Modified Conjugate Gradient Method for the Solution of Ax=b

Abstract

In this note, we examine a modified conjugate gradient procedure for solving $A\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ in which the approximation space is based upon the Krylov space ($A\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$) associated with $\sqrt A$ and $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$. We show that, given initial vectors $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ and $\sqrt A \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ (possibly computed at some expense), the best fit solution in $K^k \sqrt A ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{b}$ can be computed using a finite-term recurrence requiring only one multiplication by A per iteration. The initial convergence rate appears, as expected, to be twice as fast as that of the standard conjugate gradient method, but stability problems cause the convergence to be degraded.