# lapgsvds¶

## Purpose¶

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

## Format¶

{ C, S, R } = lapgsvds(A, B)
Parameters:
• A (MxN real or complex matrix) – data

• B (PxN real or complex 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.

## Remarks¶

1. The generalized singular value decomposition of A and B is

$U'AQ = D_1Z$
$V'BQ = D_2Z$

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, $$D_1$$ and $$D_2$$ 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  ]

1. Form the matrix

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


then

$A = U'^{-1}E_1X$
$B = V'^{-1}E_2X^{-1}$

where

E1 = [ 0  D1 ]

E2 = [ 0  D2 ]


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

$AB^{-1} = UD_1D_2^{-1}V'$

This procedure calls the LAPACK routines DGGSVD and ZGGSVD.