 b i s e c t i o n ( 0 , 1 , f )

26

27 / / s i n c e i n t he ex ampl e 2 . 4 we h ave bee n a sk ed t op e r f o r m 5 i t t e r a t i o n s ,

28

29 b i s e c t i o n 5 ( 0 , 1 , f )

30

31

32 // h e n ce t h e a p p ro xi m a te r o o t a f t e r 5 i t e r a t i o n s i s0 . 5 15 62 5 w i t i n t he p e r m i s s i b l e e r r o r o f 104 ,


regulafalsi4.sce


secant4.sce

Scilab code Exa 2.5 solution to the given equation

14


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1 // The e q ua t i on x

3

5

x+1==0ha s r e a lr o o t s .


3 xset ( window ,4);4 x = - 2 : . 0 1 : 4 ;


5 deff ( [ y]= f (x ) ,y=x35x+1 ) ; //d e f i n i n g t he c u nc t io n .

6 y = feval ( x , f ) ;

7

8 a = gca () ;

9



// i n s t r u c t i o n t o p l o t t he gr ap h14 t i t l e ( y = x35x+1 )15


17 // t he i n t e r v a l s ( 0 , 1 ) , ( 2 , 3 ) .18 // s i n c e we have be en g iv en t h e i n t e r v a l t o be

c o n si d e r e d a s ( 0 , 1 )19 / / a = 0; b =1 ,20

2122 // S o l u ti o n by

s e c a n t m et ho d23

24

15


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Vsecant.sce


regulafalsi.sce

Scilab code Exa 2.6 solution by secant and regula falsi

1 / / The e q u a t i o nc o s ( x )x%ex==0 ha s r e a l



3 xset ( window ,3);4 x = 0 : . 0 1 : 2 ;



6 y = feval ( x , f ) ;

7

8 a = gca () ;

9



// i n s t r u c t i o n t o p l o t t he gr ap h14 t i t l e ( y = c os ( x )x%ex )

17


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15

16 // from t he abov e p l o t we c an i n f r e t ha t t hef u n c t i o n h as r o o t b etw een17 / / t he i n t e r v a l ( 0 , 1 )18

19

20 / / a = 0; b =1 ,21

22

23 // S o l u ti o n bys e c a n t m et ho d

24

2526

27

28

29 / / s i n c e i n t h e ex ampl e 2 . 6 we h ave no s p e c i f i c a t i o no f t h e no . o f i t t e r a t i o n s ,

30 // we d e f i n e a f u n c t i o n s ec an t and e x e c u t e i t .31

32

33

34 s e c a n t ( 0 , 1 , f ) // we c a l l a u se r

d e f i n e df u n c ti o n s ec an t s o a s t o f i n d t he a pp ro xi ma te35 / / r o o t o f t h e e qu at i o n w i t h a d e f i ne d p e r m i s s i b l e

e r r o r .36

37

38

39 // h en ce t he a pp ro xi ma te r o o t o c cu r ed i n s e c a ntmethod w i t i n t he p e r m i s s i b l e e r r o r o f 105 i s ,

40

41

4243 // s o l u t i o n by r e g u l a r

f a l s i met hod44

45

18


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46

4748

49 / / s i n c e i n t h e ex ampl e 2 . 6 we h ave no s p e c i f i c a t i o no f t h e no . o f i t t e r a t i o n s ,

50

51

52 r e g u l a f a l s i ( 0 , 1 , f ) // we c a l l a u se rd e f i n e d f u n c t i o n r e g u l a r f a l s i s o a s t o f i n d t h e

a p p r o x i m a t e53 / / r o o t o f t h e e qu at i o n w i t h a d e f i ne d p e r m i s s i b l e

e r r o r .


Vnewton4.sce

Scilab code Exa 2.7 solution to the equation by newton raphson method



3 xset ( window ,6);4 x = - 2 : . 0 1 : 4 ;


5 deff ( [ y]= f (x ) ,y=x35x+1 ) ; //d e f i n i n g t he f u n c t i o n .

6 deff ( [ y ] = f p ( x ) , y=3x25 ) ;

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7 y = feval ( x , f ) ;

89 a = gca () ;

10







23 / / s i n c e i n t he ex ampl e 2 . 7 we h ave bee n a sk ed t op e r f o r m 4 i t t e r a t i o n s ,


25

26

27 n e w t o n 4 ( 0 . 5 , f , f p )

28

29

30 // h e n ce t h e a p p ro xi m a te r o o t a f t e r 4 i t e r a t i o n s i s0 . 2 01 64 0 w i t i n t he p e r m i s s i b l e e r r o r o f 10 15 ,


Vnewton4.sce

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1 // The e q ua t i on x317==0 ha st h r e e r e a lr o o t s .


3 xset ( window ,7);4 x = - 5 : . 0 0 1 : 5 ;


5 deff ( [ y]= f (x ) ,y=x317 ) ; //d e f i n i n g t he c u nc t io n .

6 deff ( [ y ] = f p ( x ) , y=3x2 ) ;7 y = feval ( x , f ) ;8

9 a = gca () ;

10




15 t i t l e ( y = x3

17 )1617 / / from t he abov e p l o t we c an i n f r e t ha t t h e

f u n c t i o n h as r o o t b etw een18 // t he i n t e r v a l ( 2 , 3 ) .

21


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19

2021

22 / / s o l u t i o n by n ew to n r a ph s o n s m et ho d23

24

25

26 // s i n c e i n exa mp le no . 2 . 8 we h ave b ee n a sk ed t op e r f o r m 4 i t e r a t i o n s , we d e f i n e a f u c t i o nn ew to n4 w h ic h d o e s n ew to n r a p hs o n s m et ho d o f f i n d i n g a pp ro xi ma te r o ot upto 4 i t e r a t i o n s ,

27

2829

30 n e w t o n 4 ( 2 , f , f p ) // c a l l i n g t he pred e f i n e d f u n c t i o n n ewton4 .


Vnewton.sce




o b s e r v e d h e r e .3 xset ( window ,8);4 x = - 1 : . 0 0 1 : 2 ;


22


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5 deff ( [ y]= f (x ) , y= c os ( x )x%ex ) ;

// d e f i n i n g t he c u n ct i o n .6 deff ( [ y ] = f p ( x ) , y=s i n ( x )x%ex%ex ) ;7 y = feval ( x , f ) ;

8

9 a = gca () ;

10





18 / / t he i n t e r v a l ( 0 , 1 )19

20

21 / / a = 0; b =1 ,22

23

24

25 / / s o l u t i o n by n ew to n r a ph so n s m ethodw i t h a p e r m i s s i b l e e r r o r o f 10 8.

26

27

28 // we c a l l a u se rd e f i n e d f u n c t i o n newton s o a s t of i n d t h e a p pr o xi m at e

29 / / r o o t o f t h e e qu at i o n w it hi n t h e d e f i n e dp e r m i s s i b l e e r r o r l i m i t .

30

31 n e w t o n ( 1 , f , f p )32

33

34

35

23


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36

37 / / h en ce t he a pp ro xi ma te r o ot w i t i n t he p e r m i s s i b l ee r r o r o f 1 08 i s 0 . 5 1 7 7 5 7 4 .


muller3.sce

Scilab code Exa 2.11 solution to the given equation by muller method

1 // The e q ua t i on x35x+1==0 h asr e a l r o o t s .

2 // t h e g r aph o f t h i sf u n c t i o n can beo b s e r v e d h e r e .

3 xset ( window ,10);4 x = - 2 : . 0 1 : 4 ; //

d e f i n i n g t he r a ng e o f x .5 deff ( [ y]= f (x ) ,y=x35x+1 ) ; //

d e f i n i n g t he c u nc t io n .6 y = feval ( x , f ) ;7

8 a = gca () ;

9


12 a . x _ l o c at i o n = o r i g i n ;13 plot ( x , y ) //

i n s t r u c t i o n t o p l o t t he g r ap h14 t i t l e ( y = x35x+1 )


f u n c t i o n h as r o o t s b et we en17 // t he i n t e r v a l s ( 0 , 1 ) , ( 2 , 3 ) .

24


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18 // s i n c e we h ave be en a sk ed f o r t he s m a l l e s t

p o s i t i v e r oo t o f t h e e q ua ti on ,19 / / we a re i n t r e s t e d on t h e i n t e r v a l ( 0 , 1 )20

21

22 // s o l l u t i o n by m u ll e r method t o 3 i t e r a t i o n s.

