lapgsvdst#

Purpose#

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

Format#

{ D1, D2, Z, U, V, Q } = lapgsvdst(A, B)#

Parameters:

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

Returns:

D1 (Mx(K+L) matrix) – with singular values for A on diagonal.
D2 (Px(K+L) matrix) – with singular values for B on diagonal.
Z ((K+L)xN matrix) – partitioned matrix composed of a zero matrix and 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 $M - K - L < 0$

            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

A = U^{' - 1} E_{1} X

B = V^{' - 1} E_{2} X^{- 1}

where

E1 = [ 0  D1 ]

E2 = [ 0  D2 ]

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

A B^{- 1} = U D_{1} D_{2}^{- 1} V^{'}

This procedure calls the LAPACK routines DGGSVD and ZGGSVD.