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1968•The Computer JournalRequires access

On the calculation of orthogonal vectors

M. J. D. Powell

Open publisher page 43 citations

Abstract

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.

About this research paper

What this paper is about

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.

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OpenAlex reports 43 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

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.

Key concepts: Orthonormal basis, Orthonormality, Basis (linear algebra), Orthogonal basis, Euclidean space, Property (philosophy), Mathematics, Euclidean distance

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