Numerical Methods: Principles, Analysis, and Algorithms

Paperback | December 1, 2008

bySrimanta Pal

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With an objective to provide a firm understanding of the basic concepts of numerical methods, the book provides introductory chapters on number systems, binary arithmetic, and programming tools and techniques for different programming languages such as C, C++, BASIC and FORTRAN. Subsequently,the book offers an exhaustive coverage of topics such numerical solutions of linear and non-linear equations, eigenvalues and eigenvectors, linear least squares problem with interpolation and extrapolation, numerical differentiation and integration, ordinary differential equations, partialdifferential equations, and parabolic and elliptic partial differential equations.Written in a lucid style, the book contains a large number of solved examples and numerous end-chapter exercises to make for a student-friendly book. The book will also be useful to postgraduate students as also to practising numerical analysts, statisticians, and engineers.

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With an objective to provide a firm understanding of the basic concepts of numerical methods, the book provides introductory chapters on number systems, binary arithmetic, and programming tools and techniques for different programming languages such as C, C++, BASIC and FORTRAN. Subsequently,the book offers an exhaustive coverage of to...

Dr. Srimanta Pal is currently Associate Professor, Indian Statistical Institute, Calcutta. A Phd from IIT Kharagpur, he obtained an MBA from Jadavpur University after completing his M Tech from ISI Calcutta. He has a teaching experience of close to two decades in prestigious institutions such as Indian Institute of Management, Calcutt...

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Format:PaperbackDimensions:784 pages, 10 × 7.5 × 0.68 inPublished:December 1, 2008Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0195693752

