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Elements of Numerical Analysis 2nd Revised edition [Pehme köide]

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(Indian Institute of Technology, Roorkee)
  • Formaat: Paperback / softback, 774 pages, kõrgus x laius x paksus: 241x185x30 mm, kaal: 1020 g
  • Ilmumisaeg: 14-May-2015
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107500494
  • ISBN-13: 9781107500495
Teised raamatud teemal:
Elements of Numerical Analysis 2nd Revised editionSuurem pilt
  • Formaat: Paperback / softback, 774 pages, kõrgus x laius x paksus: 241x185x30 mm, kaal: 1020 g
  • Ilmumisaeg: 14-May-2015
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107500494
  • ISBN-13: 9781107500495
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107500495.html
Märksõnad:
Numerical analysis deals with the manipulation of numbers to solve a particular problem. This book discusses in detail the creation, analysis and implementation of algorithms to solve the problems of continuous mathematics. An input is provided in the form of numerical data or it is generated as required by the system to solve a mathematical problem. Subsequently, this input is processed through arithmetic operations together with logical operations in a systematic manner and an output is produced in the form of numbers. Covering the fundamentals of numerical analysis and its applications in one volume, this book offers detailed discussion on relevant topics including difference equations, Fourier series, discrete Fourier transforms and finite element methods. In addition, the important concepts of integral equations, Chebyshev Approximation and Eigen Values of Symmetric Matrices are elaborated upon in separate chapters. The book will serve as a suitable textbook for undergraduate students in science and engineering.

Muu info

This textbook provides detailed discussion on fundamental concepts and applications of numerical analysis.
Preface xv
1 Errors in Computation
1(21)
1.1 Introduction
1(1)
1.2 Floating Point Representation of Number
1(1)
1.3 Binary Numbers
2(3)
1.3.1 Binary number representation in computer
3(2)
1.4 Significant Digits
5(1)
1.5 Rounding and Chopping a Number
5(1)
1.6 Errors due to Rounding/Chopping
6(1)
1.7 Measures of Error in Approximate Numbers
7(1)
1.8 Errors in Arithmetic Operations
8(1)
1.9 Computation of Errors Using Differentials
9(2)
1.10 Errors in Evaluation of Some Standard Functions
11(4)
1.11 Truncation Error and Taylor's Theorem
15(7)
Exercise 1
20(1)
References and Some Useful Related Books/Papers
21(1)
2 Linear Equations and Eigenvalue Problem
22(84)
2.1 Introduction
22(1)
2.2 Ill-conditioned Equations
23(1)
2.3 Inconsistency of Equations
23(1)
2.4 Linear Dependence
23(1)
2.5 Rank of a Matrix
24(1)
2.6 Augmented Matrix
24(1)
2.7 Methodology for Computing A-1 by Solving Ax = b
25(1)
2.8 Cramer's Rule
26(1)
2.9 Inverse of Matrix by Cofactors
26(1)
2.10 Definitions of Some Matrices
27(2)
2.11 Properties of Matrices
29(1)
2.12 Elementary Transformations
30(2)
2.13 Methods for Solving Equations (Direct Methods)
32(10)
2.13.1 Gaussian elimination method (Basic)
32(3)
2.13.2 Gaussian elimination (with row interchanges)
35(7)
2.14 LU Decomposition/Factorisation
42(13)
2.14.1 By Gaussian elimination method
42(2)
2.14.2 Crout's method
44(2)
2.14.3 Cholesky's method
46(4)
2.14.4 Reduction to PA = LU
50(5)
2.15 Gauss-Jordan (or Jordan's) Method
55(1)
2.16 Tridiagonal System
56(3)
2.17 Inversion of Matrix
59(3)
