lapgsvdcst#

Purpose#

Compute the generalized singular value decomposition of a pair of real or complex general matrices.

Format#

{ C, S, R, U, V, Q } = lapgsvdcst(A, B)#

Parameters:

A (MxN matrix) – data
B (PxN matrix) – data

Returns:

C (Lx1 vector) – singular values for A.
S (Lx1 vector) – singular values for B.
R ((K+L)x(K+L)) – upper triangular matrix.
U (MxM matrix) – orthogonal transformation matrix.
V (PxP matrix) – orthogonal transformation matrix.
Q (NxN matrix) – orthogonal transformation matrix.

Remarks#

The generalized singular value decomposition of A and B is

\[U'*A*Q = D_1*Z\]

\[V'*B*Q = D_2*Z\]

where U, V, and Q are orthogonal matrices (see lapgsvdcst() and lapgsvdst()). Letting K + L = the rank of \(A|B\) then R is a \((K+L)x(K+L)\) upper triangular matrix, D1 and D2 are Mx(K+L) and Px(K+L) matrices with entries on the diagonal, \(Z = [ 0\; R ]\), and if \(M-K-L \geq 0\)

                  K L
D1 =         K  [ I 0 ]
             L  [ 0 C ]
     M - K - L  [ 0 0 ]

              K L
D2 =     P  [ 0 S ]
     P - L  [ 0 0 ]

              N-K-L   K    L
[ 0 R ] = K [   0    R11  R12 ]
          L [   0     0   R22 ]

or if \(M-K-L \lt 0\)

            K  M-K  K+L-M
D1 =   K  [ I   0     0  ]
     M-K  [ 0   0     0  ]

                  N-K-L  K   M-K  K+L-M
              K [   0   R11  R12   R13  ]
[ 0 R ] =   M-K [   0    0   R22   R23  ]
          K+L-M [   0    0    0    R33  ]

Form the matrix

X = Q [ I 0  ]
      [ 0 R^-1 ]

then

\[ \begin{align}\begin{aligned}A = U'^{-1}E_1X\\B = V'^{-1}E_2X^{-1}\end{aligned}\end{align} \]

where

E1 = [ 0  D1 ]

E2 = [ 0  D2 ]

(3) The generalized singular value decomposition of A and B implicitly produces the singular value decomposition of \(AB^{-1}\):

\[AB^{-1} = UD_1D_2^{-1}V'\]

This procedure calls the LAPACK routines DGGSVD and ZGGSVD.