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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Numerical methods for computing angles between linear subspaces
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by Ȧke Björck and Gene H. Golub PDF
Math. Comp. 27 (1973), 579-594 Request permission


Assume that two subspaces F and G of a unitary space are defined as the ranges (or null spaces) of given rectangular matrices A and B. Accurate numerical methods are developed for computing the principal angles ${\theta _k}(F,G)$ and orthogonal sets of principal vectors ${u_k} \in F$ and ${v_k} \in G,k = 1,2, \cdots ,q = \dim (G) \leqq \dim (F)$. An important application in statistics is computing the canonical correlations ${\sigma _k} = \cos {\theta _k}$ between two sets of variates. A perturbation analysis shows that the condition number for ${\theta _k}$ essentially is $\max (\kappa (A),\kappa (B))$, where $\kappa$ denotes the condition number of a matrix. The algorithms are based on a preliminary QR-factorization of A and B (or ${A^H}$ and ${B^H}$), for which either the method of Householder transformations (HT) or the modified Gram-Schmidt method (MGS) is used. Then $\cos \;{\theta _k}$ and $\sin \;{\theta _k}$ are computed as the singular values of certain related matrices. Experimental results are given, which indicates that MGS gives ${\theta _k}$ with equal precision and fewer arithmetic operations than HT. However, HT gives principal vectors, which are orthogonal to working accuracy, which is not generally true for MGS. Finally, the case when A and/or B are rank deficient is discussed.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Math. Comp. 27 (1973), 579-594
  • MSC: Primary 65F30
  • DOI:
  • MathSciNet review: 0348991