23

24 m u l l e r 3 ( 0 , . 5 , 1 , f )


muller5.sce

Scilab code Exa 2.12 solution by five itrations of muller method

1 / / The e q u a t i onc o s ( x )x%ex

==0 h as r e a lr o o t s .2 // t he gr a ph o f


3 xset ( window ,8);4 x = - 1 : . 0 0 1 : 2 ;


5 deff ( [ y]= f (x ) , y= c os ( x )x%ex ) ;

// d e f i n i n g t he c u n ct i o n .6 deff ( [ y ] = f p ( x ) , y=s i n ( x )x%ex%ex ) ;7 y = feval ( x , f ) ;

8

9 a = gca () ;

25


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10

11 a . y _ l o c at i o n = o r i g i n ;1213 a . x _ l o c at i o n = o r i g i n ;14 plot ( x , y )



18 / / t he i n t e r v a l ( 0 , 1 )

1920

21 // s o l l u t i o n by m u ll e r method t o 5 i t e r a t i o n s.

22

23

24 m u l l e r 5 ( - 1 , 0 , 1 , f )


chebyshev.sce

Scilab code Exa 2.13 solution by chebeshev method

1 // The e q ua t i on x35x+1==0ha s r e a l



26


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3 xset ( window ,12);

4 x = - 2 : . 0 1 : 4 ; //d e f i n i n g t h e r an ge o f x .


6 deff ( [ y ] = f p ( x ) , y=3x25 ) ;7 deff ( [ y]= fpp (x ) , y=6x ) ;8 y = feval ( x , f ) ;

9

10 a = gca () ;

11







24

25 // s o l u t i o n by c h e b y s h e v method26


2829

30 c h e b y s h e v ( 0 . 5 , f , f p )

31

32

27


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33 // h en ce t he a pp ro xi ma te r o ot w i t i n t he p e r m i s s i b l e

e r r o r o f 10

15 i s . 2 01 6 40 2 ,check AppendixAP 59for dependency:

chebyshev.sce


12

3 / / The e q u a t i on1/ x7==0 has a

r e a l r o ot .4 // t he gr a ph o f


5 xset ( window ,13);6 x = 0 . 0 0 1 : . 0 0 1 : . 2 5 ;


7 deff ( [ y]= f (x ) , y=1/x7 ) ; //d e f i n i n g t he f u n c t i o n .

8 deff ( [ y ] = f p ( x ) , y=1/x2 ) ;9 y = feval ( x , f ) ;

10

11 a = gca () ;

12

13 a . y _ l o c at i o n = o r i g i n ;



28


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17 t i t l e ( y =1/ x7 )


f u n c t i o n h as r o o t s b et we en20 // t he i n t e r v a l ( 0 , 2 / 7 )21

22

23 / / s o l u t i o n by c h eb y s he v metho d24

25

26 c h e b y s h e v ( 0 . 1 , f , f p ) // c a l l i n g t hepred e f i n e d f u n c t i o n c h eby s he v t o f i n d t he

a pp ro xi ma te r o o t i n t he r an ge o f ( 0 , 2 / 7 ) .


chebyshev.sce





4 x = - 1 : . 0 0 1 : 2 ;//

d e f i n i n g t h e r an ge o f x .5 deff ( [ y]= f (x ) , y= c os ( x )x%ex ) ;

// d e f i n i n g t he c u n ct i o n .

29


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6 deff ( [ y ] = f p ( x ) , y=s i n ( x )x%ex%ex ) ;

7 deff ( [ y]= fpp (x ) , y=

c o s ( x )

x

%ex

2

%ex ) ;8 y = feval ( x , f ) ;9

10 a = gca () ;

11




16 t i t l e ( y = c os ( x )x%ex )17


19 / / t he i n t e r v a l ( 0 , 1 )20

21

22 / / a = 0; b =1 ,23

24

25

26 // s o l u t i o n by c he by sh ev w it h ap e r m i s s i b l e e r r o r o f 10 15.

27

28 // we c a l l a u se rd e f i n e d f u n c t i o n c he by sh ev s o a st o f i n d t he a pp ro xi ma te


30

31 c h e b y s h e v ( 1 , f , f p )

32

3334

35 / / h en ce t he a pp ro xi ma te r o ot w i t i n t he p e r m i s s i b l ee r r o r o f 1015 i s

30


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multipoint_iteration31.sce


multipoint_iteration33.sce

Scilab code Exa 2.16 multipoint iteration




4 x = - 2 : . 0 1 : 4 ;//

d e f i n i n g t h e r an ge o f x .5 deff ( [ y]= f (x ) ,y=x35x+1 ) ; //

d e f i n i n g t he f u n c t i o n .6 deff ( [ y ] = f p ( x ) , y=3x25 ) ;7 deff ( [ y]= fpp (x ) , y=6x ) ;8 y = feval ( x , f ) ;

9

10 a = gca () ;

11



31


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15 plot ( x , y )




p o s i t i v e r oo t o f t h e e q ua ti on ,21 / / we a re i n t r e s t e d on t h e i n t e r v a l ( 0 , 1 )22 / / a = 0; b =1 ,

2324

25 // s o l u t i o n by m u l t i p o i n t i t e r a t i o nmethod

26


28

29

30 m u l t i p o i n t _ i t e r a t i o n 3 1 ( 0 . 5 , f , f p )

31

32 // h en ce t he a pp ro xi ma te r o ot w i t i n t he p e r m i s s i b l ee r r o r o f 1015 i s . 2 01 6 40 ,

33

34

35

36 m u l t i p o i n t _ i t e r a t i o n 3 3 ( 0 . 5 , f , f p )

37

38 // h en ce t he a pp ro xi ma te r o ot w i t i n t he p e r m i s s i b l ee r r o r o f 1015 i s . 2 01 6 40 ,

check AppendixAP 57for dependency:multipoint_iteration33.sce

32


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Scilab code Exa 2.17 multipoint iteration



3 xset ( window ,8);4 x = - 1 : . 0 0 1 : 2 ;


5 deff ( [ y]= f (x ) , y= c os ( x )x%ex ) ;// d e f i n i n g t he f u n c t i o n .

6 deff ( [ y ] = f p ( x ) , y=s i n ( x )x%ex%ex ) ;7 deff ( [ y]= fpp (x ) , y=c o s ( x )x%ex2%ex ) ;8 y = feval ( x , f ) ;

9

10 a = gca () ;

11

12 a . y _ l o c at i o n = o r i g i n

;

13



18 / / from t he abov e p l o t we c an i n f r e t ha t t h ef u n c t i o n h as r o o t b etw een

19 / / t he i n t e r v a l ( 0 , 1 )

2021

22 / / a = 0; b =1 ,23

24

33


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25

26 // s o l u t i o n by m u l t i p o i n t i t e r a t i o nmethod u s i ng t he f or m ul a g i ve n i neq ua ti on no . 2 . 3 3 .

27

28 // we c a l l a u se rd e fi n e d f u nc t i o n m u l t i p o i n t i t e r a t i o n 3 3 s o a s t o f i n d t h ea p p r o x i m a t e


30

31 m u l t i p o i n t _ i t e r a t i o n 3 3 ( 1 , f , f p )

3233

34 / / h en ce t he a pp ro xi ma te r o ot w i t i n t he p e r m i s s i b l ee r r o r o f 1 05 i s 0 . 5 1 7 7 5 7 4 .


generaliteration.sce

Scilab code Exa 2.23 general iteration

1 / / The e q u a t i on3x3+4x2+4x+1==0 ha st h r e e r e a lr o o t s .


o b s e r v e d h e r e .3 xset ( window ,22);4 x = - 1 . 5 : . 0 0 1 : 1 . 5 ;


34


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5 deff ( [ y]= f (x ) , y=3x3+4x2+4x+1 ) ;

// d e f i n i n g t he c u n ct i o n6 y = feval ( x , f ) ;7

8 a = gca () ;

9




1415 t i t l e ( y =3x3+4x2+4x+1 )16


18 / / t h e i n t e r v a l ( 1 ,0 ) ,19

20 x 0 = - . 5 ; / / i n i t i a l a p p r o x i m a t i o n21

22

23 // l e t t h e i t e r a t i v e f u nc t i o n g ( x ) b e x+A

( 3

x3+4

x2+4x + 1) =g ( x ) ;24

25 // gp ( x )=(1+A ( 9 x2+8x +4) )26 / / we need t o c ho os e a v al ue f o r A , whi ch makes ab s

( g p ( x 0 ) )


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abs ( gp( x 0) ) ,

36 / / we c an i n f e r t h at f o r t h e v a l e s o f A i n t h er an ge ( . 8 , 0) g ( x ) w i l l be g i v i n g a c on ve rg i n gs o l u t i o n ,

37

38 / / h en ce d e l i b e r a t e l e we c ho os e a t o be 0 . 5 ,39

40 A = - 0 . 5 ;

41

42 deff ( [ y ] = g ( x ) , y= x 0 . 5( 3 x3+4x2+4x+1) ) ;43 deff ( [ y ] = gp( x ) , y= 1 0. 5( 9 x2+8x+4) ) ; //

h en ce d e f i n i n g g ( x ) and gp ( x ) ,

44 g e n e r a l i t e r a t i o n ( x 0 , g , g p )


aitken.sce

Scilab code Exa 2.24.1 solution by general iteration and aitken method

1 // The e q ua t i on x35x+1==0 h asr e a l r o o t s .