2.18 Number of Arithmetic Operations in Gaussian Elimination
62(1)
2.19 Eigenvalues and Eigenvectors
63(6)
2.20 Power Method to Find Dominant Eigenvalue/Latent Root
69(8)
2.20.1 To find smallest eigenvalue by power method
71(2)
2.20.2 Determination of subdominant eigenvalues
73(4)
2.21 Iterative Methods
77(4)
2.21.1 Gauss-Jacobi method
79(1)
2.21.2 Gauss-Seidel method
79(2)
2.22 Condition for Convergence of Iterative Methods
81(2)
2.23 Successive Over-Relaxation (S.O.R.) Method
83(2)
2.24 Norms of Vectors and Matrices
85(12)
2.24.1 Vector norm
85(1)
2.24.2 Matrix norm
86(2)
2.24.3 Forms of matrix norm
88(3)
2.24.4 Compatibility of matrix and vector norms
91(3)
2.24.5 Spectral norm
94(3)
2.25 Sensitivity of Solution of Linear Equations
97(9)
Exercise 2
100(5)
References and Some Useful Related Books/Papers
105(1)
3 Nonlinear Equations
106(29)
3.1 Introduction
106(1)
3.2 Order of Convergence of Iterative Method
107(1)
3.3 Method of Successive Substitution
108(2)
3.4 Bisection Method (Method of Halving)
110(2)
3.5 Regula--Falsi Method (or Method of False Position)
112(1)
3.6 Secant Method
113(1)
3.7 Convergence of Secant/Regula--Falsi Methods
114(3)
3.8 Newton--Raphson (N--R) Method
117(1)
3.8.1 Evaluation of some arithmetical functions
118(3)
3.8.2 Convergence of Newton--Raphson method
121(2)
3.8.3 Convergence when roots are repeated
123(1)
3.9 Simultaneous Equations
123(3)
3.9.1 Method of successive substitution
124(1)
3.9.2 Newton-Raphson method
124(2)
3.10 Complex Roots
126(3)
3.11 Bairstow's Method
129(6)
Exercise 3
133(1)
References and Some Useful Related Books/Papers
134(1)
4 Interpolation
135(63)
4.1 Introduction
135(1)
4.2 Some Operators and their Properties
136(7)
4.2.1 Linearity and commutativity of operators
136(1)
4.2.2 Repeated application and exponentiation of operators
137(1)
4.2.3 Interrelations between operators
137(3)
4.2.4 Application of operators on some functions
140(3)
4.3 Finite Difference Table
143(6)
4.3.1 Propagation of error in a difference table
147(2)
4.4 Error in Approximating a Function by Polynomial
149(2)
4.4.1 Justification for approximation by polynomial
151(1)
4.5 Newton's (Newton--Gregory) Forward Difference (FD) Formula
151(4)
4.5.1 Error in Newton's FD formula
153(2)
4.6 Newton's (Newton--Gregory) Backward Difference (BD) Formula
155(1)
4.7 Central Difference (CD) Formulae
155(15)
4.7.1 Gauss's Backward (GB) formula
156(3)
4.7.2 Gauss's Forward (GF) formula
159(1)
4.7.3 Stirling's formula
160(2)
4.7.4 Bessel's formula
162(2)
4.7.5 Everett's formula
164(2)
4.7.6 Steffensen's formula
166(3)
4.7.7 Comments on central difference formulae
169(1)
4.8 General Comments on Interpolation
170(10)
4.9 Lagrange's Method
180(1)
4.10 Divided Differences (DD)
181(7)
4.10.1 Divided differences are independent of order of arguments
183(3)
4.10.2 Newton's Divided Difference (DD) formula
186(2)
4.11 Lagrange's Formula Versus Newton's DD Formula
188(10)
4.1.2 Hermite's Interpolation
191(3)
Exercise 4
194(3)
References and Some Useful Related Books/Papers
197(1)
5 Numerical Differentiation
198(25)
5.1 Introduction
198(1)
5.2 Methodology for Numerical Differentiation
198(1)
5.3 Differentiation by Newton's FD Formula
199(3)
5.3.1 Error in differentiation
200(2)
5.4 Differentiation by Newton's BD Formula
202(6)
5.5 Differentiation by Central Difference Formulae
208(8)
5.5.1 At tabular- points
208(3)
5.5.2 At non-tabular points
211(5)
5.6 Method of Undetermined Coefficients
216(2)
5.7 Comments on Differentiation
218(1)
5.8 Derivatives with Unequal Intervals
