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## 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 2.3.5. 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 A12 and B13 are nonsingular upper triangular, and A23 is upper triangular. If m-k-l < 0, the bottom zero block of U1T A Q1 does not appear, and A23 is upper trapezoidal. U1, V1 and Q1 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 A23 and B13 is computed using xTGSJA2.2: Here U2, V2 and Q2 are orthogonal (or unitary) matrices, C and S are both real nonnegative diagonal matrices satisfying C2 + S2 = I, S is nonsingular, and R is upper triangular and nonsingular.

 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  for details.

The generalized singular value decomposition of two triangular matrices, performed by xTGSJA, is done using a Jacobi-like method as described in [83,10].     Next: Performance of LAPACK Up: Computational Routines Previous: Deflating Subspaces and Condition   Contents   Index
Susan Blackford
1999-10-01