3 xset ( window ,2);4 x = - 2 : . 0 1 : 4 ; //

d e f i n i n g t he r a ng e o f x .


6 y = feval ( x , f ) ;

7

8 a = gca () ;

36


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9

10 a . y _ l o c at i o n = o r i g i n ;1112 a . x _ l o c at i o n = o r i g i n ;13 plot ( x , y ) //

i n s t r u c t i o n t o p l o t t he g r ap h14 t i t l e ( y = x35x+1 )15



p o s i t i v e r oo t o f t h e e q ua ti on ,19 / / we a re i n t r e s t e d on t h e i n t e r v a l ( 0 , 1 )20

21 x 0 = . 5 ;

22

23 // s o l u t i o n u s in g l i n e a r i t e r a t i o n methodf o r t he f i r s t two i t e r a t i o n s and a it ke n s p r oc e ss two t im es f o r t he t h i r di t e r a t i o n .

24

25 deff ( [ y ] = g ( x )

, y = 1 /5

(x 3+1) ) ;

26 deff ( [ y ] = gp( x ) , y = 1 /5(3x 2 ) ) ;27

28

29 g e n e r a l i t e r a t i o n 2 ( x 0 , g , g p )

30

31

32 / / f rom t he abov e i t e r a t i o n s p er fo rm ed we c an i n f e rt h a t

33 x 1 = 0 . 2 2 5 ;

34 x 2 = 0 . 2 0 2 2 7 8 ;

3536

37

38

39 a i t k e n ( x 0 , x 1 , x 2 , g ) // c a l l i n g t h e a i t k e n

37


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method f o r one i t e r a t i o n


generaliteration2.sce


aitken.sce

Scilab code Exa 2.24.2 solution by general iteration and aitken method

1 / / The e q u a t i on x%ex==0 ha s r e a l r o o t s .


3 xset ( window ,24);4 x = - 3 : . 0 1 : 4 ;

//d e f i n i n g t h e r an ge o f x .5 deff ( [ y]= f (x ) , y=x%ex ) ; //

d e f i n i n g t he c u nc t io n .6 y = feval ( x , f ) ;

7

8 a = gca () ;

9



13 plot ( x , y )

// i n s t r u c t i o n t o p l o t t he gr ap h14 t i t l e ( y = x%ex )15

38


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f u n c t i o n h as r o o t b etw een17 / / t he i n t e r v a l ( 0 , 1 )18

19 x 0 = 1 ;

20

21 // s o l u t i o n u s in g l i n e a r i t e r a t i o n methodf o r t he f i r s t two i t e r a t i o n s and a i t ke n s p r o c e s s two t im es f o r t he t h i r di t e r a t i o n .

22

23

2425 deff ( [ y ] = g ( x ) , y=%ex ) ;26 deff ( [ y ] = gp( x ) , y=%ex ) ;27

28

29 g e n e r a l i t e r a t i o n 2 ( x 0 , g , g p )

30

31


33 x 1 = 0 . 3 6 7 8 7 9 ;

34 x 2 = 0 . 6 9 2 2 0 1 ;

35

36

37

38

39 a i t k e n ( x 0 , x 1 , x 2 , g ) // c a l l i n g t h e a i t k e nmethod f o r one i t e r a t i o n




aitken.sce


39


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Scilab code Exa 2.25 solution by general iteration and aitken method



3 xset ( window ,25);4 x = 0 : . 0 1 : 2 ;



6 y = feval ( x , f ) ;

7

8 a = gca () ;

9



// i n s t r u c t i o n t o p l o t t he gr ap h14 t i t l e ( y = c os ( x )x%ex )

1516 / / from t he abov e p l o t we c an i n f r e t ha t t h e

f u n c t i o n h as r o o t b etw een17 / / t he i n t e r v a l ( 0 , 1 )18

40


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19 x 0 = 0 ;

2021 // s o l u t i o n u s in g l i n e a r i t e r a t i o n method

f o r t he f i r s t two i t e r a t i o n s and a it ke n s p r oc e ss two t im es f o r t he t h i r di t e r a t i o n .

22

23 deff ( [ y ] = g ( x ) , y=x +1/2( c o s ( x )x%ex ) ) ;24 deff ( [ y ] = gp( x ) , y=1+1/2( s i n ( x )x%ex

%ex ) ) ;25

26

27 g e n e r a l i t e r a t i o n 2 ( x 0 , g , g p )28

29


31 x 1 = 0 . 5 0 0 0 0 0 0 0 ;

32 x 2 = 0 . 5 2 6 6 1 1 0 ;

33

34

35

36 a i t k e n ( x 0 , x 1 , x 2 , g ) // c a l l i n g t h e a i t k e nmethod f o r one i t e r a t i o n


Vnewton.sce


modified_newton.sce

Scilab code Exa 2.26 solution to the eq with multiple roots

41


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1 // The e q ua t i on x

3

7

x2+16

x12==0 ha sr e a l r o o t s .


3 xset ( window ,25);4 x = 0 : . 0 0 1 : 4 ;


5 deff ( [ y]= f (x ) ,y=x37x2+16x12 ) ;// d e f i n i n g t he c u n ct i o n .

6 deff ( [ y ] = f p ( x ) , y=3x 2 14x+16 ) ;7 y = feval ( x , f ) ;

8

9 a = gca () ;

10



// i n s t r u c t i o n t o p l o t t he gr ap h15 t i t l e ( y = x37x2+16x12 )16

17

18

19

20 / / g i v e n t ha t t he e q u a ti o n ha s d ou bl e r o o t s a t x=2he nc e m= 2;

21

22 m = 2 ;23

24 // s o l u t i o n by newton r ap hs onmethod

25

42


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26

27 n e w t o n ( 1 , f , f p ) // c a l l i n g t h e u se rd e f i n e d f u n c t i o n28

29

30

31

32 // s o l u t i o n by m o d if i edn ew to n r a p h s o n s m at ho d

33

34

35

36 m o d i f i e d _ n e w t o n ( 1 , f , f p )


Vnewton.sce


Vnewton4.sce


newton63.sce


secant64.sce


secant65.sce

Scilab code Exa 2.27 solution to the given transcendental equation

43


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1 / / The e q u a t i on

27

x5+27

x4+36x3+28x2+9x+1==0ha s r e a lr o o t s .


3 xset ( window ,26);4 x = - 2 : . 0 0 1 : 3 ;


5 deff ( [ y]= f (x ) , y=27x5+27x4+36x3+28x2+9x+1 ) ; // d e f i n i n g t he c u n ct i o n .

6 deff ( [ y ] = f p ( x ) , y = 275x4+274x3+363x2+282x+9 ) ;

7 deff ( [ y]= fpp (x ) , y = 2754x 3+ 2743x2+3632x+2 82 ) ;

8 y = feval ( x , f ) ;

9

10 a = gca () ;

11



// i n s t r u c t i o n t o p l o t t he gr ap h16 t i t l e ( y = 27x5+27x4+36x3+28x2+9x+1 )17

18

1920

21 // s o l u t i o n by newton r ap hs onmethod a s p e r t h e e q u a t i o n no .

2 . 1 4

44


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22

2324 n e w t o n ( - 1 , f , f p ) // c a l l i n g t h e u se r

d e f i n e d f u n c t i o n25

26

27 n e w t o n 4 ( - 1 , f , f p )

28

29

30 // s o l u t i o n by newtonr a ph s on method a s p e r t h e

e qu at i o n no . 2 . 6 3

3132 n e w t o n 6 3 ( - 1 , f , f p , f p p ) // c a l l i n g

t h e u se r d e f i ne d f u nc t i o n33 // s o l u t i o n by t he s e c a n t

method d e f i n e d t os a t i s f y t he e qu at io nno . 2 . 6 4 .

