Solving A$\underline x$=$\underline b$ Using a Modified Conjugate Gradient Method Based on Roots of A

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.