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Generalized (or Quotient) Singular Value Decomposition
The generalized (or quotient) singular value decomposition
of an mbyn matrix A and a pbyn matrix B is described
in section 2.3.5.
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 A_{12} and B_{13} are nonsingular upper triangular, and
A_{23} is upper triangular.
If mkl < 0, the bottom zero block of U_{1}^{T} A Q_{1} does not appear,
and A_{23} is upper trapezoidal.
U_{1}, V_{1} and Q_{1} 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 lbyl
upper triangular matrices A_{23} and B_{13} is computed using
xTGSJA^{2.2}:
Here U_{2}, V_{2} and Q_{2} are orthogonal (or unitary) matrices,
C and S are both real
nonnegative diagonal matrices satisfying C^{2} + S^{2} = I, S is nonsingular,
and R is upper triangular and nonsingular.
Table 2.16:
Computational routines for the generalized singular value decomposition
Operation 
Single precision 
Double precision 

real 
complex 
real 
complex 
triangular reduction of A and B 
SGGSVP 
CGGSVP 
DGGSVP 
ZGGSVP 
GSVD of a pair of triangular matrices 
STGSJA 
CTGSJA 
DTGSJA 
ZTGSJA 
The reduction to triangular form, performed by
xGGSVP, uses QR decomposition with column pivoting
for numerical rank determination. See [8] for details.
The generalized singular value decomposition of two
triangular matrices, performed by xTGSJA, is done
using a Jacobilike method as described in [83,10].
Next: Performance of LAPACK
Up: Computational Routines
Previous: Deflating Subspaces and Condition
Contents
Index
Susan Blackford
19991001