218(5)
5.8.1 Forward Difference formulae
219(1)
5.8.2 Backward Difference formulae
220(1)
5.8.3 Central Difference formulae
221(1)
Exercise 5
221(1)
References and Some Useful Related Books/Papers
222(1)
6 Numerical Integration
223(52)
6.1 Introduction
223(1)
6.2 Methodology for Numerical Integration
223(2)
6.3 Rectangular Rule
225(3)
6.4 Trapezoidal Rule
228(3)
6.5 Simpson's 1/3rd Rule
231(4)
6.5.1 Comments on Simpson's 1/3rti rule
234(1)
6.6 Simpson's 3/8th Rule
235(1)
6.7 Weddle's Rule
235(5)
6.8 Open-Type Formulae
240(2)
6.9 Newton-Cotes (or Cotes) Formulae
242(3)
6.10 Method of Undetermined Coefficients
245(9)
6.11 Euler-Maclaurin Formula
249(5)
6.12 Richardson's Extrapolation
254(2)
6.13 Romberg Integration
256(3)
6.14 Comments on Numerical Integration
259(1)
6.15 Gaussian Quadrature
259(16)
6.15.1 Gauss--Legendre quadrature formula
260(9)
6.15.2 Gauss--Chebyshev quadrature formulae
269(1)
6.15.3 Gauss--Laguerre formula
270(1)
6.15.4 Gauss--Hermite formula
271(1)
Exercise 6
272(2)
References and Some Useful Related Books/Papers
274(1)
7 Ordinary Differential Equations
275(38)
7.1 Introduction
275(1)
7.2 Initial Value and Boundary Value Problems (IVP and BVP): Solution of IVP
276(1)
7.3 Reduction of Higher-Order IVP to System of First Order Equations
277(1)
7.4 Picard's Method (Method of Successive Approximations)
277(2)
7.5 Taylor's Series Method
279(3)
7.6 Numerical Method, its Order and Stability
282(1)
7.7 Euler's Method
283(4)
7.8 Modified (Improved) Euler's Method
287(2)
7.9 Runge--Kutta (R-K) Methods
289(6)
7.9.1 Application to first order simultaneous equations
292(3)
7.10 Predictor--Corrector (P--C) Methods
295(7)
7.10.1 Milne's method
296(3)
7.10.2 Adams--Bashforth method
299(3)
7.11 Boundary Value Problem (BVP)
302(6)
7.12 BVP as an Eigenvalue Problem
308(5)
Exercise 7
309(2)
References and Some Useful Related Books/Papers
311(2)
8 Splines and their Applications
313(41)
8.1 Introduction
313(1)
8.2 A Piece-Wise Polynomial
314(1)
8.3 Spline Approximation
314(1)
8.4 Uniqueness of Cubic Spline
315(1)
8.5 Construction of Cubic Spline (Second Derivative Form)
316(3)
8.6 Construction of Cubic Spline (First Derivative Form)
319(3)
8.7 Minimal Property of a Cubic Spline
322(9)
8.8 Application to Differential Equations
331(5)
8.9 Cubic Spline: Parametric Form
336(10)
8.10 Introduction to B-Splines
346(1)
8.11 Bezier Spline Curves
347(2)
8.12 Convex Polygon and Convex Hull
349(5)
Exercise 8
351(1)
References and Some Useful Related Books/Papers
352(2)
9 Method of Least Squares and Chebyshev Approximation
354(30)
9.1 Introduction
354(1)
9.2 Least Squares Method
354(3)
9.3 Normal Equations in Matrix Form
357(2)
9.4 Approximation by Standard Functions
359(4)
9.5 Over-Determined System of Linear Equations
363(3)
9.6 Approximation by Linear Combination of Functions
366(1)
9.7 Approximation by Orthogonal Polynomials
367(3)
9.8 Chebyshev Approximation
370(14)
Exercise 9
382(1)
References and Some Useful Related Books/Papers
383(1)
10 Eigenvalues of Symmetric Matrices
384(53)
10.1 Introduction
384(1)
10.2 Compact Form of Eigenvalues and Eigenvectors
385(1)
10.3 Eigenvalues of Powers of a Matrix
386(1)
10.4 Eigenvalues of Transpose of a Matrix
387(1)
10.5 Theorem: Eigenvectors of A and AT are Biorthogonal
387(1)
10.6 Corrollary: Eigenvectors of Symmetric Matrix form Orthogonal Set
388(1)
10.7 Theorem: Eigenvalues of Hermitian Matrix are Real
388(1)
10.8 Product of Orthogonal Matrices is an Orthogonal Matrix
389(1)
10.9 Eigenvalues of STAS when S is Orthogonal
