book

Numerical Analysis, 1/e

Name: Numerical Analysis, 1/e
Author: Das Siva Ramakrishna
ISBN: 9789332540804

by Das Siva Ramakrishna

April 2014

Intermediate to advanced

768 pages

15h 41m

English

Pearson Education India

Read now

Unlock full access

Cover
Title Page
Contents
Preface
About the Authors
Acknowledgements
CHAPTER 1 ERRORS IN NUMERICAL COMPUTATIONS
1.0. Introduction 1.1. Accuracy of Numbers 1.1.1 Significant Figures1.1.2 Rounding Off of Numbers1.1.3 A Safe Rule1.2. Errors and their Analysis 1.2.1 Classification of ErrorsWorked Examples1.3. A General Formula for Error Worked Examples1.4. Error in Series Approximation 1.4.1 Error in Some Important SeriesWorked ExamplesExercises 1.1Short Answer Questions
CHAPTER 2 SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
2.0. Introduction 2.1. Bisection Method or Bolzano Method Worked Examples2.2. Method of False Position or Regula-Falsi (in Latin) Worked Examples2.3. The Secant Method or the Chord Method Worked Examples2.4. The Method of Iteration or Fixed Point Iteration: x = f (x) Method Worked Examples2.5. Newton-Raphson Method or Newton’s Method of Finding a Root of f(x) = 0 Worked ExamplesExercises 2.12.6. Generalised Newton-Raphson Method or Modified Newton’s Method Worked Examples2.7. Ramanujan’s Method Worked Examples2.8. Muller’s Method Worked Examples2.9. Chebyshev’s Method Worked ExamplesExercises 2.22.10. Convergence of Iteration Methods Worked Examples2.11. Newton-Raphson Method for Non-Linear Equations in Two Variables Exercises 2.32.12 Solution of Polynomial Equations 2.12.1 Horner’s MethodWorked ExamplesExercises 2.42.12.2 Graffe’s Root-Squaring MethodWorked ExamplesExercises 2.52.12.3 Lin-Bairstow’s MethodWorked ExamplesExercises 2.6Short Answer Questions
CHAPTER 3 SOLUTION OF SYSTEM OF LINEAR ALGEBRAIC EQUATIONS
3.0. Introduction 3.1. Direct Methods 3.1.1 Matrix Inverse MethodWorked Examples3.1.2 Gauss Elimination MethodWorked Examples3.1.3 Gauss-Jordan MethodWorked Examples3.1.4 Matrix Inverse by Gauss-Jordan MethodWorked ExamplesExercises 3.13.2. Iterative Methods 3.2.1 Gauss-Jacobi MethodWorked Examples3.2.2 Gauss-Seidel MethodWorked ExamplesExercises 3.23.3. Eigen Value Problem 3.3.1 Power Method Worked ExamplesExercises 3.33.3.2 Jacobi’s Method to Find the Eigen Values of a Symmetric MatrixWorked ExamplesExercises 3.43.4. Method of Factorisation or Method of Triangularisation 3.4.1 Doolittle’s MethodWorked Examples3.4.2 Crout’s MethodWorked Examples3.4.3 Cholesky DecompositionWorked ExamplesExercises 3.5Short Answer Questions
CHAPTER 4 POLYNOMIAL INTERPOLATION
4.0. Introduction 4.1. Finite Difference Operators 4.1.1 Forward Difference Operator ?4.1.3 Shift Operator or Displacement Operator E4.1.4 Relation Between the Operators E, ?, ?4.1.5 Properties of ? and EWorked ExamplesExercises 4.14.1.6 Factorial Polynomial4.2. Interpolation with Equally Spaced Arguments or Interpolation with Equal Intervals 4.2.1 Newton’s Forward Formula for Interpolation4.2.2 Newton’s Backward Formula for InterpolationWorked ExamplesExercises 4.24.3. Central Difference Interpolation Formulae 4.3.1 Gauss’s Forward Formula for InterpolationWorked Examples4.3.2 Gauss’s Backward Formula for InterpolationWorked ExamplesExercises 4.34.3.3 Stirling’s Formula for InterpolationWorked Examples4.3.4 Bessel’s Formula for InterpolationWorked Examples4.3.5 Laplace-Everett Formula for InterpolationWorked ExamplesExercises 4.44.4. Interpolation with Unequal Intervals 4.4.1 Lagrange’s Interpolation FormulaWorked ExamplesExercises 4.54.4.2 Divided DifferencesWorked Examples4.4.3 Newton’s General Interpolation Formula or Newton’s Divided Dif-ference Formula for InterpolationWorked ExamplesExercises 4.64.5. Errors in Interpolation Formulae 4.5.1 Remainder Term in Interpolation FormulaeWorked Examples4.6. Interpolation with a Cubic Spline 4.6.0 Introduction4.6.1 Cubic Spline InterpolationWorked ExamplesExercises 4.7Short Answer Questions

