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| Taylor Polynomials | |
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| The Taylor Polynomial | |
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| The Error in Taylor's Polynomials | |
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| Infinite Series | |
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| Polynomial Evaluation | |
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| An Example Program | |
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| Error and Computer Arithmetic | |
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| Floating-Point Numbers | |
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| Accuracy of Floating-Point Representation | |
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| Rounding and Chopping | |
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| Consequences for Programming of Floating-Point Arithmetic | |
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| Errors: Definitions, Sources, and Examples | |
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| Sources of Error | |
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| Loss-of-Significance Errors | |
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| Noise in Function Evaluation | |
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| Underflow and Overflow Errors | |
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| Propagation of Error | |
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| Propagated Error in Function Evaluation | |
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| Summation | |
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| Rounding versus Chopping | |
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| A Loop Error | |
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| Calculation of Inner Products | |
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| Rootfinding | |
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| The Bisection Method | |
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| Error Bounds | |
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| Newton's Method | |
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| Error Analysis | |
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| Error Estimation | |
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| Secant Method | |
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| Error Analysis | |
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| Comparison of Newton and Secant Methods | |
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| The MATLAB Function | |
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| Fixed Point Iteration | |
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| Aitken Error Estimation and Extrapolation | |
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| Higher-Order Iteration Formulas | |
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| Ill-Behaving Rootfinding Problems | |
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| Stability of Roots | |
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| Interpolation and Approximation | |
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| Polynomial Interpolation | |
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| Linear Interpolation | |
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| Quadratic Interpolation | |
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| Higher-Degree Interpolation | |
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| Divided Differences | |
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| Properties of Divided Differences | |
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| Newton's Divided Difference Interpolation | |
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| Error in Polynomial Interpolation | |
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| Another Error Formula | |
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| Behavior of the Error | |
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| Interpolation Using Spline Functions | |
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| Spline Interpolation | |
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| Construction of the Interpolating Natural Cubic Spline | |
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| Other Interpolating Spline Functions | |
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| The MATLAB Program spline | |
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| The Best Approximation Problem | |
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| Accuracy of the Minimax Approximation | |
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| Chebyshev Polynomials | |
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| The Triple Recursion Relation | |
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| The Minimum Size Property | |
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| A Near-Minimax Approximation Method | |
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| Odd and Even Functions | |
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| Least Squares Approximation | |
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| Legendre Polynomials | |
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| Solving for the Least Squares Approximation | |
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| Generalizations of Least Squares Approximation | |
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| Numerical Integration and Differentiation | |
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| The Trapezoidal and Simpson Rules | |
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| Simpson's Rule | |
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| Error Formulas | |
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| An Asymptotic Estimate of the Trapezoidal Error | |
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| Error Formulas for Simpson's Rule | |
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| Richardson Extrapolation | |
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| Periodic Integrands | |
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| Gaussian Numerical Integration | |
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| Weighted Gaussian Quadrature | |
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| Numerical Differentiation | |
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| Differentiation Using Interpolation | |
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| The Method of Undetermined Coefficients | |
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| Effects of Error in Function Values | |
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| Solution of Systems of Linear Equations | |
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| Systems of Linear Equations | |
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| Matrix Arithmetic | |
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| Arithmetic Operations | |
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| Elementary Row Operations | |
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| The Matrix Inverse | |
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| Matrix Algebra Rules | |
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| Solvability Theory of Linear Systems | |
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| Gaussian Elimination | |
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| Partial Pivoting | |
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| Calculation of Inverse Matrices | |
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| Operations Count | |
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| The LU Factorization | |
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| Compact Variants of Gaussian Elimination | |
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| Tridiagonal Systems | |
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| Matlab Built-in Functions for Solving Linear Systems | |
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| Error in solving Linear Systems | |
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| The Residual Correction Method | |
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| Stability in Solving Linear Systems | |
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| Iteration Methods | |
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| Jacobi Method and Gauss-Seidel Method | |
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| General Schema | |
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| The Residual Correction Method | |
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| Numerical Linear Algebra: Advanced Topics | |
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| Least Squares Data Fitting | |
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| The Linear Least Squares Approximation | |
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| Polynomial Least Squares Approximation | |
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| The Eigenvalue Problem | |
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| The Characteristic Polynomial | |
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| Eigenvalues for Symmetric Matrices | |
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| The Nonsymmetric Eigenvalue Problem | |
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| The Power Method | |
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| Convergence of the Power Method | |
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| Matlab Eigenvalue Calculations | |
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| Nonlinear Systems | |
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| Newton's Method | |
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| The General Newton Method | |
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| A Modified Newton's Method | |
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| Ordinary Differential Equations | |
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| Theory of Differential Equations: An Introduction | |
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| General Solvability Theory | |
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| Stability of the Initial Value Problem | |
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| Direction Fields | |
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| Euler's Method | |
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| Convergence Analysis of Euler's Method | |
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| Asymptotic Error Analysis | |
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| Richardson Extrapolation | |
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| Numerical Stability, Implicit Methods | |
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| The Backward Euler Method | |
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| The Trapezoidal Method | |
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| Taylor and Runge-Kutta Methods | |
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| Runge-Kutta Methods | |
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| Error Prediction and Control | |
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| Matlab Built-in Functions | |
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| Multistep Methods | |
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| Systems of Differential Equations | |
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| Higher-Order Differential Equations | |
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| Numerical Methods for Systems | |
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| Finite Difference Method for Two-Point Boundary Value Problems | |
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| Finite Difference Method for PDEs | |
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| The Poisson Equation | |
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| One-Dimensional Heat Equation | |
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| Semidiscretization | |
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| Explicit Full Discretization | |
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| Implicit Full Discretization | |
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| One-Dimensional Wave Equation | |
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| Mean Value Theorems | |
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| Mathematical Formulas | |
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| Algebra | |
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| Geometry | |
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| Trigonometry | |
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| Calculus | |
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| Numerical Analysis Software Packages | |
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| Commercial Packages | |
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| Public Domain Packages | |
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| Interactive Numerical Computation Environments | |
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| Symbolic Computation Environments | |
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| Literature of Mathematical Software | |
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| Matlab: An Introduction | |
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| The Binary Number System | |
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| Conversion from Decimal to Binary | |
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| Hexadecimal Numbers | |
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| Answers to Selected Problems | |
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| Bibliography | |
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| Index | |