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Numerical Analysis, 1/e

Book Description

A text book designed exclusively for undergraduate students, Numerical Analysis presents the theoretical and numerical derivations amply supported by rich pedagogy for practice. With exhaustive theory to reinforce practical computations, the book delves into the concepts of errors in numerical computation, algebraic and transcendental equations, solution of linear system of equation, curve fitting, initial-value problem for ordinary differential equations, boundary-value problems of second order partial differential equations and solution of difference equations with constant coefficient.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Preface
  5. About the Authors
  6. Acknowledgements
  7. CHAPTER 1 ERRORS IN NUMERICAL COMPUTATIONS
    1. 1.0. Introduction
    2. 1.1. Accuracy of Numbers
      1. 1.1.1 Significant Figures
      2. 1.1.2 Rounding Off of Numbers
      3. 1.1.3 A Safe Rule
    3. 1.2. Errors and their Analysis
      1. 1.2.1 Classification of Errors
      2. Worked Examples
    4. 1.3. A General Formula for Error
      1. Worked Examples
    5. 1.4. Error in Series Approximation
      1. 1.4.1 Error in Some Important Series
      2. Worked Examples
      3. Exercises 1.1
    6. Short Answer Questions
  8. CHAPTER 2 SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
    1. 2.0. Introduction
    2. 2.1. Bisection Method or Bolzano Method
      1. Worked Examples
    3. 2.2. Method of False Position or ­Regula-Falsi (in Latin)
      1. Worked Examples
    4. 2.3. The Secant Method or the Chord Method
      1. Worked Examples
    5. 2.4. The Method of Iteration or Fixed Point Iteration: x = f (x) Method
      1. Worked Examples
    6. 2.5. Newton-Raphson Method or Newton’s Method of Finding a Root of f(x) = 0
      1. Worked Examples
      2. Exercises 2.1
    7. 2.6. Generalised Newton-Raphson Method or Modified Newton’s Method
      1. Worked Examples
    8. 2.7. Ramanujan’s Method
      1. Worked Examples
    9. 2.8. Muller’s Method
      1. Worked Examples
    10. 2.9. Chebyshev’s Method
      1. Worked Examples
      2. Exercises 2.2
    11. 2.10. Convergence of Iteration Methods
      1. Worked Examples
    12. 2.11. Newton-Raphson Method for Non-Linear Equations in Two Variables
      1. Exercises 2.3
    13. 2.12 Solution of Polynomial Equations
      1. 2.12.1 Horner’s Method
      2. Worked Examples
      3. Exercises 2.4
      4. 2.12.2 Graffe’s Root-Squaring Method
      5. Worked Examples
      6. Exercises 2.5
      7. 2.12.3 Lin-Bairstow’s Method
      8. Worked Examples
      9. Exercises 2.6
    14. Short Answer Questions
  9. CHAPTER 3 SOLUTION OF SYSTEM OF LINEAR ALGEBRAIC EQUATIONS
    1. 3.0. Introduction
    2. 3.1. Direct Methods
      1. 3.1.1 Matrix Inverse Method
      2. Worked Examples
      3. 3.1.2 Gauss Elimination Method
      4. Worked Examples
      5. 3.1.3 Gauss-Jordan Method
      6. Worked Examples
      7. 3.1.4 Matrix Inverse by Gauss-Jordan Method
      8. Worked Examples
      9. Exercises 3.1
    3. 3.2. Iterative Methods
      1. 3.2.1 Gauss-Jacobi Method
      2. Worked Examples
      3. 3.2.2 Gauss-Seidel Method
      4. Worked Examples
      5. Exercises 3.2
    4. 3.3. Eigen Value Problem
      1. 3.3.1 Power Method
      2. Worked Examples
      3. Exercises 3.3
      4. 3.3.2 Jacobi’s Method to Find the Eigen Values of a Symmetric Matrix
      5. Worked Examples
      6. Exercises 3.4
    5. 3.4. Method of Factorisation or Method of Triangularisation
      1. 3.4.1 Doolittle’s Method
      2. Worked Examples
      3. 3.4.2 Crout’s Method
      4. Worked Examples
      5. 3.4.3 Cholesky Decomposition
      6. Worked Examples
      7. Exercises 3.5
    6. Short Answer Questions
  10. CHAPTER 4 POLYNOMIAL INTERPOLATION
    1. 4.0. Introduction
    2. 4.1. Finite Difference Operators
      1. 4.1.1 Forward Difference Operator ?
