ISSN:
1573-7691
Keywords:
modified conjugate gradient method
;
conjugate gradient method
;
Krylov space
;
convergence rate
;
stability
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
Notes:
Abstract We consider the modified conjugate gradient procedure for solving A $$\underline x $$ = $$\underline b $$ in which the approximation space is based upon the Krylov space associated with A 1/p and $$\underline b $$ , for any integer p. For the square-root MCG (p=2) we establish a sharpened bound for the error at each iteration via Chebyshev polynomials in $$\sqrt A$$ . We discuss the implications of the quickly accumulating effect of an error in $$\sqrt A$$ $$\underline b $$ in the initial stage, and find an error bound even in the presence of such accumulating errors. Although this accumulation of errors may limit the usefulness of this method when $$\sqrt A$$ $$\underline b $$ is unknown, it may still be successfully applied to a variety of small, “almost-SPD” problems, and can be used to jump-start the conjugate gradient method. Finally, we verify these theoretical results with numerical tests.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1011132730828
