ABSTRACT
The ideal algorithm would be one that had finite termination but would give a useful approximate solution. Magnus Hestenes and Eduard Stiefel succeeded in developing an algorithm with exactly these characteristics, the method of conjugate gradients. Hestenes was a faculty member at UCLA who became associated with this Institute, and Stiefel was a visitor from the Eidgenossischen Technischen Hochschule (ETH) in Zurich, Switzerland. There were two commonly used types of algorithms for solving linear systems. The first, like Gauss elimination, modified a tableau of matrix entries in a systematic way in order to compute the solution. The second type of algorithm used "relaxation techniques" to develop a sequence of iterates converging to the solution. Hestenes and Stiefel's paper remains the classic reference on this algorithm, explaining the use of the algorithm as both a finitely terminating method and as a way of obtaining an approximate solution if halted before termination.
