ISSN:
1070-5325
Keywords:
Orthogonal
;
polynomials
;
Residual polynomials
;
Conjugate
;
gradient
;
method
;
Ritz values
;
Field of values
;
Engineering
;
Engineering General
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mathematics
Notes:
In this paper we explore two sets of polynomials, the orthogonal polynomials and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal section of the matrix C A while the roots of the residual polynomials, also known as pseudo-Ritz values (or roots of kernel polynomials), are the reciprocals of the eigenvalues of an orthogonal section of the matrix (C A)-1. When C A is selfadjoint positive definite, this distinction is minimal, but for the indefinite or nonselfadjoint case this distinction becomes important. We use these two sets of roots to form possibly nonconvex regions in the complex plane that describe the spectrum of C A.
Additional Material:
4 Ill.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/nla.1680010503
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