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