THE APPLICATION OF THE CONJUGATE GRADIENT METHOD TO THE SOLUTION OF OPERATOR EQUATIONS -- AN UNCONVENTIONAL PERSPECTIVE

摘要

This narrative presents an alternate philosophy for the accurate solution of operator equations, you might say "both singular and nonsingular" in general. In this approach, we try to solve the exact operator equation in an approximate way, quite differently from the matrix methods which try to solve the approximate operator equation in an exact fashion. The advantage of this new philosophy is that convergence is assured and a priori error estimates are available. The conjugate gradient methods are numerical methods which provide a means to reach this new goal, as opposed to an efficient means of just solving matrix equations, which some researchers have assumed them to be. We thereby take the position that there is a heaven-and-hell difference between the application of the conjugate gradient method to solve an operator equation and its application to the solution of matrix equations. [Vol. 2, No. 2, pp. 131-135 (1987)]