390(1)
10.10 Eigenvectors of STAS when S is Orthogonal
390(1)
10.11 Methods to find Eigenvalues of Symmetric Matrix
390(1)
10.12 Jacobi's Method (Classical)
391(9)
10.12.1 Convergence of Jacobi method
396(1)
10.12.2 Cyclic Jacobi method
397(3)
10.13 Givens Method
400(5)
10.14 Householder's Method
405(10)
10.14.1 Matrix S is symmetric
405(1)
10.14.2 Matrix S is orthogonal
406(1)
10.14.3 Similarity transformation
406(1)
10.14.4 First transformation
407(3)
10.14.5 General procedure
410(5)
10.15 Sturm Sequence and its Properties
415(3)
10.15.1 Sturm sequence
415(1)
10.15.2 Theorem
416(2)
10.16 Eigenvalues of Symmetric Tridiagonal Matrix
418(2)
10.17 Upper and Lower Bounds of Eigenvalues
420(2)
10.17.1 Gerschgorin's theorem
420(1)
10.17.2 Corollary
421(1)
10.17.3 Brauer's theorem
421(1)
10.18 Determination of Eigenvectors
422(3)
10.19 LR Method
425(1)
10.20 QR Method
426(11)
Exercise 10
435(1)
References and Some Useful Related Books/Papers
436(1)
11 Partial Differential Equations
437(95)
11.1 Introduction
437(1)
11.2 Some Standard Forms
438(1)
11.3 Boundary Conditions
439(1)
11.4 Finite Difference Approximations for Derivatives
440(1)
11.5 Methods for Solving Parabolic Equation
441(37)
11.5.1 Explicit method/scheme/formula
442(1)
11.5.2 Fully Implicit scheme/method
443(1)
11.5.3 Crank--Nicolson's (C--N) scheme
444(1)
11.5.4 Comparison of three schemes
445(1)
11.5.5 Compatibility, stability and convergence
446(1)
11.5.6 Compatibility of explicit scheme
447(1)
11.5.7 Stability of" explicit scheme
448(5)
11.5.8 Stability of C--N scheme
453(2)
11.5.9 Further comparison of schemes
455(1)
11.5.10 Derivative boundary conditions
456(10)
11.5.11 Zero-time discontinuity at endpoints
466(3)
11.5.12 Parabolic equation in two dimensions
469(3)
11.5.13 Alternating Direction Implicit (ADI) method
472(5)
11.5.14 Non-rectangular space domains
477(1)
11.6 Methods for Solving Elliptic Equations
478(12)
11.6.1 Solution by Gauss-Seidel and Gaussian elimination
479(6)
11.6.2 Solution by SOR method
485(4)
11.6.3 Solution of elliptic equation by ADI method
489(1)
11.7 Methods for Solving Hyperbolic Equations
490(18)
11.7.1 Finite difference methods
491(1)
11.7.2 Explicit method
491(1)
11.7.3 Implicit method
492(1)
11.7.4 Stability analysis
493(4)
11.7.5 Characteristics of a partial differential equation
497(1)
11.7.6 Significance of characteristics
498(2)
11.7.7 Method of characteristics for solving hyperbolic equations
500(8)
11.8 Hyperbolic Equation of First Order
508(24)
11.8.1 Finite difference methods
510(1)
11.8.2 Lax--Wendroff's method
511(3)
11.8.3 Wendroff's method
514(1)
11.8.4 Other explicit/implicit methods
515(4)
11.8.5 Solving second order equation by simultaneous equations of first order
519(2)
11.8.6 Solution of first order hyperbolic equation by method of characteristics
521(4)
Exercise 11
525(6)
References and Some Useful Related Books/Papers
531(1)
12 Finite Element Method
532(88)
12.1 Introduction
532(1)
12.2 Weighted Residual Methods
533(7)
12.2.1 Galerkin's method
534(1)
12.2.2 Least squares method
534(1)
12.2.3 Subdomain method
534(1)
12.2.4 Collocation method
535(5)
12.3 Nom-homogeneous Boundary Conditions
540(1)
12.4 Variational Methods
541(5)
12.4.1 Functional and its variation
542(1)
12.4.2 Rayleigh--Ritz (or Ritz) method
543(3)
12.5 Equivalence of Rayleigh-Ritz and Galerkin Methods (1--D)
546(1)
12.6 Construction of Functional
547(9)
12.6.1 Preliminaries from vector calculus
545(4)
12.6.2 Minimum Functional Theorem (MFT)
549(6)