34

35

36 s e c a n t 6 4 ( 0 , - 1 , f , f p )

37

38

39

40

41

42

43 // s o l u t i o n by t he s e c a n tmethod d e f i n e d t o

s a t i s f y t he e qu at io nno . 2 . 6 5 .

44

4546 s e c a n t 6 5 ( 0 , - . 5 , f )

45


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Chapter 3

SYSTEM OF LINEAR

ALGEBRIC EQUATIONS

AND EIGENVALUE

PROBLEMS

Scilab code Exa 3.1 determinent

1 / / e xa mp le 3 . 12 // P o s i t i v e d e f i n i t e3

4 A =[12 4 -1;4 7 1; -1 1 6];

5 // c he ck i f t he d et er mi na nt o f t he l e a d i ng m in or s i sa p o s i t i v e v a lu e

6

7 A 1 = A ( 1 ) ;

8 det ( A 1 )

9 A 2 = A ( 1 : 2 , 1 : 2 ) ;

10 det ( A 2 )11 A 3 = A ( 1 : 3 , 1 : 3 ) ;

12 det ( A 3 )

13

14 / /we o b se rv e t ha t t he d et er mi na nt o f t he l e a d i ng

46


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m i n o r s i s a p o s i t i v e , h e n c e t he g i v en m at r i x A i s

a p o s i t i v e d e f i n i t e .

Scilab code Exa 3.2 property A in the book

1 / / e xa mp le no . 3 . 22 // show i f a g i v e n m at ri x A p o s s e s s e s p r op e r t y A3

4 A = [2 -1 0 -1; -1 2 -1 0;0 -1 2 0;0 0 -1 2 ]

56 P = [ 0 0 0 1 ; 0 1 0 0 ; 0 0 1 0 ; 1 0 0 0 ] //l e t us t ak e t he p em ut at io n m at ri x a s P

7

8 / / t h en we f i n d t ha t9

10 P * A * P

11

12 / / i s o f t h e form ( 3 . 2 ) . h e n c e t h e m a t r i x A ha sp r o p e r t y A

Scilab code Exa 3.4 solution to the system of equations

1 / / e xa mp le no . 3 . 42 // s o l v e t he s ys te m o f e q ua t i on s3

4 / / ( a ) . by u s i n g cr am er s r u l e ,5

6 A =[1 2 -1;3 6 1;3 3 2]

78 B1 =[2 2 -1;1 6 1;3 3 2]

9

10 B2 =[1 2 -1;3 1 1;3 3 2]

11

47


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12 B 3 = [ 1 2 2 ; 3 6 1 ; 3 3 3 ]

1314

15 //we know ;16

17 X1 = det ( B 1 ) / det ( A )

18 X2 = det ( B 2 ) / det ( A )

19 X3 = det ( B 3 ) / det ( A )

20

21

22 / / ( b ) . by d e t e m i n in g t h e i n v e r s e o f t h e c o e f f i c i e n tm a t r i x

2324 A =[1 2 -1;3 6 1;3 3 2]

25

26 b =[2 ;1 ; 3]

27

28 //we know ;29

30 X = inv ( A ) * b


Vgausselim.sce

Scilab code Exa 3.5 solution by gauss elimination method

1 //e x ampl e 3.52 // c ap t i o n s o l u t i o n by g a us s e l i m i b n a t i o n method3

4 A = [1 0 -1 2;1 10 -1;2 3 2 0] / / m a t r i c e s

A and b f ro m t h e a bo ve5 //

s y s t e m

o f

48


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e q u a t i o n s

6 b = [ 4 ; 3 ; 7 ]

7

8 g a u s s e l i m ( A , b ) // c a l l g au ss e l i m i n a t i o nf u n c t i o n t o s o l v e t he

9 // m a t r i c e s A and b


pivotgausselim.sci

Scilab code Exa 3.6 solution by pivoted gauss elimination method

1 //e x ampl e 3.62 // c ap t i o n s o l u t i o n by g a us s e l i m i b n a t i o n method3

4 A = [ 1 1 1 ; 3 3 4 ; 2 1 3 ] / / m a t r i c e s A andb fro m t he a bo ve

5 //

s y s t e m

o f

e q u a t i o n s

6

7 b = [ 6 ; 2 0 ; 1 3 ]

8

9 p i v o t g a u s s e l i m ( A , b ) // c a l l g a us s

e l i m i n a t i o n f u nc t i o n t o s o l v e t he10 // m a t r i c e s A and b


pivotgausselim.sci

49


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Scilab code Exa 3.8 solution by pivoted gauss elimination method

1 / / e xa mp le no . 3 . 82 / / s o l v i n g t h e m at r i x e qu at i o n w i t h p a r t i a l p i v o t i n g

i n g au ss e l i m i n a t i o n3

4 A =[2 1 1 -2;4 0 2 1;3 2 2 0;1 3 2 -1]

5

6 b = [ - 1 0 ; 8 ; 7 ; - 5 ]7

8 p i v o t g a u s s e l i m ( A , b )


jordan.sce

Scilab code Exa 3.9 solution using the inverse of the matrix

1 / / e xa mp le no . 3 . 92 / / s o l v i n g t h e s ys te m u s i n g i n v e r s e o f t h e c o f f i c i e n t

m a t r i x3

4 A = [1 1 1;4 3 -1;3 5 3]

5

6 I = [ 1 0 0 ; 0 1 0 ; 0 0 1 ]

7

8 b =[1 ;6 ; 4]

9

10 M = j o r d e n ( A , I )11

12 I A = M ( 1 : 3 , 4 : 6 )

13

14 X = I A * b

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LandU.sce


back.sce


fore.sce

Scilab code Exa 3.10 decomposition method

1 / / e xa mp le no . 3 . 1 02 / / s o l v e s y st em by d e c o m p o s i t i o n metho d3

4 A = [1 1 1;4 3 -1;3 5 3]

5 n = 3 ;

6

7 b = [ 1 ; 6 ; 4 ]8

9 [ U , L ] = L a n d U ( A , 3 )

10

11 Z = f o r e ( L , b )

12

13 X = b a c k ( U , Z )


LandU.sce

Scilab code Exa 3.11 inverse using LU decoposition

51


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1 / / e xa mp le no . 3 . 1 1

2 / / c a p t i o n : I n v e r s e u s i n g LU d e c o mp o s i t i on34 A = [ 3 2 1 ; 2 3 2 ; 1 2 2 ]

5

6 [ U , L ] = L a n d U ( A , 3 ) // c a l l LandU f u n c t i o n t oe v a l u a t e U , L o f A ,

7

8 / / s i n c e A=LU ,9 // in v (A)=in v (U) i nv ( L)

10 / / l e t i n v (A)=AI11

12 A I = U ^ - 1 * L ^ - 1


LandU.sce


back.sce


fore.sce

Scilab code Exa 3.12 solution by decomposition method

1 / / e xa mp le no . 3 . 1 22 / / s o l v e s y st em by d e c o m p o s i t i o n metho d3

4 A = [1 1 -1;2 2 5;3 2 -3]

56 b = [ 2 ; - 3 ; 6 ]

7

8

9

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10 // h en ce we c an o b s e r v e t h a t LU d e c o m po s i t i on

method f a i l s t o s o l v e t h i s s ys te m s i n c et h e p i v o t L ( 2 , 2 ) =0;11

12

13 / / we n o t e t h a t t h e c o e f f i c i e n t m a t r i x i s n o ta p o s i t i v e d e f i n i t e m a tr ix and h e n ce i t sLU d e c o m po s i t i on i s n ot g u ar a nt ee d ,

14

15

16 // i f we i n t e r ch a n g e t h e r ow s o f A a s shownb el o w t h e LU d e c o m p o s i t i o n w ou ld work ,

1718 A =[3 2 -3;2 2 5;1 1 -1]

19

20 b = [ 6 ; - 3 ; 2 ]

21

22 [ U , L ] = L a n d U ( A , 3 ) / / c a l l LandUf u nc t i o n t o e v a l ua t e U, L o f A ,

23

24 n = 3 ;

25 Z = f o r e ( L , b ) ;

26

27 X = b a c k ( U , Z )


LandU.sce


back.sce


fore.sce

Scilab code Exa 3.13 LU decomposition

53


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1 / / e xa mp le no . 3 . 1 3

2 / / s o l v e s y st em by LU d e c o m p o s i t i o n metho d34 A =[2 1 1 -2;4 0 2 1;3 2 2 0;1 3 2 -1]

5

6 b = [ - 1 0 ; 8 ; 7 ; - 5 ]

7

8 [ U , L ] = L a n d U ( A , 4 )

