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A modified direct preconditioner for indefinite symmetric Toeplitz systems
Author(s) -
Concus Paul,
Saylor Paul
Publication year - 1995
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1680020504
Subject(s) - preconditioner , toeplitz matrix , mathematics , block matrix , inverse , linear system , matrix (chemical analysis) , positive definite matrix , mathematical analysis , eigenvalues and eigenvectors , pure mathematics , geometry , physics , materials science , quantum mechanics , composite material
A modification is presented of the classical O ( n 2 ) algorithm of Trench for the direct solution of Toeplitz systems of equations. The Trench algorithm is stable for symmetric matrices that are positive definite, but can be unstable for the indefinite case (and for general matrices). Our modification permits extension of the algorithm to compute approximate inverses of indefinite symmetric Toeplitz matrices, for which the unmodified algorithm breaks down when principal submatrices are singular. As a preconditioner, the approximate inverse has an advantage that only matrix—vector multiplications are required for the solution of a linear system, without forward and backward solves. The approximate inverse so obtained can be sufficiently accurate, moreover, that, when it is used as a preconditioner for the applications investigated, subsequent iteration may not even be necessary. Numerical results are given for several test matrices. The perturbation to the original matrix that defines the modification is related to a perturbation in a quantity generated in the Trench algorithm; the associated stability of the Trench algorithm is discussed.