Mathematical Methods for Numerical Analysis and Optimization

8/10/2019 Mathematical Methods for Numerical Analysis and Optimization

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Biyani's Think TankConcept based notes

Mathematical Methods for

NumericalAnalysisandOptimization

(BCA Part-II)

Varsha GuptaRevised by: Poonam Fatehpuria

Deptt. of Information Technology

Biyani Girls College, Jaipur


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For free study notes log on: www.gurukpo.com

Published by :

Think TanksBiyani Group of Colleges

Concept & Copyright :

Biyani Shikshan SamitiSector-3, Vidhyadhar Nagar,

Jaipur-302 023 (Rajasthan)

Ph : 0141-2338371, 2338591-95 Fax : 0141-2338007E-mail : [email protected] :www.gurukpo.com; www.biyanicolleges.org

ISBN : 978-93-81254-42-4

Edition : 2011Price :

Leaser Type Setted by :

While every effort is taken to avoid errors or omissions in this Publication, any

mistake or omission that may have crept in is not intentional. It may be taken note ofthat neither the publisher nor the author will be responsible for any damage or loss of

any kind arising to anyone in any manner on account of such errors and omissions.


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Mathematical Methods for Numerical Analysis and Optimization 3


Biyani College Printing Department


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Preface

I am glad to present this book, especially designed to serve the needs of thestudents. The book has been written keeping in mind the general weakness inunderstanding the fundamental concepts of the topics. The book is self-explanatoryand adopts the Teach Yourself style. It is based on question -answer pattern. Thelanguage of book is quite easy and understandable based on scientific approach.

This book covers basic concepts related to the microbial understandings aboutdiversity, structure, economic aspects, bacterial and viral reproduction etc.

Any further improvement in the contents of the book by making corrections,omission and inclusion is keen to be achieved based on suggestions from thereaders for which the author shall be obliged.

I acknowledge special thanks to Mr. Rajeev Biyani, Chairman & Dr. SanjayBiyani, Director(Acad.) Biyani Group of Colleges, who are the backbones and mainconcept provider and also have been constant source of motivation throughout thisEndeavour. They played an active role in coordinating the various stages of thisEndeavour and spearheaded the publishing work.

I look forward to receiving valuable suggestions from professors of variouseducational institutions, other faculty members and students for improvement of thequality of the book. The reader may feel free to send in their comments andsuggestions to the under mentioned address.

Author


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Syllabus

B.C.A. Part-II

Mathematical Methods for

Numerical Analysis and Optimization

Computer arithmetics and errors. Algorithms and programming for numerical

solutions. The impact of parallel computer : introduction to parallel architectures.

Basic algorithms Iterative solutions of nonlinear equations : bisection method,

Newton-Raphson method, the Secant method, the method of successive

approximation. Solutions of simultaneous algebraic equations, the Gauss

elimination method. Gauss-Seidel Method, Polynomial interpolation and other

interpolation functions, spline interpolation system of linear equations, partial

pivoting, matrix factorization methods. Numerical calculus : numerical

differentiating, interpolatory quadrature. Gaussian integration. Numerical solutions

of differential equations. Euler's method. Runge-Kutta method. Multistep method.Boundary value problems : shooting method.


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Content

S.No. Name of Topic Page No.

1. Computer Arithmetic and Errors 7-9

2. Bisection Method 10-15

3. Regula Falsi Method 16-22

4. Secant Method 23-26

5. Newton Raphson Method 27-31

6. Iterative Method 32-37

7. Gauss Elimination Method 38-41

8. Gauss Jordan Elimination Method 42-43

9. Matrix Inversion Method 44-48

10. Matrix Factorization Method 49-57

11. Jacobi Method 58-61

12. Gauss Seidel Method 62-66

13. Forward Difference 67-67

14. Backward Difference 68-69

15. Newton Gregory Formula for Forward Interpolation 70-73


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S.No. Name of Topic Page No.

16. Newtons Formula for Backward Interpolation 74-75

17. Divided Difference Interpolation 76-79

18. Lagranges Interpolation 80-81

19. Spline Interpolation 82-82

20. Quadratic Splines 83-84

21. Cubic Splines 85-85

22. Numerical Differentiation 85-88

23. Numerical Integration 89-91

24. Eulers Method 92-93

25. Eulers Modified Method 94-97

26. Rungs Kutta Method 98-102

27.

28.

Shooting Method

Unsolved Papers 2011 to 2006

103-104

105-170


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Chapter-1

Computer Arithmetic and Errors

Q.1. An approximate value of is given by x1 = 22/7 = 3.1428571 and its truevalue is x = 3.1415926. Find the absolute and relative errors.

Ans.: True value of (x) = 3.1415926

Approximate value of (x1) = 3.1428571

Absolute erroris given by

Ea=x x1

=3.1415926 3.1428571

= 0.0012645

Relative erroris given by

Er = 1x x

x

=3.1415926 3.1428571

3.1415926

=

0.0012645

3.1415926

= 0.0004025


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Q.2. Let x = 0.00458529 find the absolute error if x is truncated to three decimaldigits.

Ans.: x = 0.00458529 = 0.458529 x 10-2 [in normalized floating point form]

x1 = 0.458 x 10-2[after truncating to three decimal places]

Absolute error =x x1

= 0.458529 x 10-20.458 x 10-2

= 0.000529 x 10-2

= 0.000529 E 2

= 0.529 E 5

Q.3. Let the solution of a problem be xa= 35.25 with relative error in the solution

atmost 2% find the range of values upto 4 decimal digits, within which the

exact value of the solution must lie.

Ans.: We are given that the approximate solution of the problem is (xa) = 35.25 and

it has relative error upto 2% so

35.25x

x< 0.02

= -0.02 < 35.25x

x< 0.02

Case-I :if -0.02x < 35.25x

x

-0.02x < x 35.25

35.25 < x + 0.02x

35.25 < x (1 + 0.02)

35.25 < x (1.02)35.25 < 1.02x

35.25

1.02< x


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x > 34.5588 _ _ _ (1)

Case-II: if 35.25xx

< 0.02

x 35.25 < 0.02x

x 0.02x < 35.25

0.98x < 35.25

x

Mathematical Methods for Numerical Analysis and Optimization

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