      2. 4.1.3 Shift Operator or Displacement Operator E
      3. 4.1.4 Relation Between the Operators E, ?, ?
      4. 4.1.5 Properties of ? and E
      5. Worked Examples
      6. Exercises 4.1
      7. 4.1.6 Factorial Polynomial
    3. 4.2. Interpolation with Equally Spaced Arguments or Interpolation with Equal Intervals
      1. 4.2.1 Newton’s Forward Formula for Interpolation
      2. 4.2.2 Newton’s Backward Formula for Interpolation
      3. Worked Examples
      4. Exercises 4.2
    4. 4.3. Central Difference Interpolation Formulae
      1. 4.3.1 Gauss’s Forward Formula for Interpolation
      2. Worked Examples
      3. 4.3.2 Gauss’s Backward Formula for Interpolation
      4. Worked Examples
      5. Exercises 4.3
      6. 4.3.3 Stirling’s Formula for Interpolation
      7. Worked Examples
      8. 4.3.4 Bessel’s Formula for Interpolation
      9. Worked Examples
      10. 4.3.5 Laplace-Everett Formula for Interpolation
      11. Worked Examples
      12. Exercises 4.4
    5. 4.4. Interpolation with Unequal Intervals
      1. 4.4.1 Lagrange’s Interpolation Formula
      2. Worked Examples
      3. Exercises 4.5
      4. 4.4.2 Divided Differences
      5. Worked Examples
      6. 4.4.3 Newton’s General Interpolation Formula or Newton’s Divided Dif-ference Formula for Interpolation
      7. Worked Examples
      8. Exercises 4.6
    6. 4.5. Errors in Interpolation Formulae
      1. 4.5.1 Remainder Term in Interpolation Formulae
      2. Worked Examples
    7. 4.6. Interpolation with a Cubic Spline
      1. 4.6.0 Introduction
      2. 4.6.1 Cubic Spline Interpolation
      3. Worked Examples
      4. Exercises 4.7
    8. Short Answer Questions
  11. CHAPTER 5 INVERSE INTERPOLATION
    1. 5.0. Introduction
    2. 5.1. Lagrange’s Inverse Interpolation Formula
      1. Worked Examples
      2. Exercises 5.1
    3. 5.2. Successive Approximation Method or Iteration Method
      1. Worked Examples
      2. Exercises 5.2
    4. 5.3. Reversion of Series Method
      1. Worked Examples
      2. Exercises 5.3
    5. Short Answer Questions
  12. CHAPTER 6 NUMERICAL DIFFERENTIATION
    1. 6.0. Introduction
    2. 6.1. Numerical Differentiation
      1. 6.1.1 Derivative Using Newton’s Forward Difference Interpo-lating Formula
      2. 6.1.2 Derivative Using Newton’s Backward Difference Interpo-lating Formula
      3. Worked Examples
      4. Exercises 6.1
    3. 6.2. Maxima and Minima of Tabulated Function
      1. Worked Examples
      2. Exercises 6.2
    4. Short Answer Questions
  13. CHAPTER 7 NUMERICAL INTEGRATION
    1. 7.0. Introduction
    2. 7.1. A General Quadrature Formula or Newton-Cotes Quadrature Formula
    3. 7.2. Trapezoidal Rule
      1. 7.2.1 Geometrical Meaning
      2. Worked Examples
    4. 7.3. Simpson’s Rule or Simpson’s Rule
      1. 7.3.1 Geometrical Meaning
      2. Worked Examples
    5. 7.4. Simpson’s Rule
      1. 7.4.1 Geometrical Meaning
      2. Worked Examples
    6. 7.5. Boole’s Rule
      1. Worked Examples
    7. 7.6. Weddle’s Rule
      1. Worked Examples
    8. 7.7. Error in Numerical Integration Formulae
      1. 7.7.1 Error in Trapezoidal Rule
      2. 7.7.2 Error in Simpson’s Rule
      3. Exercises 7.1
    9. 7.8. Romberg’s Method for Integration
      1. 7.8.1 Romberg’s Integration Formula Based on Trapezoidal Rule
      2. Worked Examples
      3. 7.8.2 Romberg Integration Formula Based on Simpson’s Rule
      4. Worked Examples
    10. 7.9. Two and Three Point Gaussian Quadrature Formulae
      1. 7.9.0 Introduction
      2. 7.9.1 Two Point Gaussian Quadrature Formula
      3. 7.9.2 Three Point Gaussian Quadrature Formula
      4. Worked Examples
      5. Exercises 7.2
    11. 7.10. Euler-Maclaurin Formula for Numerical Integration
      1. Worked Examples
      2. 7.10.1 Application of Euler-Maclaurin Formula
