An iterative solution method for solving f(A)x = b,using Krylov subspace information obtained for the symmetric positive definite matrix A

The conjugate gradients method generates successive approximations x_i for the solution of the linear system Ax = b, where A is symmetric positive definite and usually sparse. It will be shown how intermediate information obtained by the conjugate gradients (cg) algorithm (or by the closely related Lanczos algorithm) can be used to solve f(A)x = b iteratively in an efficient way, for suitable functions f. The special case f(A) = A² is discussed in particular. We also consider the problem of solving Ax = b for different right-hand sides b. A variant on a well-known algorithm for that problem is proposed, which does not seem to suffer from the usual loss of orthogonality in the standard cg and Lanczos algorithms.