9 n = 4 ;

10 Z = f o r e ( L , b ) ;

11

12 X = b a c k ( U , Z )

1314 / / s i n c e A=LU ,15 // in v (A)=in v (U) i nv ( L)16 / / l e t i n v (A)=AI17

18 A I = U ^ - 1 * L ^ - 1


back.sce


cholesky.sce


fact.sci

Scilab code Exa 3.14 cholesky method

1 / / e xa mp le no . 3 . 1 42 / / s o l v e s ys te m by c h o l e s k y method3

4 A =[1 2 3;2 8 22;3 22 82]

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5

6 b = [ 5 ; 6 ; - 1 0 ]7

8 L = c h o l e s ky ( A , 3 ) // c a l l c ho l es k y f u nc t i o n t oe v al u a t e t he r o ot o f t he s yst em

9 n = 3 ;

10 Z = f o r e ( L , b ) ;

11

12 X = b a c k ( L , Z )


back.sce


cholesky.sce


fore.sce

Scilab code Exa 3.15 cholesky method

1 / / e xa mp le no . 3 . 1 52 / / s o l v e s ys te m by c h o l e s k y method3

4 A =[4 -1 0 0; -1 4 -1 0;0 -1 4 -1;0 0 -1 4]

5

6 b = [ 1 ; 0 ; 0 ; 0 ]

7

8 L = c h o l e s ky ( A , 4 ) // c a l l c ho l es k y f u nc t i o n t o

e v al u a t e t he r o ot o f t he s yst em9

10 n = 4 ;

11 Z = f o r e ( L , b ) ;

12

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13 X = b a c k ( L , Z )

1415 / / s i n c e A=LL ,16 / / i n v ( A)=i n v ( L ) i nv ( L)17 / / l e t i n v (A)=AI18

19 A I = L ^ - 1 * L ^ - 1


jacobiiteration.sce

Scilab code Exa 3.21 jacobi iteration method

1 / / e xa mp le no . 3 . 2 12 // s o l v e t he s ys te m by j a c o b i i t e r a t i o n method3

4 A = [ 4 1 1 ; 1 5 2 ; 1 2 3 ]

5

6 b = [2 ; - 6; - 4]

7

8 N = 3 ; // no . o f i e r a t i o n s9 n = 3 ; // o rd er o f t h e m a t r i x i s nn

10

11 X = [ . 5 ; - . 5 ; - . 5 ] // i n i t i a l app r ox i ma t i o n12

13

14 j a c o b i i t e r a t i o n ( A , n , N , X , b ) // c a l l t hef u n c ti o n which p er fo rm s j a c o b i i t e r a t i o n methodt o s o l v e t he s ys te m

check AppendixAP 36for dependency:Vgaussseidel.sce

56


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Scilab code Exa 3.22 solution by gauss siedal method

1 / / e xa mp le no . 3 . 2 22 // s o l v e t he s ys te m by g a us s s e i d e l method3

4 A = [2 -1 0; -1 2 -1;0 -1 2]

5

6 b = [ 7 ; 1 ; 1 ]

7

8 N = 3 ; // no . o f i e r a t i o n s9 n = 3 ; // o rd er o f t h e m a t r i x i s nn

10

11 X = [ 0 ; 0 ; 0 ] // i n i t i a l app r ox i ma t i o n12

13

14 g a u s s s e i d e l ( A , n , N , X , b ) // c a l l t hef u n c t i o n wh ich p e rf o rm s g a us s s e i d e l method t os o l v e t he s ys te m


geigenvectors.sci

Scilab code Exa 3.27 eigen vale and eigen vector

1 / / e xa mp le 3 . 2 72 // a ) f i n d e i ge n v a l u e and e i g e n v e c t o r ;3 // b ) v e r i f y i nv ( S )AS i s a d i ag o na l m at r i x ;4

5 / / 1 )6 A =[1 2 -2 ;1 1 1;1 3 -1];

78 B = [ 1 0 0 ; 0 1 0 ; 0 0 1 ] ;

9

10 [ x , l a m ] = g ei g e n ve c t o rs ( A , B ) ;

11

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12 inv ( x ) * A * x

1314 // 2)15 A =[3 2 2;2 5 2;2 2 3];

16

17 B = [ 1 0 0 ; 0 1 0 ; 0 0 1 ] ;

18

19

20 [ x , l a m ] = g ei g e n ve c t o rs ( A , B ) ;

21

22 inv ( x ) * A * x

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Chapter 4

INTERPOLATION AND

APPROXIMATION


NDDinterpol.sci


aitkeninterpol.sci


legrangeinterpol.sci

Scilab code Exa 4.3 linear interpolation polinomial

1 / / ex am pl e 4 . 32 // f i n d t h e l i n e a r i n t e r p o l a t i o n p o l i n o m i a l3 // u s in g

45 disp ( f (2 ) =4 ) ;6 disp ( f ( 2 . 5 ) = 5. 5 ) ;7 // 1 ) l a g ra n ge i n t e r p o l a t i o n ,8

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9 P 1 = l e g r a ng e i n te r p o l ( 2 , 2 . 5 , 4 , 5 .5 )

10 // 2 ) a it ke n s i t e r a t e d i n t e r p o l a t i o n ,1112 P 1 = a i t k en i n t e rp o l ( 2 , 2 . 5 , 4 , 5 . 5)

13

14 // 3 ) n ewton d ev id ed d i f f e r a n c e i n t e r p o l a t i o n ,15

16 P 1 = N D D i nt e r po l ( 2 , 2 . 5 , 4 , 5 . 5)

17

18 // h en ce a pp ro xi ma te v al u e o f f ( 2 . 2 ) = 4 . 6 ;



Scilab code Exa 4.4 linear interpolation polinomial

1 / / ex am pl e 4 . 42 // f i n d t h e l i n e a r i n t e r p o l a t i o n p o l i n o m i a l3 // u s in g l a g ra n g e i n t e r p o l a t i o n ,

45 disp ( s i n ( . 1 ) =. 0 9 9 8 3 ; s i n ( . 2 ) = .1 9 86 7 ) ;6

7

8 P 1 = l e g r a ng e i n te r p o l ( . 1 , . 2 , . 0 99 8 3 , . 1 98 6 7)

9

10 / / h e n c e ;11 disp ( P ( . 1 5 ) = .0 0 0 9 9 +. 9 8 8 4 . 1 5 )12 disp ( P ( 0 . 1 5 ) = 0. 1 4 9 25 ) ;



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Scilab code Exa 4.6 legrange linear interpolation polinomial

1 // ex ampl e : 4 . 62 // c ap ti on : o bt ai n t he l e g r a n g e l i n e a r

i n t e r p o l a t i n g p o l i n om i al3

4

5 // 1 ) o bt a in t h e l e g r a n g e l i n e a r i n t e r p o l a t i n gp o l i n om i a l i n t h e i n t e r v a l [ 1 , 3 ] and o bt ai na p pr o xi m at e v a l u e o f f ( 1 . 5 ) , f ( 2 . 5 ) ;

6 x 0 = 1 ; x 1 = 2 ; x 2 = 3 ; f 0 = . 8 4 1 5 ; f 1 = . 9 0 9 3 ; f 2 = . 1 4 1 1 ;

7

8 P 13 = l e g r an g e i nt e r p o l ( x 0 , x2 , f 0 , f 2 ) // int h e r a n g e [ 1 , 3 ]

9

10

11 P 12 = l e g r an g e i nt e r p o l ( x 0 , x1 , f 0 , f 1 ) //i n t h e r a ng e [ 1 , 2 ]

12

13 P 23 = l e g r an g e i nt e r p o l ( x 1 , x2 , f 1 , f 2 )

// i n th e r a ng e [ 2 , 3 ]14

15 / / from P23 we f i n d t ha t ; where a s e xa ct v al ue i ss i n ( 2 . 5 ) =0 . 5 9 8 5 ;

16 disp ( P ( 1 . 5 ) = 0 . 87 5 4 ) ;17 disp ( e x a c t v a l u e o f s i n ( 1 . 5 ) = .9 97 5 )18 disp ( P ( 2 . 5 ) = 0 . 52 5 2 ) ;


lagrangefundamentalpoly.sci

Scilab code Exa 4.7 polynomial of degree two

1 / / ex am pl e 4 . 72 // p o l i no m i a l o f d e g r ee 2 ;

61


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3

4 // f (0 ) =1; f (1 ) =3; f (3 ) =55;56 // u s i ng l e g r a n g e f un da me nt al p o l i n o m i a l r u le ,7

8 x =[0 1 3 ]; // a r r a i n g i n g t h ei n p ut s o f t h e f u n c t i o n a s e le me nt s o f a row ,

9 f =[1 3 5 5] ; // a r r a i n g i ng t h eo ut pu ts o f t he f u n c ti o n a s e l e me nt s o f a r ow ,

10 n = 2 ; // d eg re e o f t h ep o l i n o m i a l ;

11

1213 P 2 = l a g r a n g e f u n d a m e n t a l p o l y ( x , f , n )



Scilab code Exa 4.8 solution by quadratic interpolation

1

2 / / ex am pl e 4 . 83 // c ap t i o n : s o l u t i o n by q u a d r a ti c i n t e r p o l a t i o n ;4

5 // xd e g re e s : [ 1 0 20 3 0 ]6 // h en c e x i n r ad i a n s i s7 x = [ 3 . 1 4/ 1 8 3 . 1 4 /9 3 . 1 4 / 6 ];

8 f = [ 1 . 1 58 5 1 . 2 8 17 1 . 3 66 0 ] ;

9 n = 2 ;

10


13

14 // h en ce from P2 , t he e xa c t v al u e o f f ( 3 . 1 4 /1 2 ) i s1 . 2 2 4 6 ;

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15 // w here a s e xa ct v al ue i s 1 . 2 2 4 7 ;


NDDinterpol2.sci


iteratedinterpol.sci

Scilab code Exa 4.9polinomial of degree two

1 / / e xa mp le 4 . 92 // c a p t i o n : o bt ai n t h e p o l i n om i al o f d eg re e 23

4 x = [0 1 3];

5 f = [1 3 5 5] ;

6 n = 2 ;

7

8 // 1 ) i t e r a t e d i n t e r p o l a t i o n ;9

1011 [ L 01 2 , L 0 2 , L 0 1 ] = i t e r a te d i n t er p o l ( x , f , n )

12

13 // 2 ) newton d i v i d e d d i f f r e n c e s i n t e r p o l a t i o n ;14

15

16 P 2 = N D D i nt e r p ol 2 ( x , f )


NBDP.sci

Scilab code Exa 4.15 forward and backward difference polynomial

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1 / / exa mp le 4 . 1 5 :

2 / / o bt ai n t he i n t e r p o l a t e u si ng backward d i f f e r e n c e sp o l i n o m i a l3

4

5 xL =[.1 .2 .3 .4 .5 ]

6

7 f = [ 1. 4 1 .5 6 1 .7 6 2 2 .2 8]

8 n = 2 ;

9

10

11 / / h e n c e ;

12 disp (P=1.4+(x . 1) ( . 1 6 / . 1 ) +(x . 5 ) ( x . 4 ) ( . 0 4 / . 0 2 ) )

13 disp (P=2x2+x +1. 28 ) ;14

15 // 1 ) o bt ai n t h e i n t e r p o l a t e a t x = 0 .2 5;16 x = 0 . 2 5 ;

17 [ P ] = N B D P ( x , n , x L , f ) ;

18 P

19 disp ( f ( . 2 5 ) = 1 . 6 5 5 ) ;20

21

22 // 2 ) o bt ai n t h e i n t e r p o l a t e a t x = 0 .3 5;23 x = 0 . 3 5 ;

24 [ P ] = N B D P ( x , n , x L , f ) ;

25 P

26 disp ( f ( . 3 5 ) = 1 . 8 7 5 ) ;


hermiteinterpol.sci

Scilab code Exa 4.20 hermite interpolation

1 / / exa mp le 4 . 2 0 ;

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2 // h er mi te i n t e r p o l a t i o n :

34 x =[ -1 0 1 ];

5

6 f=[1 1 3];

7

8 f p = [ -5 1 7 ];

9

10 P = h e r m i te i n t er p o l ( x , f , fp ) ;

11

12 / / h e n c e ;13 disp ( f ( 0.5)=3/8 ) ;

14 disp ( f (0 . 5 ) =11/8 ) ;



Scilab code Exa 4.21 piecewise linear interpolating polinomial

1 / / e xa mp le : 4 . 2 1 ;2 // p i e c e w i s e l i n e a r i n t e r p o l a t i n g p o l i n o m i a l s :

3

4 x 1 = 1 ; x 2 = 2 ; x 3 = 4; x 4 = 8 ;

5 f 1 = 3 ; f 2 =7 ; f 3 = 21 ; f 4 = 73 ;

6 // we ne ed t o a pp ly l e g r a n g e s i n t e r p o l a t i o n i n subra n g es [ 1 , 2 ] ; [ 2 , 4 ] , [ 4 , 8 ] ;

7

8 x = poly (0 , x ) ;9

10 P 1 = l e g r a ng e i n t er p o l ( x1 , x 2 , f 1 , f 2 ) ; // i n t h e

r an ge [ 1 , 2 ]11 P1

12

13 P 1 = l e g r a ng e i n t er p o l ( x2 , x 3 , f 2 , f 3 ) ; // i n t h er an ge [ 2 , 4 ]

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14 P1

1516 P 1 = l e g r a ng e i n t er p o l ( x3 , x 4 , f 3 , f 4 ) ; // i n t h e

r an ge [ 4 , 8 ]17 P1



Scilab code Exa 4.22 piecewise quadratic interpolating polinomial

1 / / e xa mp le : 4 . 2 2 ;2 // p i e c e w i s e q u a d r a ti c i n t e r p o l a t i n g p o l i n o m i a l s :3

4 X =[ -3 -2 -1 1 3 6 7];

5 F = [ 36 9 2 22 1 71 1 65 2 07 9 90 1 77 9] ;

6 // we ne ed t o a pp ly l e g r a n g e s i n t e r p o l a t i o n i n subr an g es [3 , 1 ] ; [ 1 , 3 ] , [ 3 , 7 ] ;

7

8 x = poly (0 , x

) ;

9

10 // 1 ) i n t he r an g e [ 3 , 1]11 x = [ -3 -2 -1 ];

12 f = [ 36 9 2 22 17 1] ;

13 n = 2 ;

14 P 2 = l a g r a n g e f u n d a m e n t a l p o l y ( x , f , n ) ;

15

16 // 2 ) i n t he r a n g e [ 1 ,3 ]17 x =[ -1 1 3];

18 f = [ 17 1 1 6 5 2 07 ];

19 n = 2 ;20 P 2 = l a g r a n g e f u n d a m e n t a l p o l y ( x , f , n )

21

22 // 3 ) i n t h e r a n g e [ 3 , 7 ]23 x =[3 6 7];

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24 f = [ 20 7 9 9 0 1 7 79 ] ;

25 n = 2 ;26 P 2 = l a g r a n g e f u n d a m e n t a l p o l y ( x , f , n )

27

28

29

30 // hence , we o b ta i n t he v a l ue s o f f ( 2 .5 ) =48 ; f ( 6 . 5 ) = 1 3 5 1 . 5 ;



Scilab code Exa 4.23 piecewise cubical interpolating polinomial

1 / / e xa mp le : 4 . 2 3 ;2 / / p i e c e w i s e c u b i c a l i n t e r p o l a t i n g p o l i n o m i a l s :3

4 X =[ -3 -2 -1 1 3 6 7];

5 F = [ 36 9 2 22 1 71 1 65 2 07 9 90 1 77 9] ;

6 // we ne ed t o a pp ly l e g r a n g e s i n t e r p o l a t i o n i n sub

r an g es [3 , 1 ] ; [ 1 , 7 ] ;7

8

9 x = poly (0 , x ) ;10

11 // 1 ) i n t he r an g e [ 3 ,1 ]12 x =[ -3 -2 -1 1];

13 f = [ 36 9 22 2 1 71 1 65 ];

14 n = 3 ;

15 P 2 = l a g r a n g e f u n d a m e n t a l p o l y ( x , f , n ) ;

1617 // 2 ) i n t h e r a n g e [ 1 , 7 ]18 x =[1 3 6 7];

19 f = [ 16 5 2 0 7 9 9 0 1 7 79 ];

20 n = 3 ;

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2223

24

25 / / h en c e ,26 disp ( f ( 6 . 5 ) = 1 3 3 9 . 2 5 ) ;

Scilab code Exa 4.31 linear approximation

1 / / e xa mp le 4 . 3 12 // o b ta i n t he l i n e a r p o l i no m i a l a pp ro xi ma ti on t o t he

f u n c t i o n f ( x ) =x 33

4 / / l e t P ( x )=a0x+a15

6 / / h en ce I ( a 0 , a1 )= i n t e g r a l ( x 3( a0x+a1 ) ) 2 i n t hei n t e r v a l [ 0 , 1 ]

