### Abstract

Original language | English |
---|---|

Pages (from-to) | 17-33 |

Number of pages | 16 |

Journal | Numerische Mathematik |

Volume | 62 |

DOIs | |

Publication status | Published - Dec 1992 |

### Fingerprint

### Keywords

- Vandermonde matrices
- numerical mathematics
- structured matrices
- Vandermonde
- vectors

### Cite this

*Numerische Mathematik*,

*62*, 17-33. https://doi.org/10.1007/BF01396218

}

*Numerische Mathematik*, vol. 62, pp. 17-33. https://doi.org/10.1007/BF01396218

**The structured sensitivity of Vandermonde-like systems.** / Bartels, S.G.; Higham, D.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The structured sensitivity of Vandermonde-like systems

AU - Bartels, S.G.

AU - Higham, D.J.

PY - 1992/12

Y1 - 1992/12

N2 - We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.

AB - We consider a general class of structured matrices that includes (possibly confluent) Vandermonde and Vandermonde-like matrices. Here the entries in the matrix depend nonlinearly upon a vector of parameters. We define, condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right-hand side. Convenient expressions are derived for the infinity norm based condition numbers, and order-of-magnitude estimates are given for condition numbers defined in terms of a general vector norm. We then discuss the computation of the corresponding backward errors. After linearising the constraints, we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity-norm solution of an underdetermined linear system. Exact componentwise condition numbers are also derived for matrix inversion and the least squares problem, and the linearised least squares backward error is characterised.

KW - Vandermonde matrices

KW - numerical mathematics

KW - structured matrices

KW - Vandermonde

KW - vectors

UR - http://dx.doi.org/10.1007/BF01396218

U2 - 10.1007/BF01396218

DO - 10.1007/BF01396218

M3 - Article

VL - 62

SP - 17

EP - 33

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

ER -