CHAPTER 5 INVERSE INTERPOLATION
5.0. Introduction 5.1. Lagrange’s Inverse Interpolation Formula Worked ExamplesExercises 5.15.2. Successive Approximation Method or Iteration Method Worked ExamplesExercises 5.25.3. Reversion of Series Method Worked ExamplesExercises 5.3Short Answer Questions
CHAPTER 6 NUMERICAL DIFFERENTIATION
6.0. Introduction 6.1. Numerical Differentiation 6.1.1 Derivative Using Newton’s Forward Difference Interpo-lating Formula6.1.2 Derivative Using Newton’s Backward Difference Interpo-lating FormulaWorked ExamplesExercises 6.16.2. Maxima and Minima of Tabulated Function Worked ExamplesExercises 6.2Short Answer Questions
CHAPTER 7 NUMERICAL INTEGRATION
7.0. Introduction 7.1. A General Quadrature Formula or Newton-Cotes Quadrature Formula 7.2. Trapezoidal Rule 7.2.1 Geometrical MeaningWorked Examples7.3. Simpson’s Rule or Simpson’s Rule 7.3.1 Geometrical MeaningWorked Examples7.4. Simpson’s Rule 7.4.1 Geometrical MeaningWorked Examples7.5. Boole’s Rule Worked Examples7.6. Weddle’s Rule Worked Examples7.7. Error in Numerical Integration Formulae 7.7.1 Error in Trapezoidal Rule7.7.2 Error in Simpson’s RuleExercises 7.17.8. Romberg’s Method for Integration 7.8.1 Romberg’s Integration Formula Based on Trapezoidal RuleWorked Examples7.8.2 Romberg Integration Formula Based on Simpson’s RuleWorked Examples7.9. Two and Three Point Gaussian Quadrature Formulae 7.9.0 Introduction7.9.1 Two Point Gaussian Quadrature Formula7.9.2 Three Point Gaussian Quadrature FormulaWorked ExamplesExercises 7.27.10. Euler-Maclaurin Formula for Numerical Integration Worked Examples7.10.1 Application of Euler-Maclaurin FormulaWorked ExamplesExercises 7.37.11. Double Integration 7.11.1 Trapezoidal Rule for Double IntegralWorked Examples7.11.2 Simpson’s Rule for Double IntegralWorked ExamplesExercises 7.4Short Answer Questions
CHAPTER 8 CURVE FITTING
8.0. Introduction 8.1. Method of Least Squares 8.1.1 Fit a Straight Line by the Method of Least SquaresWorked Examples8.1.1 (a) Fitting Other Type of Equations Reducible to the Form Worked Examples8.1.1 (b) Fit a Parabola y ? ax2 ? bx ? c by the Method of Least SquaresWorked ExamplesExercises 8.18.2. Method of Group Averages Worked ExamplesExercises 8.28.3. Method of the Sum of Exponentials Worked ExamplesExercises 8.38.4. Method of Moments Worked ExamplesExercises 8.4Short Answer Questions
CHAPTER 9 INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS
9.0. Introduction 9.1. Taylor’s Series Method Worked Examples9.2. Euler’s Method and Modified Euler’s Method Worked ExamplesExercises 9.19.3. Runge-Kutta Method (R-K Method) Worked Examples9.4. Runge-Kutta Method for the Solution of Simultaneous Equations and Second Order Equations 9.4.1 Runge-Kutta Method for Simultaneous EquationsWorked Examples9.4.2 Runge-Kutta Method for Second Order EquationsWorked ExamplesExercises 9.29.5. Milne’s Predictor–Corrector Method Worked Examples9.6. Adam’s Predictor and Corrector Method Worked ExamplesExercises 9.39.7. Picard’s Method 9.7.1 Picard’s Method of Successive ApproximationsWorked ExamplesExercises 9.4 Short Answer Questions
CHAPTER 10 BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATION
10.0. Introduction 10.1 Finite Difference Methods for Solution of Second Order Ordinary Differential Equations Worked ExamplesExercises 10.110.2. Numerical Solution of Partial Differential Equations 10.2.1 Classifications of Second Order Partial Differential EquationsWorked Examples10.2.2 Finite Difference Approximations to Partial Derivatives10.2.3 Solution of Laplace Equation Worked Examples10.2.4 Poisson EquationWorked Examples10.3. One Dimensional Heat Equation 10.3.1 Schmidt’s Method [Explicit Method]10.3.2 Crank-Nicolson Method [Implicit Method]Worked Examples10.4. One-Dimensional Wave Equation Worked ExamplesExercises 10.2Short Answer Questions
CHAPTER 11 DIFFERENCE EQUATIONS
11.0 Introduction 11.1 Linear Difference Equation 11.2 Solution of a Difference Equation 11.3 Formation of a Difference Equation Worked ExamplesExercises 11.111.4. Linear Homogeneous Difference Equation with Constant Coefficients 11.4.1 Working Rule11.5. Some Basic Results of Difference Operator to Solve Difference Equations Worked ExamplesExercises 11.211.6. Non-Homogeneous Linear Difference Equations with Constant Coefficients 11.6.1 Evaluation of Particular IntegralsWorked ExamplesExercises 11.311.7. First Order Linear Difference Equation with Variable Coefficients 11.7.1 First Order Linear Homogeneous Difference Equation with Variable CoefficientsShort Answer Questions
Bibliography

Content preview from Numerical Analysis, 1/e

Numerical Analysis

P. Sivaramakrishna Das Retired Professor and Head, Department of Mathematics Ramakrishna Mission Vivekananda College Chennai Currently Head, Department of Science and Humanities Professor of Mathematics KCG College of Technology Chennai

C. Vijayakumari Professor of Mathematics (Retired) Queen Mary’s College (Autonomous) Chennai

Chennai • Delhi

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.

Read now

Unlock full access

More than 5,000 organizations count on O’Reilly

O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.

Julian F.

Head of Cybersecurity

I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.

Addison B.

Field Engineer

I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.

Amir M.

Data Platform Tech Lead

I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.

Mark W.

Embedded Software Engineer

Introduction to Numerical Analysis and Scientific Computing

Publisher Resources

ISBN: 9789332540804

Cloud Computing

Data Engineering

Data Science

AI & ML

Programming Languages

Software Architecture

IT/Ops

Security

Design

Business

Soft Skills

Numerical Analysis, 1/e

by Das Siva Ramakrishna

Numerical Analysis

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.