7

8 printf ( I =1/72(a0 /5+a1 /4)+a0 2/3+a 0a1+a12 )9 printf ( d I / d a 0 = 2/5+2/3a0+a1=0 )

1011 printf ( d I / d a 1 = 1/2+a0+2a1 =0 )12

13 / / h e nc e14

15 printf ( [ 2/ 3 1 ; 1 2 ] [ a 0 ; a 1 ] = [ 2 / 5 ; 1 / 2 ] )16

17 // s o l v i n g f o r a0 and a1 ;18

19 a 0 = 9 / 1 0 ;

20 a 1 = - 1 / 5 ;21 // h en ce c o n s i d e r i n g t he p o l i n om i al w i t h i n t e r c e p tP 1 ( x ) = ( 9x2 ) / 1 0 ;

22

23 // c o n s i d e r i n g t he p o l i n o m i a l a p pr ox im a ti o n t hr ou gh

68


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28

29 printf ( d I / d c 0 = ( 2/ 3

c0

c1/2

c2 /2 )=0 )3031 printf ( dI /dc 1 =(2/5 c0/2c1/3c2 /4 ) =0 )32

33 printf ( dI /dc 2 =(2/7 c0/3c1/4c2 /5 ) =0 )34

35

36 / / h e nc e37

38 printf ( [ 1 1/2 1 / 2; 1 /2 1/3 1 /4 ; 1/ 3 1/4 1 / 5 ] [ c 0 ; c 1 ;c2 ]=[ 2 / 3 ; 2/5; 2/7] )

3940 // h en ce s o l v i n g f o r c0 , c1 and c 2 ;41

42

43 / / t h e f i r s t d e g r e e s q u a r e a p p r o x im a t i o n P ( x ) = (6+48x20x 2 ) / 3 5 ;


straightlineapprox.sce

Scilab code Exa 4.34 least square straight fit

1 / / e xa mp le 4 . 3 42

3 // o bt ai n l e a s t s qu ar e s t r a i g h t l i n e f i t4

5 x = [. 2 .4 .6 .8 1] ;

6 f = [. 44 7 . 63 2 . 77 5 . 89 4 1 ];

78

9 [ P ] = s t r a i g h t l i n e a p p r o x ( x , f ) // c a l l o f t h ef u n c t i o n t o g et t h e d e s i r e d s o l u t i o n

70


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quadraticapprox.sci

Scilab code Exa 4.35 least square approximation

1 / / e xa mp le 4 . 3 52

3 / / o b ta i n l e a s t s qu ar e a pp ro xi ma ti on o f s ec on dd e g r e e ;

4 x =[ -2 -1 0 1 2];5 f = [15 1 1 3 19];

6

7 [ P ] = q u a d r a t i c a p p r o x ( x , f ) // c a l l o f t h ef u n c t i o n t o g et t h e d e s i r e d s o l u t i o n

Scilab code Exa 4.36 least square fit

1 / / e xa mp le 4 . 3 62 // method o f l e a s t s q u a r e s t o f i t t h e d a t a t o t h e

c u r v e P ( x ) =c 0X+c1 / s q u r t (X)3

4 x =[.2 .3 .5 1 2];

5 f =[16 14 11 6 3];

6

7 / / I ( c 0 , c 1 )= s um m at io n o f ( f ( x )( c0 X+c1 / sq r t (X) ) ) 28

9 / / h en ce on p a r c i a l l y d e r i v a t i n g t he summation ,

1011 n = length ( x ) ; m = length ( f ) ;

12 if m < > n then

13 error ( l i n r e g Ve ct or s x and f a re no t o f t h es am e l e n g t h . ) ;

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14 abort ;

15 end ;16

17 s 1 = 0 ; / / s 1= s um ma ti on o f x ( i) f ( i )

18 s 2 = 0 ; / / s 2= s um ma ti on o f f ( i) / sq r t ( x ( i ) )

19 s 3 = 0 ;

20 for i = 1 : n

21 s 1 = s 1 + x ( i ) * f ( i ) ;

22 s 2 = s 2 + f ( i ) / sqrt ( x ( i ) ) ;

23 s 3 = s 3 + 1 / x ( i ) ;

24 end25

26 c0 = det ([s1 sum ( sqrt ( x ) ); s 2 s 3 ]) / det ([ sum ( x ^ 2 ) sum (

sqrt ( x ) ) ; sum ( sqrt ( x ) ) s 3 ])

27

28 c1 = det ([ sum ( x ^ 2) s 1 ;sum ( sqrt ( x ) ) s 2 ]) / det ([ sum ( x ^ 2 )

sum ( sqrt ( x ) ) ; sum ( sqrt ( x ) ) s 3 ])

29 X = poly (0 , X ) ;30 P = c 0 * X + c 1 / X ^ 1 / 2

31 // h e n ce c o n s i d e r i n g t he p o l i no m i a l P( x ) =7 .5 96 1X

1/2

1. 1836

X

Scilab code Exa 4.37 least square fit

1 / / e xa mp le 4 . 3 72 // method o f l e a s t s q u a r e s t o f i t t h e d a t a t o t h e

c u r v e P ( x ) =a%e(3 t )+b%e(2 t ) ;3

4 t = [. 1 .2 .3 . 4] ;5 f = [ .7 6 . 58 . 44 . 35 ];

6

7 / / I ( c 0 , c 1 )= s um m at io n o f ( f ( x )a%e(3 t )+b%e(2 t ) )

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8

9 / / h en ce on p a r c i a l l y d e r i v a t i n g t he summation ,1011 n = length ( t ) ; m = length ( f ) ;

12 if m < > n then

13 error ( l i n r e g V ec to rs t and f a re no t o f t h es am e l e n g t h . ) ;

14 abort ;

15 end ;

16

17 s 1 = 0 ; / / s 1= s um ma ti on o f f ( i) %e(3 t ( i ) ) ;

18 s 2 = 0 ; / / s 2= s um ma ti on o f f ( i) %e(2 t ( i ) ) ;

19

20 for i = 1 : n

21 s 1 = s 1 + f ( i ) * % e ^ ( - 3 * t ( i ) ) ;

22 s 2 = s 2 + f ( i ) * % e ^ ( - 2 * t ( i ) ) ;

23

24 end

25

26 a = det ([s1 sum ( % e ^ ( - 5* t ) ) ; s 2 sum ( % e ^ ( - 4 * t ) ) ] ) / det ([

sum ( % e ^ ( - 6 * t ) ) sum ( % e ^ ( - 5 * t ) ) ; sum ( % e ^ ( - 5 * t ) ) sum

( % e ^ ( - 4 * t ) ) ] )

27

28 b = det ([ sum ( % e ^( - 6* t ) ) s 1 ; sum ( % e ^ ( - 5* t ) ) s 2 ] ) /det ([

sum ( % e ^ ( - 6 * t ) ) sum ( % e ^ ( - 5 * t ) ) ; sum ( % e ^ ( - 5 * t ) ) sum

( % e ^ ( - 4 * t ) ) ] )

29

30 // h en ce c o n s i d e r i n g t he p o l i no m i a l P( t ) =. 06 85 3%e(3 t ) +0. 305 8%e(2 t )

Scilab code Exa 4.38 gram schmidt orthogonalisation

1 / / e xa mp le 4 . 3 8

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2 / / g ram s ch mi dt o r t h o g o n a l i s a t i o n

34 W = 1 ;

5 x = poly (0 , x ) ;6 P 0 = 1 ;

7 p h i 0 = P 0 ;

8 a 1 0 = i n t e g r a t e (Wx p h i 0 , x ,0,1)/ i n t e g r a t e (W1p h i 0 , x ,0,1)

9 P 1 = x - a 1 0 * p h i 0

10 p h i 1 = P 1 ;

11

12 a 2 0 = i n t e g r a t e (Wx 2p h i 0 , x ,0,1)/ i n t e g r a t e (

W1p h i 0 , x ,0,1)13

14 a 2 1 = i n t e g r a t e ( ( x 2 ) ( x1/2) , x ,0,1)/ i n t e g r a t e( ( x1/2)2 , x ,0,1)

15

16 P 2 = x ^ 2 - a 2 0 * x - a 2 1 * p h i 1

17

18 // s i n c e , I= i n t g r a l [ x ( 1 /2 )c0 P0c1 P1c 2 P 2 ] 2i n t h e r an ge [ 0 , 1 ]