      3. Worked Examples
      4. Exercises 7.3
    12. 7.11. Double Integration
      1. 7.11.1 Trapezoidal Rule for Double Integral
      2. Worked Examples
      3. 7.11.2 Simpson’s Rule for Double Integral
      4. Worked Examples
      5. Exercises 7.4
    13. Short Answer Questions
  14. CHAPTER 8 CURVE FITTING
    1. 8.0. Introduction
    2. 8.1. Method of Least Squares
      1. 8.1.1 Fit a Straight Line by the Method of Least Squares
      2. Worked Examples
      3. 8.1.1 (a) Fitting Other Type of Equations Reducible to the Form
      4. Worked Examples
      5. 8.1.1 (b) Fit a Parabola y ? ax2 ? bx ? c by the Method of Least Squares
      6. Worked Examples
      7. Exercises 8.1
    3. 8.2. Method of Group Averages
      1. Worked Examples
      2. Exercises 8.2
    4. 8.3. Method of the Sum of Exponentials
      1. Worked Examples
      2. Exercises 8.3
    5. 8.4. Method of Moments
      1. Worked Examples
      2. Exercises 8.4
    6. Short Answer Questions
  15. CHAPTER 9 INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS
    1. 9.0. Introduction
    2. 9.1. Taylor’s Series Method
      1. Worked Examples
    3. 9.2. Euler’s Method and Modified Euler’s Method
      1. Worked Examples
      2. Exercises 9.1
    4. 9.3. Runge-Kutta Method (R-K Method)
      1. Worked Examples
    5. 9.4. Runge-Kutta Method for the Solution of Simultaneous Equations and Second Order Equations
      1. 9.4.1 Runge-Kutta Method for Simultaneous Equations
      2. Worked Examples
      3. 9.4.2 Runge-Kutta Method for Second Order Equations
      4. Worked Examples
      5. Exercises 9.2
    6. 9.5. Milne’s Predictor–Corrector Method
      1. Worked Examples
    7. 9.6. Adam’s Predictor and Corrector Method
      1. Worked Examples
      2. Exercises 9.3
    8. 9.7. Picard’s Method
      1. 9.7.1 Picard’s Method of Successive Approximations
      2. Worked Examples
      3. Exercises 9.4
    9. Short Answer Questions
  16. CHAPTER 10 BOUNDARY VALUE PROBLEMS IN ORDINARY AND PARTIAL DIFFERENTIAL EQUATION
    1. 10.0. Introduction
    2. 10.1 Finite Difference Methods for Solution of Second Order Ordinary Differential Equations
      1. Worked Examples
      2. Exercises 10.1
    3. 10.2. Numerical Solution of Partial Differential Equations
      1. 10.2.1 Classifications of Second Order Partial Differential Equations
      2. Worked Examples
      3. 10.2.2 Finite Difference Approximations to Partial Derivatives
      4. 10.2.3 Solution of Laplace Equation
      5. Worked Examples
      6. 10.2.4 Poisson Equation
      7. Worked Examples
    4. 10.3. One Dimensional Heat Equation
      1. 10.3.1 Schmidt’s Method [Explicit Method]
      2. 10.3.2 Crank-Nicolson Method [Implicit Method]
      3. Worked Examples
    5. 10.4. One-Dimensional Wave Equation
      1. Worked Examples
      2. Exercises 10.2
    6. Short Answer Questions
  17. CHAPTER 11 DIFFERENCE EQUATIONS
    1. 11.0 Introduction
    2. 11.1 Linear Difference Equation
    3. 11.2 Solution of a Difference Equation
    4. 11.3 Formation of a Difference Equation
      1. Worked Examples
      2. Exercises 11.1
    5. 11.4. Linear Homogeneous Difference Equation with Constant Coefficients
      1. 11.4.1 Working Rule
    6. 11.5. Some Basic Results of Difference Operator to Solve Difference Equations
      1. Worked Examples
      2. Exercises 11.2
    7. 11.6. Non-Homogeneous Linear Difference Equations with Constant Coefficients
      1. 11.6.1 Evaluation of Particular Integrals
      2. Worked Examples
      3. Exercises 11.3
    8. 11.7. First Order Linear Difference Equation with Variable Coefficients
      1. 11.7.1 First Order Linear Homogeneous Difference Equation with Variable Coefficients
    9. Short Answer Questions
  18. Bibliography