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**Effectively well-conditioned linear systems.**
*(English)*
Zbl 0664.65041

The authors discuss effective well-conditioning of the linear system \(Ax=b\). The condition number \(K(A)=\| A\| \| A^{-1}\|\) is often an overly conservative measure of the sensitivity of x under perturbations of \(\Delta\) A and \(\Delta\) b to A and b respectively. Two practical cases in which the sensitivity of x may be significantly less than the worst case predicted by K(A) are presented. The first characterizes a class of Vandermonde matrices and right-hand-sides and the second a FFT-based fast Poisson solver, for each of which accurate solutions may be obtained.

For Vandermonde systems N. J. Higham [Numer. Math. 50, 613-532 (1987; Zbl 0595.65029)] has shown that the algorithm of A. Björck and V. Pereyra [Math. Comput. 24, 893-903 (1971; Zbl 0221.65054)] gives relative errors in the non-zero components of x which are independent of K(A) provided that scalars \(\alpha_ j\) of the Vandermonde matrix are in ascending order and that the elements of the right-hand side b oscillate in sign.

The computation of accurate solutions to the discretized one-dimensional Poisson problem is discussed and the authors’ experiments indicate that a fast Poisson solver composed entirely of fast sine transforms will have better numerical performance than the more common fast transform with tridiagonal solving.

For Vandermonde systems N. J. Higham [Numer. Math. 50, 613-532 (1987; Zbl 0595.65029)] has shown that the algorithm of A. Björck and V. Pereyra [Math. Comput. 24, 893-903 (1971; Zbl 0221.65054)] gives relative errors in the non-zero components of x which are independent of K(A) provided that scalars \(\alpha_ j\) of the Vandermonde matrix are in ascending order and that the elements of the right-hand side b oscillate in sign.

The computation of accurate solutions to the discretized one-dimensional Poisson problem is discussed and the authors’ experiments indicate that a fast Poisson solver composed entirely of fast sine transforms will have better numerical performance than the more common fast transform with tridiagonal solving.

Reviewer: A.Swift

### MSC:

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65F05 | Direct numerical methods for linear systems and matrix inversion |

65G50 | Roundoff error |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |