BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-71-225
       ENTRY:: November 01, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Numerical methods for computing angles between linear
               subspaces
        TYPE:: Technical Report
      AUTHOR:: Bjoerck, Ake
      AUTHOR:: Golub, Gene H.
        DATE:: July 1971
       PAGES:: 33
    ABSTRACT:: Assume that two subspaces F and G of unitary space are
               defined as the ranges (or nullspaces) 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 $u_k\ \epsilon\ F$ and
               $v_k\ \epsilon\ G$, k = 1,2,..., q = dim(G) $\leq$ 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 in general true for MGS. Finally the
               case when A and/or B are rank deficient is discussed.
       NOTES:: [Adminitrivia V1/Prg/19951101]
         END:: STAN//CS-TR-71-225