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Numerical analysis with algorithms and programming / Santanu Saha Ray

Author
Saha Ray, Santanu
Published
Boca Raton : CRC Press, Taylor & Francis Group, [2016]
Physical Description
xix, 685 pages : illustrations ; 26 cm

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Contents
Machine generated contents note: 1.1.Introduction -- 1.2.Preliminary Mathematical Theorems -- 1.3.Approximate Numbers and Significant Figures -- 1.3.1.Significant Figures -- 1.3.1.1.Rules of Significant Figures -- 1.4.Rounding Off Numbers -- 1.4.1.Absolute Error -- 1.4.2.Relative and Percentage Errors -- 1.4.2.1.Measuring Significant Digits in xA -- 1.4.3.Inherent Error -- 1.4.4.Round-Off and Chopping Errors -- 1.5.Truncation Errors -- 1.6.Floating Point Representation of Numbers -- 1.6.1.Computer Representation of Numbers -- 1.7.Propagation of Errors -- 1.8.General Formula for Errors -- 1.9.Loss of Significance Errors -- 1.10.Numerical Stability, Condition Number, and Convergence -- 1.10.1.Condition of a Problem -- 1.10.2.Stability of an Algorithm -- 1.11.Brief Idea of Convergence -- Exercises -- 2.1.Introduction -- 2.2.Basic Concepts and Definitions -- 2.2.1.Sequence of Successive Approximations -- 2.2.2.Order of Convergence -- 2.3.Initial Approximation -- 2.3.1.Graphical Method -- 2.3.2.Incremental Search Method -- 2.4.Iterative Methods -- 2.4.1.Method of Bisection -- 2.4.1.1.Order of Convergence of the Bisection Method -- 2.4.1.2.Advantage and Disadvantage of the Bisection Method -- 2.4.1.3.Algorithm for the Bisection Method -- 2.4.2.Regula-Falsi Method or Method of False Position -- 2.4.2.1.Order of Convergence of the Regula-Falsi Method -- 2.4.2.2.Advantage and Disadvantage of the Regula-Falsi Method -- 2.4.2.3.Algorithm for the Regula-Falsi Method -- 2.4.3.Fixed-Point Iteration -- 2.4.3.1.Condition of Convergence for the Fixed-Point Iteration Method -- 2.4.3.2.Acceleration of Convergence: Aitken's A2-Process -- 2.4.3.3.Advantage and Disadvantage of the Fixed-Point Iteration Method -- 2.4.3.4.Algorithm of the Fixed-Point Iteration Method -- 2.4.4.Newton-Raphson Method -- 2.4.4.1.Condition of Convergence -- 2.4.4.2.Order of Convergence for the Newton-Raphson Method -- 2.4.4.3.Geometrical Significance of the Newton-Raphson Method -- 2.4.4.4.Advantage and Disadvantage of the Newton-Raphson Method -- 2.4.4.5.Algorithm for the Newton-Raphson Method -- 2.4.5.Secant Method -- 2.4.5.1.Geometrical Significance of the Secant Method -- 2.4.5.2.Order of Convergence for the Secant Method -- 2.4.5.3.Advantage and Disadvantage of the Secant Method -- 2.4.5.4.Algorithm for the Secant Method -- 2.5.Generalized Newton's Method -- 2.5.1.Numerical Solution of Simultaneous Nonlinear Equations -- 2.5.1.1.Newton's Method -- 2.5.1.2.Fixed-Point Iteration Method -- 2.6.Graeffe's Root Squaring Method for Algebraic Equations -- Exercises -- 3.1.Introduction -- 3.2.Polynomial Interpolation -- 3.2.1.Geometric Interpretation of Interpolation -- 3.2.2.Error in Polynomial Interpolation -- 3.2.3.Finite Differences -- 3.2.3.1.Forward Differences -- 3.2.4.Shift, Differentiation, and Averaging Operators -- 3.2.4.1.Shift Operator -- 3.2.4.2.Differentiation Operator -- 3.2.4.3.Averaging Operator -- 3.2.5.Factorial Polynomial -- 3.2.5.1.Forward Differences of Factorial Polynomial -- 3.2.6.Backward