Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time

Hargreaves, Gareth I. (2006) Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time. SIAM Journal On Matrix Analysis and Applications, 27 (3). pp. 801-820. ISSN 1095-7162

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Abstract

For an $n \times n$ tridiagonal matrix we exploit the structure of its QR factorization to devise two new algorithms for computing the 1-norm condition number in $O(n)$ operations. The algorithms avoid underflow and overflow, and are simpler than existing algorithms since tests are not required for degenerate cases. An error analysis of the first algorithm is given, while the second algorithm is shown to be competitive in speed with existing algorithms. We then turn our attention to an $n \times n$ diagonal-plus-semiseparable matrix, $A$, for which several algorithms have recently been developed to solve $Ax=b$ in $O(n)$ operations. We again exploit the QR factorization of the matrix to present an algorithm that computes the 1-norm condition number in $O(n)$ operations.

Item Type:	Article
Uncontrolled Keywords:	condition number, tridiagonal matrix, diagonal-plus-semiseparable matrix, QR factorization
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Nick Higham
Date Deposited:	16 Feb 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/157

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