A remark on band-Toeplitz preconditions for Hermitian Toeplitz systems

Conference · Sat Dec 31 04:00:00 EST 1994

OSTI ID:219616

Parter, S V; You, J ^[1]

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1. Univ. of Wisconsin, Madison, WI (United States)

This note presents a modification of an idea of R.H. Chan and P.T.P. Tang. Let f(0) {>=} 0 be a real valued, bounded, continuous function defined on ({minus}{pi}, {pi}). Let T{sub n}[f] be the Toeplitz matrix of order n + 1 generated by f(0). Chan and Tang suggest that for given {ell} {>=} 1 one chose g{sub {ell}}(0) {>=} 0 as a real valued function of fixed degree {ell} which minimizes a particular matrix. They construct g(0) via the Remez algorithm. Then T{sub n}[g{sub {ell}}]{sup {minus}1} is used as the precondition for T{sub n}[f]. The author suggests that g(0) be chosen as the even trigonometric polynomial of minimal degree which {open_quotes}matches{close_quotes} f(0) at all points {theta}{sub j} at which f({theta}) assumes its minimum. Clearly this g({theta}) is much easier to determine that the g{sub {ell}}({theta}). This choice is based on earlier work on the extreme eigenvalues of Hermitian Toeplitz matrices and the more recent work of Manteuffel and Parter on Preconditioning finite element discretizations of elliptic operators. It is shown that the condition number of T{sub n}[g]{sup {minus}1} T{sub N}[F] is uniformly bounded for all n. Experimental results demonstrate the efficacy of these preconditioners.

• Research Organization: Front Range Scientific Computations, Inc., Boulder, CO (United States); USDOE, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
• OSTI ID: 219616
• Report Number(s): CONF-9404305--Vol.2; ON: DE96005736
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
HERMITIAN MATRIX
ITERATIVE METHODS