Generalized (or Quotient) Singular Value Decomposition     Next: Performance of LAPACK Up: Computational Routines Previous: Generalized Nonsymmetric Eigenproblems

Generalized (or Quotient) Singular Value Decomposition

The generalized (or quotient) singular value decomposition of an m-by-n matrix A and a p-by-n matrix B is described in section 2.2.5. The routines described in this section, are used to compute the decomposition. The computation proceeds in the following two stages:

1. xGGSVP     is used to reduce the matrices A and B to triangular form: where and are nonsingular upper triangular, and is upper triangular. If m - k - 1 < 0, the bottom zero block of does not appear, and is upper trapezoidal. , and are orthogonal matrices (or unitary matrices if A and B are complex). l is the rank of B, and k + l is the rank of .

2. The generalized singular value decomposition of two l-by-l upper triangular matrices and is computed using xTGSJA : Here , and are orthogonal (or unitary) matrices, C and S are both real nonnegative diagonal matrices satisfying , S is nonsingular, and R is upper triangular and nonsingular.

--------------------------------------------------------
Single precision  Double precision
Operation             real     complex  real     complex
--------------------------------------------------------
triangular reduction  SGGSVP   CGGSVP   DGGSVP   ZGGSVP
of A and B
--------------------------------------------------------
GSVD of a pair of      STGSJA   CTGSJA   DTGSJA   ZTGSJA
triangular matrices
--------------------------------------------------------
Table 2.16: Computational routines for the generalized singular value decomposition

The reduction to triangular form, performed by xGGSVP, uses QR decomposition with column pivoting   for numerical rank determination. See  for details.

The generalized singular value decomposition of two triangular matrices, performed by xTGSJA, is done using a Jacobi-like method as described in .

Tue Nov 29 14:03:33 EST 1994