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| Preface | |
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| Numerical Algorithms | |
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| Finding roots | |
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| Analyzing Heron's algorithm | |
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| Where to start | |
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| An unstable algorithm | |
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| General roots: effects of floating-point | |
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| Exercises | |
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| Solutions | |
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| Nonlinear Equations | |
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| Fixed-point iteration | |
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| Particular methods | |
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| Complex roots | |
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| Error propagation | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Linear Systems | |
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| Gaussian elimination | |
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| Factorization | |
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| Triangular matrices | |
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| Pivoting | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Direct Solvers | |
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| Direct factorization | |
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| Caution about factorization | |
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| Banded matrices | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Vector Spaces | |
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| Normed vector spaces | |
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| Proving the triangle inequality | |
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| Relations between norms | |
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| Inner-product spaces | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Operators | |
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| Operators | |
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| Schur decomposition | |
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| Convergent matrices | |
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| Powers of matrices | |
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| Exercises | |
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| Solutions | |
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| Nonlinear Systems | |
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| Functional iteration for systems | |
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| Newton's method | |
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| Limiting behavior of Newton's method | |
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| Mixing solvers | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Iterative Methods | |
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| Stationary iterative methods | |
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| General splittings | |
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| Necessary conditions for convergence | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Conjugate Gradients | |
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| Minimization methods | |
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| Conjugate Gradient iteration | |
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| Optimal approximation of CG | |
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| Comparing iterative solvers | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Polynomial Interpolation | |
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| Local approximation: Taylor's theorem | |
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| Distributed approximation: interpolation | |
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| Norms in infinite dimensional spaces | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Chebyshev and Hermite Interpolation | |
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| Error term � | |
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| Chebyshev basis functions | |
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| Lebesgue function | |
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| Generalized interpolation | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Approximation Theory | |
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| Best approximation by polynomials | |
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| Weierstrass and Bernstein | |
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| Least squares | |
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| Piecewise polynomial approximation | |
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| Adaptive approximation | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Numerical Quadrature | |
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| Interpolatory quadrature | |
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| Peano kernel theorem | |
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| Gregorie Euler Maclaurin formulas | |
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| Other quadrature rules | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Eigenvalue Problems | |
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| Eigenvalue examples | |
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| Gershgorin's theorem | |
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| Solving separately | |
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| How not to eigen | |
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| Reduction to Hessenberg form | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Eigenvalue Algorithm | |
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| Power method | |
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| Inverse iteration | |
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| Singular value decomposition | |
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| Comparing factorizations | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Ordinary Differential Equations | |
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| Basic theory of ODEs | |
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| Existence and uniqueness of solutions | |
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| Basic discretization methods | |
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| Convergence of discretization methods | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Higher-order ODE Discretization Methods | |
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| Higher-order discretization | |
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| Convergence conditions | |
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| Backward differentiation formulas | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Floating Point | |
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| Floating-point arithmetic | |
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| Errors in solving systems | |
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| More reading | |
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| Exercises | |
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| Solutions | |
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| Notation | |
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| Bibliography | |
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| Index | |