Differences -- 3.2.6.1.Relation between the Forward and Backward Difference Operators -- 3.2.6.2.Backward Difference Table -- 3.2.7.Newton's Forward Difference Interpolation -- 3.2.7.1.Error in Newton's Forward Difference Interpolation -- 3.2.7.2.Algorithm for Newton's Forward Difference Interpolation -- 3.2.8.Newton's Backward Difference Interpolation -- 3.2.8.1.Error in Newton's Backward Difference Interpolation -- 3.2.8.2.Algorithm for Newton's Backward Difference Interpolation -- 3.2.9.Lagrange's Interpolation Formula -- 3.2.9.1.Error in Lagrange's Interpolation -- 3.2.9.2.Advantages and Disadvantages of Lagrange's Interpolation -- 3.2.9.3.Algorithm for Lagrange's Interpolation -- 3.2.10.Divided Difference -- 3.2.10.1.Some Properties of Divided Differences -- 3.2.10.2.Newton's Divided Difference Interpolation Formula -- 3.2.10.3.Divided Difference Table -- 3.2.10.4.Algorithm for Newton's Divided Difference Interpolation -- 3.2.10.5.Some Important Relations -- 3.2.11.Gauss's Forward Interpolation Formula -- 3.2.12.Gauss's Backward Interpolation Formula -- 3.2.13.Central Difference -- 3.2.13.1.Central Difference Table -- 3.2.13.2.Stirling's Interpolation Formula -- 3.2.13.3.Bessel's Interpolation Formula -- 3.2.13.4.Everette's Interpolation Formula -- 3.2.14.Hermite's Interpolation Formula -- 3.2.14.1.Uniqueness of Hermite Polynomial -- 3.2.14.2.The Error in Hermite Interpolation -- 3.2.15.Piecewise Interpolation -- 3.2.15.1.Piecewise Linear Interpolation -- 3.2.15.2.Piecewise Quadratic Interpolation -- 3.2.15.3.Piecewise Cubic Interpolation -- 3.2.16.Cubic Spline Interpolation -- 3.2.16.1.Cubic Spline -- 3.2.16.2.Error in Cubic Spline -- 3.2.17.Interpolation by Iteration -- 3.2.17.1.Aitken's Interpolation Formula -- 3.2.17.2.Neville's Interpolation Formula -- 3.2.18.Inverse Interpolation -- Exercises -- 4.1.Introduction -- 4.2.Errors in Computation of Derivatives -- 4.3.Numerical Differentiation for Equispaced Nodes -- 4.3.1.Formulae Based on Newton's Forward Interpolation -- 4.3.1.1.Error Estimate -- 4.3.2.Formulae Based on Newton's Backward Interpolation -- 4.3.2.1.Error Estimate -- 4.3.3.Formulae Based on Stirling's Interpolation -- 4.3.3.1.Error Estimate -- 4.3.4.Formulae Based on Bessel's Interpolation -- 4.3.4.1.Error Estimate -- 4.4.Numerical Differentiation for Unequally Spaced Nodes -- 4.4.1.Formulae Based on Lagrange's Interpolation -- 4.4.1.1.Error Estimate -- 4.4.2.Formulae Based on Newton's Divided Difference Interpolation -- 4.4.2.1.Error Estimate -- 4.5.Richardson Extrapolation -- Exercises -- 5.1.Introduction -- 5.2.Numerical Integration from Lagrange's Interpolation -- 5.3.Newton-Cotes Formula for Numerical Integration (Closed Type) -- 5.3.1.Deduction of Trapezoidal, Simpson's One-Third, Weddle's, and Simpson's Three-Eighth Rules from the Newton-Cotes Numerical Integration Formula -- 5.3.1.1.Trapezoidal Rule and Its Error Estimate -- 5.3.1.2.Simpson's One-Third Rule or Parabolic Rule with Error Term -- 5.3.1.3.Weddle's Rule -- 5.3.1.4.Simpson's Three-Eighth Rule with Error Term -- 5.4.Newton-Cotes Quadrature Formula (Open Type) -- 5.5.Numerical Integration Formula from Newton's Forward Interpolation Formula -- 5.6.Richardson Extrapolation -- 5.7.Romberg Integration -- 5.7.1.Algorithm for Romberg's Integration -- 5.8.Gauss Quadrature Formula -- 5.8.1.Guass-Legendre Integration Method -- 5.9.Gaussian Quadrature: Determination of Nodes and Weights through Orthogonal Polynomials -- 5.9.1.Gauss-Legendre Quadrature Method -- 5.9.2.Gauss-Chebyshev Quadrature Method -- 5.9.3.Gauss-Laguerre Quadrature Method -- 5.9.4.Gauss-Hermite Quadrature Method -- 5.10.Lobatto Quadrature Method -- 5.11.Double Integration -- 5.11.1.Trapezoidal Method -- 5.11.1.1.Algorithm for the Trapezoidal Method -- 5.11.2.Simpson's One-Third Method -- 5.11.2.1.Algorithm for Simpson's Method -- 5.12.Bernoulli Polynomials and Bernoulli Numbers -- 5.12.1.Some Properties of Bernoulli Polynomials -- 5.13.Euler-Maclaurin Formula -- Exercises -- 6.1.Introduction -- 6.2.Vector and Matrix Norm -- 6.2.1.Vector Norm -- 6.2.2.Matrix Norm -- 6.2.3.Condition Number of a Matrix -- 6.2.4.Spectral Radius and Norm Convergence -- 6.2.5.Jordan Block -- 6.2.6.Jordan Canonical Form -- 6.3.Direct Methods -- 6.3.1.Gauss Elimination Method -- 6.3.1.1.Pivoting in the Gauss Elimination Method -- 6.3.1.2.Operation Count in the Gauss Elimination Method -- 6.3.1.3.Algorithm for the Gauss Elimination Method -- 6.3.2.Gauss-Jordan Method -- 6.3.2.1.Algorithm for the Gauss-Jordan Method -- 6.3.3.Triangularization Method -- 6.3.3.1.Doolittle's Method -- 6.3.3.2.Crout's Method -- 6.3.3.3.Cholesky's Method -- 6.4.Iterative Method -- 6.4.1.Gauss-Jacobi Iteration -- 6.4.1.1.Convergence of the Gauss-Jacobi Iteration Method -- 6.4.1.2.Algorithm for the Gauss-Jacobi Method -- 6.4.2.Gauss-Seidel Iteration Method -- 6.4.2.1.Convergence of the Gauss-Seidel Iteration Method -- 6.4.2.2.Algorithm for the Gauss-Seidel Method -- 6.4.3.SOR Method -- 6.4.3.1.Convergence of the SOR Method -- 6.4.3.2.Algorithm for the SOR Method -- 6.5.Convergent Iteration Matrices -- 6.6.Convergence of Iterative Methods -- 6.6.1.Rate of Convergence -- 6.7.Inversion of a Matrix by the Gaussian Method -- 6.8.Ill-Conditioned Systems -- 6.9.Thomas Algorithm -- 6.9.1.Operational Count for Thomas Algorithm -- 6.9.2.Algorithm -- Exercises -- 7.1.Introduction -- 7.2.Single-Step Methods -- 7.2.1.Picard's Method of Successive Approximations -- 7.2.2.Taylor's Series Method -- 7.2.2.1.Error Estimate -- 7.2.2.2.Alternatively -- 7.2.3.General Form of a Single-Step Method -- 7.2.3.1.Error Estimate -- 7.2.3.2.Convergence of the Single-Step Method -- 7.2.4.Euler Method -- 7.2.4.1.Local Truncation Error -- 7.2.4.2.Geometrical Interpretation -- 7.2.4.3.Backward Euler Method -- 7.2.4.4.Midpoint Method -- 7.2.4.5.Algorithm for Euler's Method -- 7.2.5.Improved Euler Method -- 7.2.5.1.Algorithm of the Improved Euler Method -- 7.2.6.Runge-Kutta Methods -- 7.2.6.1.Algorithm for R-K Method of Order 4 -- 7.2.6.2.A General Form for Explicit R-K Methods -- 7.2.6.3.Estimation of the Truncation Error and Control -- 7.2.6.4.R-K-Fehlberg Method -- 7.3.Multistep Methods -- 7.3.1.Adams-Bashforth and Adams-Moulton Predictor-Corrector Method -- 7.3.1.1.Error Estimate -- 7.3.1.2.Algorithm of Adams Predictor-Corrector Method -- 7.3.2.Milne's Method -- 7.3.2.1.Error Estimate -- 7.3.2.2.Algorithm of Milne's Method -- 7.3.3.Nystrom Method -- 7.4.System of Ordinary Differential Equations of First-Order -- 7.4.1.Algorithm of R-K Method of the Fourth Order for Solving System of Ordinary Differential Equations -- 7.5.Differential Equations of Higher Order -- 7.6.Boundary Value Problems -- 7.6.1.Finite Difference Method -- 