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| List of Figures | |
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| List of Tables | |
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| Preface | |
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| Numerical Algorithms | |
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| Scientific computing | |
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| Numerical algorithms and errors | |
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| Algorithm properties | |
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| Exercises | |
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| Additional notes | |
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| Roundoff Errors | |
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| The essentials | |
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| Floating point systems | |
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| Roundoff error accumulation | |
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| The IEEE standard | |
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| Exercises | |
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| Additional notes | |
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| Nonlinear Equations in One Variable | |
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| Solving nonlinear equations | |
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| Bisection method | |
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| Fixed point iteration | |
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| Newton's method and variants | |
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| Minimizing a function in one variable | |
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| Exercises | |
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| Additional notes | |
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| Linear Algebra Background | |
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| Review of basic concepts | |
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| Vector and matrix norms | |
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| Special classes of matrices | |
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| Singular values | |
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| Examples | |
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| Exercises | |
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| Additional notes | |
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| Linear Systems: Direct Methods | |
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| Gaussian elimination and backward substitution | |
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| LU decomposition | |
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| Pivoting strategies | |
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| Efficient implementation | |
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| The Cholesky decomposition | |
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| Sparse matrices | |
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| Permutations and ordering strategies | |
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| Estimating errors and the condition number | |
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| Exercises | |
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| Additional notes | |
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| Linear Least Squares Problems | |
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| Least squares and the normal equations | |
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| Orthogonal transformations and QR | |
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| Householder transformations and Gram-Schmidt orthogonalization | |
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| Exercises | |
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| Additional notes | |
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| Linear Systems: Iterative Methods | |
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| The need for iterative methods | |
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| Stationary iteration and relaxation methods | |
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| Convergence of stationary methods | |
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| Conjugate gradient method | |
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| *Krylov subspace methods | |
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| *Multigrid methods | |
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| Exercises | |
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| Additional notes | |
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| Eigenvalues and Singular Values | |
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| The power method and variants | |
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| Singular value decomposition | |
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| General methods for computing eigenvalues and singular values | |
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| Exercises | |
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| Additional notes | |
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| Nonlinear Systems and Optimization | |
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| Newton's method for nonlinear systems | |
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| Unconstrained optimization | |
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| *Constrained optimization | |
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| Exercises | |
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| Additional notes | |
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| Polynomial Interpolation | |
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| General approximation and interpolation | |
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| Monomial interpolation | |
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| Lagrange interpolation | |
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| Divided differences and Newton's form | |
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| The error in polynomial interpolation | |
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| Chebyshev interpolation | |
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| Interpolating also derivative values | |
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| Exercises | |
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| Additional notes | |
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| Piecewise Polynomial Interpolation | |
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| The case for piecewise polynomial interpolation | |
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| Broken line and piecewise Hermite interpolation | |
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| Cubic spline interpolation | |
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| Hat functions and B-splines | |
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| Parametric curves | |
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| *Multidimensional interpolation | |
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| Exercises | |
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| Additional notes | |
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| Best Approximation | |
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| Continuous least squares approximation | |
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| Orthogonal basis functions | |
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| Weighted least squares | |
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| Chebyshev polynomials | |
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| Exercises | |
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| Additional notes | |
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| Fourier Transform | |
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| The Fourier transform | |
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| Discrete Fourier transform and trigonometric interpolation | |
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| Fast Fourier transform | |
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| Exercises | |
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| Additional notes | |
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| Numerical Differentiation | |
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| Deriving formulas using Taylor series | |
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| Richardson extrapolation | |
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| Deriving formulas using Lagrange polynomial interpolation | |
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| Roundoff and data errors in numerical differentiation | |
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| *Differentiation matrices and global derivative approximation | |
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| Exercises | |
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| Additional notes | |
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| Numerical Integration | |
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| Basic quadrature algorithms | |
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| Composite numerical integration | |
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| Gaussian quadrature | |
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| Adaptive quadrature | |
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| Romberg integration | |
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| *Multidimensional integration | |
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| Exercises | |
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| Additional notes | |
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| Differential Equations | |
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| Initial value ordinary differential equations | |
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| Euler's method | |
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| Runge-Kutta methods | |
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| Multistep methods | |
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| Absolute stability and stiffness | |
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| Error control and estimation | |
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| *Boundary value ODEs | |
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| *Partial differential equations | |
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| Exercises | |
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| Additional notes | |
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| Bibliography | |
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| Index | |