## Fundamentals of Matrix ComputationsA significantly revised and improved introduction to a critical aspect of scientific computation Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights. This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes: * Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations * Early introduction of the singular value decomposition * A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems * An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation. |

### Contents

1 | |

2 Sensitivity of Linear Systems | 111 |

3 The Least Squares Problem | 181 |

4 The Singular Value Decomposition | 261 |

5 Eigenvalues and Eigenvectors I | 289 |

6 Eigenvalues and Eigenvectors II | 413 |

7 Iterative Methods for Linear Systems | 521 |

Some Sources of Software for Matrix Computations | 603 |

References | 605 |

611 | |

617 | |

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### Common terms and phrases

A₁ applied approximation arithmetic Arnoldi process back substitution block calculate Cholesky decomposition Cholesky factor Cholesky's method Cnxn coefficient matrix column computed condition number convergence defined denote diagonal matrix eigenvalue problem example Exercise flop count forward substitution Gauss-Seidel Gaussian elimination Gram-Schmidt process Hessenberg matrix ill conditioned inner product inverse iteration iterative methods Jacobi least squares problem linear system linearly independent main diagonal main-diagonal entries MATLAB multiply nonsingular nonzero entries norm normal O(uČ obtain operations orthogonal orthogonal matrix orthonormal partial pivoting perform perturbation polynomial positive definite preconditioner proof Prove q₁ QR algorithm QR decomposition QR step Rayleigh quotient iteration reflectors residual result Rnxm rotator roundoff errors satisfies Section sequence shift Show singular values solution span{v1 sparse sparse matrix steepest descent stored Suppose symmetric symmetric matrix system Ax Theorem tridiagonal unique unitary upper triangular v₁ vector zero

### Popular passages

Page 6 - C is the dot product of the ith row of A with the jth column of B.