Solving Ax = b using a modified conjugate gradient method based on roots of A

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