ISBN - 13:9780195693751

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Table of Contents

1. Background to Numerical Methods1.1 Introduction1.2 Number Systems and Their Conversions1.3 Representation of Negative Numbers1.4 Addition and Subtraction Rules1.5 Binary Multiplication and Division1.6 Fixed-Point Representation1.7 Floating-Point Representation1.8 Actual Computer Number System2. Scope of Mathematical and Numerical Methods2.1 Introduction2.2 Numerical Methods2.3 Mathematical Formula and Numerical Computation2.4 Stability of Numerical Solution2.5 Safe-Guard for Wrong Use of Mathematical Results2.6 Practical Guidelines for Numerical Computation2.7 Remarks3. Errors and Their Propagation3.1 Introduction3.2 Basic Sources of Errors3.3 Errors in Measurement3.4 Absolute, Relative and Percentage Err3.5 Errors in Finite-Precision Representation3.6 Error Propagation3.7 Process Graphs for Propagated Errors3.8 Extended Additions andMultiplications3.9 Error Analysis3.10 Truncation Error3.11 Error Bounds in a Number Approximation4. Programming Tools and Techniques4.1 Introduction4.2 Algorithms4.3 Flowchart4.4 BASIC4.5 FORTRAN4.6 PASCAL4.7 C4.8 MATLAB4.9 MATHEMATICA4.10 C, Pascal and FORTRAN Constructs4.11 Example 14.12 Example 25. Numerical Solution of One-Dimensional Non-Linear Equations5.1 Introduction5.2 InitialWork and First Approximation5.3 TheMethod of Bisection5.4 Regula Falsi (Method of False Position)5.5 BisectionMethod vs. Regula FalsiMethods5.6 Ridders'Method5.7 General IterativeMethods5.8 Linear IterativeMethod5.9 Aitken's?-Method5.10 Newton-RaphsonMethod5.11 SecantMethod5.12 KiznerMethod5.13 Brent'sMethod5.14 Polynomial Equations5.15 IterativeMethods for Polynomial Equations5.16 Laguerre's Method5.17 M"uller'sMethod5.18 Bairstow-HitchcockMethod for Complex Roots5.19 Bernoulli's Method5.20 Graeffe's Root-SquaringMethod5.21 The Quotient-Difference (QD) Algorithm5.22 Comments and Discussions with Examples5.23 Applications6. Numerical Solution of Linear Equations: Direct Methods6.1 Introduction6.2 Gaussian Elimination and Triangular Systems6.3 Gauss-JordanMethod6.4 Error and Sensitivity Analysis6.5 Iterative RefinementWith Gaussian Elimination6.6 Wilkinson Algorithm6.7 Cholesky Factorization6.8 Complex Systemof Linear Equations7. Numerical Solution for Matrix In7.1 Introduction7.2 Two-ArrayMethod7.3 Gauss-Jordan Two-ArrayMethodWith Pivoting7.4 Inverse in PlaceWithout Pivoting7.5 Inverse in PlaceWith Pivoting7.6 Inverses of TriangularMatrices7.7 Inverses of ComplexMatrices7.8 Iterative Procedure8. Numerical Solution of Linear Systems of Equations: Iterative Methods8.1 Introduction8.2 Nature of IterativeMethods for Linear Equations8.3 Point IterativeMethods8.4 Computational Techniques for Point Iterative Methods8.5 Block IterativeMethods9. Eigenvalues and Eigenvectors9.1 Introduction9.2 The Problem9.3 Power and Inverse PowerMethods9.4 Eigenvalues and Eigenvectors of SymmetricMatrices9.5 Generalized Eigenvalue Problem9.6 Methods for Non-SymmetricMatrices10. Numerical Solution of Systems of Nonlinear Equations10.1 Introduction10.2 Problem10.3 Generalized Linear IterativeMethod10.4 Newton'sMethod10.5 Generalized LinearMethods11. Linear Least Squares Problem11.1 Introduction11.2 Existence of Solution: Normal Equations11.3 Solution of Normal Equations11.4 Orthogonal Triangularization11.5 Solution of Linear Least Squares Problem12. Interpolation and Extrapolation12.1 Introduction12.2 LinearMethod of Interpolation12.3 LagrangianMethod of Interpolation12.4 Iterated Linear Interpolation12.5 Newton's Divided Difference Interpolation12.6 Difference Operators12.7 Equal Interval Finite Difference Methods12.8 Different Finite Difference Interpolation Formulas12.9 Correction of Tabular Values12.10 Inverse Interpolation12.11 Osculating Polynomials12.12 Chebyshev Interpolation12.13 Multi-Dimensional Interpolation12.14 Piecewise Polynomial Interpolation13. Numerical Differentiation13.1 Introduction13.2 Differentiation13.3 Differentiation Formulae for Numerical Computation13.4 Computational Problems13.5 Extrapolation in Derivative Computation13.6 Application to Solving a Differential Equation14. Numerical Integration14.1 Introduction14.2 Quadrature Formula14.3 Methods Based on Difference Polynomials14.4 Newton-Cotes Quadrature Formulae14.5 Computers in Numerical Integration14.6 Error Analysis in Trapezoidal Rule14.7 Romberg Integration14.8 Adaptive Quadrature14.9 Gaussian Quadrature14.10 Orthogonal Polynomials14.11 Lagrangian Interpolating Polynomials14.12 Gaussian Quadrature Problem14.13 Use of Gaussian Quadrature14.14 Comparison of Integration Formulae14.15 Spline Integration14.16 Integrals With Infinite Range of Integration14.17 Singular Integrals14.18 Multiple Integration14.19 Application of Quadrature Rules14.20 Numerical Integration UsingMonte CarloM15. Numerical Solution of Ordinary Differential Equations: Initial Value Problem15.1 Introduction15.2 First Order Differential Equation15.3 Intuitive Meaning of Solution of a Differential Equation15.4 Some Theoretical Results15.5 Semi-NumericMethod15.6 Simple Difference Methods15.7 Single StepMethods15.8 Runge-Kutta (RK) Methods15.9 Multi-StepMethods15.10 Predictor-Corrector (PC) Methods15.11 Multivalued Methods15.12 Choice of A Method15.13 System of First Order Ordinary Differential Equations15.14 Higher-Order Ordinary Differential Equations16. Numerical Solution of Ordinary Differential Equations: Boundary Value Problem16.1 Introduction16.2 Problem16.3 Shooting Methods16.4 Finite Difference Methods16.5 Finite Elements Methods16.6 Relaxation Methods17. Partial Differential Equations17.1 Introduction17.2 Definitions and Terminology17.3 The Classification of Partial Differential Equations17.4 Some Standard Partial Differential Equations17.5 Methodology to Solve Partial Differential Equations18. Numerical Solution of Parabolic Partial Differential Equations18.1 Introduction18.2 Derivation of Heat Equation18.3 Why Numerical Methods?18.4 Explicit Metho18.5 Implicit Method18.6 An Unified Explicit and Implicit Scheme18.7 Iterative Methods18.8 Improve Spatial Accuracy18.9 Generalization18.10 Non-Linear Parabolic Partial Differential Equations18.11 Parabolic Partial Differential Equation in 2-D or 3-D18.12 Alternating Direction Implicit (A.D.I.) Method19. Numerical Solution of Hyperbolic Partial Differential Equations19.1 Introduction19.2 Derivation ofWave Equation19.3 Simple Finite Difference Methods19.4 Second Order Hyperbolic Equations19.5 Method of Characteristics for Hyperbolic Partial Differential Equations19.6 Hyperbolic Differential Equations in 2-D or 3-D20. Numerical Solution of Elliptic Partial Differential Equations20.1 Introduction20.2 Direct Method20.3 Finite Difference Methods20.4 ADI Method20.5 Relaxation Methods20.6 Finite Element Methods21. Advances in Numerical Methods Using Parallel Computing Paradigm21.1 Introduction21.2 Parallel Computing21.3 Parallel Programming21.4 Basic Numerical Operations in Parallel Computing21.5 Root FromOne Dimensional Nonlinear Equation21.6 Interpolation21.7 Integration21.8 System of Linear Equations21.9 Differential Equations22. Advances in Numerical Methods Using Neurocomputing Paradigm22.1 Introduction22.2 Biological Neural Network22.3 Artificial Neural Network22.4 Network Architecture22.5 Learning22.6 Multilayer Perceptron (MLP)22.7 Radial Basis Function (RBF) Neural Network22.8 Hopfield Network22.9 Function Approximation UsingMLP22.10 Numerical Differentiation Using RBF22.11 Numerical Integration UsingMLP22.12 Solution of Non-Linear Equation Using Neural Networks22.13 Matrix Inversion Using Hopfield Network22.14 System of Linear Equation22.15 Eigenvalues and Eigenvectors Computation22.16 Interpolation Using Neural Networks22.17 Extrapolation Using Neural Network22.18 Neural Networks for Differential Equations23. Numerical Solution of Difference Equations23.1 Introduction23.2 Difference Equations23.3 Computational Problems23.4 Second Order Boundary-Value Problem