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

We consider the modified conjugate gradient procedure for solving A = in which the approximation space is based upon the Krylov space associated with A ^1/p and , 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 . We discuss the implications of the quickly accumulating effect of an error in 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 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.