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The Computer Journal 1968 11(3):302-304; doi:10.1093/comjnl/11.3.302
© 1968 by British Computer Society
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On the calculation of orthogonal vectors

M. J. D. Powell *

Mathematics Branch, Theoretical Physics Division, Atomic Energy Research Establishment, Harwell, UK

Given an orthonormal basis, d1, d2,..., dn of Euclidean n-space, and given some vector d0 which is not orthogonal to dn, this paper shows how to calculate, in O(n2) computer operations, a new orthonormal basis, d1*, d2*,...,dn*, having the property that dk* is a linear combination of the k vectors d0, d1,...,dk–1. The method is useful because it reduces the amount of computer time that is needed by Rosenbrock's (1960) minimisation procedure. We show that any errors do not grow if the method is applied many times.

Received May 1968.

* Mathematics Branch, Theoretical Physics Division, Atomic Energy Research Establishment, Harwell, Berkshire


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