19

20 // h e n c e p a r t i a l l y d e r i v a t i n g I21

22 c0 = i n t e g r a t e ( x ( 1 / 2 ) , x ,0,1)/ i n t e g r a t e ( 1 , x ,0,1)

23 c1 = i n t e g r a t e ( ( x ( 1 / 2 ) ) ( x (1/2 ) ) , x ,0,1)/i n t e g r a t e ( ( x(1/ 2) ) 2 , x ,0,1)

24 c1 = i n t e g r a t e ( ( x ( 1 / 2 ) ) ( x 24x/3+1/2) , x ,0,1)/i n t e g r a t e ( (x24x /3+ 1/2) 2 , x ,0,1)

Scilab code Exa 4.39 gram schmidt orthogonalisation

1 / / e xa mp le 4 . 3 92 / / g ram s ch mi dt o r t h o g o n a l i s a t i o n

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3

4 / / 1 )5 W = 1 ;6 x = poly (0 , x ) ;7 P0=1

8 p h i 0 = P 0 ;

9 a10=0;

10 P 1 = x - a 1 0 * p h i 0

11 p h i 1 = P 1 ;

12

13 a 2 0 = i n t e g r a t e ( x 2 , x ,-1,1)/ i n t e g r a t e (W1p h i 0 , x ,-1,1);

1415 a 2 1 = i n t e g r a t e ( ( x 3 ) , x ,-1,1)/ i n t e g r a t e ( (x ) 2

, x ,-1,1);16

17 P 2 = x ^ 2 - a 2 0 * x - a 2 1 * p h i 1

18

19

20 / / 2 )21 disp ( W=1/(1x 2 ) ( 1 / 2 ) ) ;22 x = poly (0 , x ) ;23 P0=1

24 p h i 0 = P 0 ;

25 a10=0;

26 P 1 = x - a 1 0 * p h i 0

27 p h i 1 = P 1 ;

28

29 a 2 0 = i n t e g r a t e ( x 2/( 1x 2 ) ( 1 / 2 ) , x ,-1,1)/i n t e g r a t e ( 1/(1x 2 ) ( 1 / 2 ) , x ,-1,1);

30

31 a21=0; // s i n c e x 3 i san odd f u n c t i o n ;

3233 P 2 = x ^ 2 - a 2 0 * x - a 2 1 * p h i 1

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Scilab code Exa 4.41 chebishev polinomial

1 / / e xa mp le 4 . 4 12 // u s in g c he by sh ev p o l i n o m i al s o b ta i n l e a s t s q ua r es

a p pr o x im a t io n o f s e co n d d e g r e e ;3

4 // t he c he be sh ev p o l i n o m i a l s ;5 x = poly (0 , x ) ;6 T 0 = 1 ;

7 T 1 = x ;

8 T 2 = 2 * x ^ 2 - 1 ;

9

10

11 / / I= i n t e g r a t e ( 1/ (1 x 2 ) ( 1 / 2 ) (x4c0 T0c 1 T1c2T2) 2 , x , 1 ,1 )

12

13 // s i n c e ;14 c0 = i n t e g r a t e ( ( 1 / 3 . 1 4 ) ( x 4 ) / ( 1x 2 ) ( 1 / 2 ) , x

,-1,1)

1516 c1 = i n t e g r a t e ( ( 2 / 3 . 1 4 ) ( x 5 ) / ( 1x 2 ) ( 1 / 2 ) , x

,-1,1)

17

18 c2 = i n t e g r a t e ( ( 2 / 3 . 1 4 ) ( x 4 ) ( 2 x 21)/(1x 2 ) ( 1 / 2 ) , x ,-1,1)

19

20 f = ( 3 / 8 ) * T 0 + ( 1 / 2 ) * T 2 ;

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Chapter 5

DIFFERENTIATION AND

INTEGRATION


linearinterpol.sci

Scilab code Exa 5.1 linear interpolation

1 / / e xa mp le : 5 . 12 // l i n e a r and q u a d ra t i c i n t e r p o l a t i o n :3

4 / / f ( x ) =l n x ;5

6 x L = [2 2 .2 2 .6 ];

7 f = [ . 6 9 31 5 . 7 8 84 6 . 9 5 55 1 ] ;

8

9 // 1 ) f p ( 2 ) w i t h l i n e a r i n t e r p o l a t i o n ;10

11 f p = l i n e a r i n t e r p o l ( x L , f ) ;12 disp ( f p ) ;

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15 disp ( J ) ;


simpson.sci


trapezoidal.sci

Scilab code Exa 5.11solution by trapizoidal and simpsons

1 // exa mp le : 5 . 1 12 // s o l v e t h e d e f i n i t e i n t e g r a l by 1 ) t r a p e z o i d a l

r u l e , 2 ) s i mp s on s r u l e3 / / e xa ct v al ue o f t h e i n t e g r a l i s l n 2= 0 . 6 93 1 4 7 ,4

5 deff ( [ y]=F(x ) , y=1/(1+x ) )6

7 / / 1 ) t r a p e z o i d a l r u l e ,8

9 a = 0 ;10 b = 1 ;

11 I = t r a p e z oi d a l ( 0 ,1 , F )

12 disp ( error = . 7 5 - . 6 9 3 1 4 7 )

13

14 / / s im ps on s r u l e15

16 I = s i m p s o n ( a , b , F )

17

18 disp ( error = . 6 9 4 4 4 4 - . 6 9 3 1 4 7 )

Scilab code Exa 5.12 integral approximation by mid point and two point

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10 [ I ] = s i mp so n 38 ( x , f )

Scilab code Exa 5.15 quadrature formula

1 / / exa mp le : 5 . 1 52 / / f i n d t he q ua dr at ur e f or mu la o f 3 / / i n t e g r a l o f f ( x ) ( 1 / s q r t ( x (1+ x ) ) ) i n t h e r a n g e

[ 0 , 1 ] = a1f (0 )+a2f ( 1 /2 ) +a3f (1)=I4 // h en ce f i n d i n t e g r a l 1/ s q r t ( xx 3 ) i n t he r an ge

[ 0 , 1 ]5

6 // making t he method e x ac t f o r p o l i n o m i a l s o f d e g re eu p to 2 ,

7 // I=I1=a1+a2+a38 // I=I2 = ( 1/2) a2+a39 // I=I3 = ( 1/4) a2+a3

10

11 / / A=[ a 1 a 2 a 3 ] 12

13 I1 = i n t e g r a t e ( 1 / s q r t ( x(1x ) ) , x ,0,1)

14 I2 = i n t e g r a t e ( x / s q r t ( x(1x ) ) , x ,0,1)15 I3 = i n t e g r a t e ( x 2 / s q r t ( x(1x ) ) , x ,0,1)16

17 / / h e n c e18 // [ 1 1 1 ; 0 1/2 1 ; 0 1/4 1 ]A=[ I 1 I 2 I 3 ] 19

20 A = inv ([ 1 1 1;0 1/2 1 ;0 1/4 1 ]) *[ I1 I2 I3 ]

21 / / I = ( 3 . 1 4 / 4 ) ( f (0 ) +2 f ( 1/ 2) + f ( 1) ) ;22

23 / / h ence , f o r s o l v i n g t h e i n t e g r a l 1/ s q r t ( xx 3 )

i n t he r an ge [ 0 , 1 ] = I2425 deff ( [ y]= f (x ) , y =1/ sq r t (1+ x ) ) ;26 I = ( 3 . 1 4 / 4 ) * [ 1 + 2 * sqrt ( 2 / 3 ) + sqrt ( 2 ) / 2 ]

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Scilab code Exa 5.16 gauss legendary three point method

1 / / e xa mp le 5 . 1 62 // c a p t io n : g au ssl e g e n d r e t h r e e p o i n t method3 // I= i n t e g r a l 1/(1+ x ) i n t he r an ge [ 0 , 1 ] ;4 // f i r s t we ne ed t i t ra ns fo rm t he i n t e r v a l [ 0 , 1 ] t o

[ 1 , 1 ] , s i n c e g au ssl e g e n d r e t h r e e p o i n t methodi s a p p l i c a b l e i n t h e r a ng e [ 1 ,1] ,

5

6 / / l e t t=a x+b ;7 // s o l v i n g f o r a , b fr om t he two r an ge s , we g e t a =2;

b=1; t=2x1;8

9 / / h en ce I= i n t e g r a l 1/(1+ x ) i n t he r an ge [ 0 , 1 ] =i n t e g r a l 1 /( t +3) i n t he r an ge [ 1 , 1 ] ;

10

11

12 deff ( [ y ]= f ( t ) , y = 1/( t +3) ) ;13 / / s i n c e , from g au ss l e

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