7.6.1.1.Boundary Conditions Involving the Derivative -- 7.6.1.2.Nonlinear Second-Order Differential Equation -- 7.6.2.Shooting Method -- 7.6.3.Collocation Method -- 7.6.4.Galerkin Method -- 7.7.Stability of an Initial Value Problem -- and Contents note continued: 7.7.1.Stability Analysis of Single Step Methods -- 7.7.1.1.Stability of Euler's Method -- 7.7.1.2.Stability of the Backward Euler Method -- 7.7.1.3.Stability of R-K Methods -- 7.7.2.Stability Analysis of General Multistep Methods -- 7.7.2.1.General Methods for Finding the Interval of Absolute Stability -- 7.8.Stiff Differential Equations -- 7.9.A-stability and L-stability -- 7.9.1.A-stability -- 7.9.2.L-stability -- Exercises -- 8.1.Introduction -- 8.1.1.Characteristic Equation, Eigenvalue, and Eigenvector of a Square Matrix -- 8.1.2.Similar Matrices and Diagonalizable Matrix -- 8.2.Inclusion of Eigenvalues -- 8.2.1.Gerschgorin's Discs -- 8.2.2.Gerschgorin's Theorem -- 8.3.Householder's Method -- 8.3.1.Algorithm for Householder's Method -- 8.4.The QR Method -- 8.4.1.Algorithm for the QR Method -- 8.4.2.The QR Method with Shift -- 8.5.Power Method -- 8.5.1.Algorithm of Power Method -- 8.6.Inverse Power Method -- 8.6.1.Algorithm of Inverse Power Method -- 8.7.Jacobi's Method -- 8.8.Givens Method -- 8.8.1.Eigenvalues of a Symmetric Tridiagonal Matrix -- Exercises -- 9.1.Introduction -- 9.1.1.Bernstein Polynomials and Its Properties -- 9.2.Least Square Curve Fitting -- 9.2.1.Straight Line Fitting -- 9.2.2.Fitting of kth Degree Polynomial -- 9.3.Least Squares Approximation -- 9.4.Orthogonal Polynomials -- 9.4.1.Weight Function -- 9.4.2.Gram-Schmidt Orthogonalization Process -- 9.5.The Minimax Polynomial Approximation -- 9.5.1.Characterization of the Minimax Polynomial -- 9.5.2.Existence of the Minimax Polynomial -- 9.5.3.Uniqueness of the Minimax Polynomial -- 9.5.4.The Near-Minimax Polynomial -- 9.6.B-Splines -- 9.6.1.Function Approximation by Cubic B-Spline -- 9.7.Padé Approximation -- Exercises -- 10.1.Introduction -- 10.2.Classification of PDEs of Second Order -- 10.3.Types of Boundary Conditions and Problems -- 10.4.Finite Difference Approximations to Partial Derivatives -- 10.5.Parabolic PDEs -- 10.5.1.Explicit FDM -- 10.5.1.1.Algorithm for Solving Parabolic PDE by FDM -- 10.5.2.Crank-Nicolson Implicit Method -- 10.5.2.1.Algorithm for Solving Parabolic PDE by the Crank-Nicolson Method -- 10.6.Hyperbolic PDEs -- 10.6.1.Explicit Central Difference Method -- 10.6.1.1.Algorithm for Solving Hyperbolic PDE by the Explicit Central Difference Method -- 10.6.2.Implicit FDM -- 10.7.Elliptic PDEs -- 10.7.1.Laplace Equation -- 10.7.2.Algorithm for Solving Laplace Equation by SOR Method -- 10.8.Alternating Direction Implicit Method -- 10.8.1.Algorithm for Two-Dimensional Parabolic PDE by ADI Method -- 10.9.Stability Analysis of the Numerical Schemes -- Exercises -- 11.1.Introduction -- 11.2.Piecewise Linear Basis Functions -- 11.3.The Rayleigh-Ritz Method -- 11.3.1.Algorithm of Rayleigh-Ritz Method -- 11.4.The Galerkin Method -- Further Reading -- Exercises.
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ISBN
9781498741743 (hardback : acid-free paper)
1498741746 (hardback : acid-free paper)
Note
"A Chapman & Hall book."
Bibliography Note
Includes bibliographical references (pages 673-674) and index.
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