## 2006.180: Computing a nearest symmetric positive semidefinite matrix

2006.180:
Nicholas J. Higham
(1988)
*Computing a nearest symmetric positive semidefinite matrix.*
Linear Algebra and its Applications, 103.
pp. 103-118.
ISSN 0024-3795

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DOI: 10.1016/0024-3795(88)90223-6

## Abstract

The nearest symmetric positive semidefinite matrix in the Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2, where H is the symmetric polar factor of B=(A + AT)/2. In the 2-norm a nearest symmetric positive semidefinite matrix, and its distance δ2(A) from A, are given by a computationally challenging formula due to Halmos. We show how the bisection method can be applied to this formula to compute upper and lower bounds for δ2(A) differing by no more than a given amount. A key ingredient is a stable and efficient test for positive definiteness, based on an attempted Choleski decomposition. For accurate computation of δ2(A) we formulate the problem as one of zero finding and apply a hybrid Newton-bisection algorithm. Some numerical difficulties are discussed and illustrated by example.

Item Type: | Article |
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Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2006.180 |

Deposited By: | Miss Louise Stait |

Deposited On: | 05 July 2006 |

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