Modified gram-schmidt process vs. classical gram-schmidt

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Abstract
Gram-Schmidt Orthogonalization has long been recognized for its numerical stability. Other virtues include ease of programming, and facility of obtaining useful side statistics. It has been argued in the literature for years that while Classical Gram-Schmidt Orthogonalization always requires reinforcement, Modified Gram-Schmidt never requires reorthogonalization. It is true that the Classical approach forms the upper triangular matrix T , n × n, by columns and that the Modified approach forms T by rows, which might dictate a choice of one over the other for different reasons. It is easy to show, however, that the Classical approach can be programmed in such a way that the numerical stability is the same as that produced by Modified Gram-Schmidt, which is to say that the Gram-Schmidt process can be modified so that the results on the computer are the same whether T is formed by rows or by columns. Program listings of each approach without iterative refinement are available which demonstrate that the results produced by Longley's version of the Classical approach are identical to those produced by the Modified approach. Hint: To change from rows to columns, change the order of the subscripts.

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