12.6.3 Application of MFT to one-dimension problem
555(1)
12.7 Equivalence of Rayleigh--Ritz and Galerkin Methods (2--D)
556(3)
12.8 Pre-requisites for Finite Element Method
559(8)
12.8.1 Shape functions
559(5)
12.8.2 Normalised/natural coordinates
564(3)
12.9 Finite Element Method
567(53)
12.9.1 Ordinary differential equation
567(16)
12.9.2 Elliptic equation
583(15)
12.9.3 Node-wise (point-wise) assembly
598(1)
12.9.4 Higher order elements
599(4)
12.9.5 Element of rectangular shape
603(2)
12.9.6 Parabolic equation (one dimension)
605(8)
12.9.7 Parabolic equation (two dimensions)
613(3)
12.9.8 Hyperbolic equation
616(1)
Exercise 12
616(3)
References and Some Useful Related Books/Papers
619(1)
13 Integral Equations
620(39)
13.1 Introduction
620(1)
13.2 Fredholm Integral Equations
620(1)
13.3 Volterra Integral Equations
621(1)
13.4 Green's Function
622(3)
13.5 Solution of Differential Equation Represented by Integral and Vice-Versa
625(2)
13.6 Reduction of Differential Equation to Integral Equation
627(4)
13.6.1 Reduction of a BVP to Fredholm equation
628(3)
13.6.2 Reduction of IVP to Volterra equation
631(1)
13.7 Methods for Solving Fredholm Equations
631(16)
13.7.1 Analytical method
632(5)
13.7.2 Classical iterative method
637(3)
13.7.3 Numerical method
640(7)
13.8 Methods for Solving Volterra Equation
647(12)
13.8.1 Numerical method
647(1)
13.8.2 Taylor's series method
648(2)
13.8.3 Iterative method
650(7)
Exercise 13
657(1)
References and Some Useful Related Books/Papers
658(1)
14 Difference Equations
659(17)
14.1 Introduction
659(1)
14.2 Method of Solution
660(8)
14.2.1 To find yH
660(2)
14.2.2 To find yp
662(6)
14.3 Simultaneous Difference Equations and Exponentiation of Matrix
668(8)
14.3.1 Property of constant Row-sum (Column-sum)
673(1)
Exercise 14
674(1)
References and Some Useful Related Books/Papers
675(1)
15 Fourier Series, Discrete Fourier Transform and Fast Fourier Transform
676(33)
15.1 Introduction
676(1)
15.2 Fourier Series
676(2)
15.3 Fourier Series with Other Intervals
678(1)
15.4 Half-Range Fourier Series
679(2)
15.5 Fourier Series for Discrete Data
681(4)
15.6 Fourier Transform
685(3)
15.7 Discrete Fourier Transform (DFT)
688(2)
15.8 Representation of Transforms in Matrix Form
690(1)
15.9 Complex Roots of Unity
691(5)
15.10 Fast Fourier Transform (FFT)
696(3)
15.11 Fast Fourier Transform via Inverse Transform (Author's Comments)
699(10)
Exercise 15
707(1)
References and some useful related books/papers
708(1)
16 Free and Moving Boundary Problems: A Brief Introduction
709(14)
16.1 Introduction
709(1)
16.2 Moving Boundary Problems
710(5)
16.3 Moving Grid Method (MGM)
715(5)
16.3.1 MGM with interpolations
716(3)
16.3.2 MGM without interpolations
719(1)
16.4 Free Boundary Problem
720(3)
References and Some Useful Related Books/Papers
721(2)
Appendices
723(7)
Appendix A Some Theorems and Formulae
723(3)
Appendix B Expansions of Some Functions
726(1)
Appendix C Graphs of Some Functions
727(3)
Answers to Exercises 730(25)
Index 755
Radhey S. Gupta is former professor and head in the Department of Mathematics at Indian Institute of Technology, Roorkee. He also served as Director at the Trinity Institute of Higher Education (Guru Gobind Singh Indraprastha University), New Delhi. Gupta was a visiting professor at Asian Institute of Technology, Bangkok and University of Florence, Italy. He has more than thirty-five years of experience in teaching and research in the field of numerical analysis with specialization in moving boundary problems.