Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived using different tools and language. My question is, who was the first person to observe that Conjugate Gradient is a Krylov subspace method? I have heard Youssef Saad given this credit, but when I search for references to back up the assertion, I cannot find relevant references. I figure someone must know or that perhaps the understanding came gradually and cannot be attributed to any one author (or even a small group).


Krylov Subspace Methods for Solving Large Unsymmetric Linear Systems, Y. Saad, Mathematics of Computation 37, 105-126 (1981).

The purpose of the present paper is to generalize the conjugate gradient method regarded as a projection process onto the Krylov subspace K„. We shall say of a method realizing such a process that it belongs to the class of Krylov subspace methods. It will be seen that these methods can be efficient for solving large nonsymmetric systems.

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  • $\begingroup$ Thank you! I don't know why this did not come up in my searches. $\endgroup$ – Kirk S. Apr 23 '13 at 11:22

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