| List of Figures | p. xi |
| List of Tables | p. xix |
| Preface | p. xxi |
| Numerical Algorithms | p. 1 |
| Scientific computing | p. 1 |
| Numerical algorithms and errors | p. 3 |
| Algorithm properties | p. 9 |
| Exercises | p. 14 |
| Additional notes | p. 15 |
| Roundoff Errors | p. 17 |
| The essentials | p. 17 |
| Floating point systems | p. 21 |
| Roundoff error accumulation | p. 26 |
| The IEEE standard | p. 29 |
| Exercises | p. 32 |
| Additional notes | p. 36 |
| Nonlinear Equations in One Variable | p. 39 |
| Solving nonlinear equations | p. 39 |
| Bisection method | p. 43 |
| Fixed point iteration | p. 45 |
| Newton's method and variants | p. 50 |
| Minimizing a function in one variable | p. 55 |
| Exercises | p. 58 |
| Additional notes | p. 64 |
| Linear Algebra Background | p. 65 |
| Review of basic concepts | p. 65 |
| Vector and matrix norms | p. 73 |
| Special classes of matrices | p. 78 |
| Singular values | p. 80 |
| Examples | p. 83 |
| Exercises | p. 89 |
| Additional notes | p. 92 |
| Linear Systems: Direct Methods | p. 93 |
| Gaussian elimination and backward substitution | p. 94 |
| LU decomposition | p. 100 |
| Pivoting strategies | p. 105 |
| Efficient implementation | p. 110 |
| The Cholesky decomposition | p. 114 |
| Sparse matrices | p. 117 |
| Permutations and ordering strategies | p. 122 |
| Estimating errors and the condition number | p. 127 |
| Exercises | p. 133 |
| Additional notes | p. 139 |
| Linear Least Squares Problems | p. 141 |
| Least squares and the normal equations | p. 141 |
| Orthogonal transformations and QR | p. 151 |
| Householder transformations and Gram-Schmidt orthogonalization | p. 157 |
| Exercises | p. 163 |
| Additional notes | p. 166 |
| Linear Systems: Iterative Methods | p. 167 |
| The need for iterative methods | p. 167 |
| Stationary iteration and relaxation methods | p. 173 |
| Convergence of stationary methods | p. 179 |
| Conjugate gradient method | p. 182 |
| *Krylov subspace methods | p. 191 |
| *Multigrid methods | p. 204 |
| Exercises | p. 210 |
| Additional notes | p. 218 |
| Eigenvalues and Singular Values | p. 219 |
| The power method and variants | p. 219 |
| Singular value decomposition | p. 229 |
| General methods for computing eigenvalues and singular values | p. 236 |
| Exercises | p. 245 |
| Additional notes | p. 249 |
| Nonlinear Systems and Optimization | p. 251 |
| Newton's method for nonlinear systems | p. 251 |
| Unconstrained optimization | p. 258 |
| *Constrained optimization | p. 271 |
| Exercises | p. 286 |
| Additional notes | p. 293 |
| Polynomial Interpolation | p. 295 |
| General approximation and interpolation | p. 295 |
| Monomial interpolation | p. 298 |
| Lagrange interpolation | p. 302 |
| Divided differences and Newton's form | p. 306 |
| The error in polynomial interpolation | p. 313 |
| Chebyshev interpolation | p. 316 |
| Interpolating also derivative values | p. 319 |
| Exercises | p. 323 |
| Additional notes | p. 330 |
| Piecewise Polynomial Interpolation | p. 331 |
| The case for piecewise polynomial interpolation | p. 331 |
| Broken line and piecewise Hermite interpolation | p. 333 |
| Cubic spline interpolation | p. 337 |
| Hat functions and B-splines | p. 344 |
| Parametric curves | p. 349 |
| *Multidimensional interpolation | p. 353 |
| Exercises | p. 359 |
| Additional notes | p. 363 |
| Best Approximation | p. 365 |
| Continuous least squares approximation | p. 366 |
| Orthogonal basis functions | p. 370 |
| Weighted least squares | p. 373 |
| Chebyshev polynomials | p. 377 |
| Exercises | p. 379 |
| Additional notes | p. 382 |
| Fourier Transform | p. 383 |
| The Fourier transform | p. 383 |
| Discrete Fourier transform and trigonometric interpolation | p. 388 |
| Fast Fourier transform | p. 396 |
| Exercises | p. 405 |
| Additional notes | p. 406 |
| Numerical Differentiation | p. 409 |
| Deriving formulas using Taylor series | p. 409 |
| Richardson extrapolation | p. 413 |
| Deriving formulas using Lagrange polynomial interpolation | p. 415 |
| Roundoff and data errors in numerical differentiation | p. 420 |
| *Differentiation matrices and global derivative approximation | p. 426 |
| Exercises | p. 434 |
| Additional notes | p. 438 |
| Numerical Integration | p. 441 |
| Basic quadrature algorithms | p. 442 |
| Composite numerical integration | p. 446 |
| Gaussian quadrature | p. 454 |
| Adaptive quadrature | p. 462 |
| Romberg integration | p. 469 |
| *Multidimensional integration | p. 472 |
| Exercises | p. 475 |
| Additional notes | p. 479 |
| Differential Equations | p. 481 |
| Initial value ordinary differential equations | p. 481 |
| Euler's method | p. 485 |
| Runge-Kutta methods | p. 493 |
| Multistep methods | p. 500 |
| Absolute stability and stiffness | p. 507 |
| Error control and estimation | p. 515 |
| *Boundary value ODEs | p. 520 |
| *Partial differential equations | p. 524 |
| Exercises | p. 531 |
| Additional notes | p. 537 |
| Bibliography | p. 539 |
| Index | p. 543 |
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