## Accuracy and Stability of Numerical Algorithms: Second EditionThis book gives a thorough, up-to-date treatment of the behaviour of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis, all enlivened by historical perspective and informative quotations. The coverage of the first edition has been expanded and updated, involving numerous improvements. Two new chapters treat symmetric indefinite systems and skew-symmetric systems, and nonlinear systems and Newton's method. Twelve new sections include coverage of additional error bounds for Gaussian elimination, rank revealing LU factorizations, weighted and constrained least squares problems, and the fused multiply-add operation found on some modern computer architectures. This new edition is a suitable reference for an advanced course and can also be used at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises. In addition the thorough indexes and extensive, up-to-date bibliography are in a readily accessible form. |

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### Contents

Principles of Finite Precision Computation | 1 |

Floating Point Arithmetic | 35 |

Basics | 61 |

Summation | 79 |

Polynomials | 93 |

Norms | 105 |

Perturbation Theory for Linear Systems | 119 |

Triangular Systems | 139 |

Matrix Powers | 339 |

QR Factorization | 353 |

The Least Squares Problem | 381 |

Underdetermined Systems | 407 |

Vandermonde Systems | 415 |

Fast Matrix Multiplication | 433 |

The Fast Fourier Transform and Applications | 451 |

Nonlinear Systems and Newtons Method | 459 |

LU Factorization and Linear Equations | 157 |

Cholesky Factorization | 195 |

Symmetric Indefinite and SkewSymmetric Systems | 213 |

Iterative Refinement | 231 |

Block LU Factorization | 245 |

Matrix Inversion | 259 |

Condition Number Estimation | 287 |

The Sylvester Equation | 305 |

Stationary Iterative Methods | 321 |

Automatic Error Analysis | 471 |

Software Issues in Floating Point Arithmetic | 489 |

A Gallery of Test Matrices | 511 |

A Solutions to Problems | 527 |

B Acquiring Software | 573 |

The Matrix Computation Toolbox | 583 |

657 | |

### Common terms and phrases

accuracy algorithm applied backward stable Bjorck BLAS block LDLT factorization chapter Cholesky factorization column complete pivoting componentwise relative computed solution condition estimation condition number convergence defined Demmel diagonally dominant eigenvalues elements error analysis evaluation example floating point arithmetic floating point numbers floating-point formula Fortran forward error bound function Gaussian elimination GEPP given growth factor Hence Higham IEEE arithmetic implementation inequality ISBN Kahan LAPACK least squares problems Lemma Linear Algebra Appl linear systems LINPACK LS problem LU factorization Math Mathematics MATLAB Matrix Anal nonnegative nonsingular norm normwise backward Note Numerical Analysis numerical stability obtain optimal orthogonal partial pivoting pivoting strategy polynomial precision arithmetic proof recursive relative error residual result Rnxn rook pivoting rounding errors roundoff routines satisfies scaling shows SIAM singular value Software Strassen's method Sylvester equation symmetric positive definite Theorem triangular matrix tridiagonal tridiagonal matrix underflow upper triangular Vandermonde vector zero

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