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### Generalized Singular Value Decomposition (GSVD)

The generalized (or quotient) singular value decomposition of an m-by-n matrix A and a p-by-n matrix B is given by the pair of factorizations The matrices in these factorizations have the following properties:
• U is m-by-m, V is p-by-p, Q is n-by-n, and all three matrices are orthogonal. If A and B are complex, these matrices are unitary instead of orthogonal, and QT should be replaced by QH in the pair of factorizations.
• R is r-by-r, upper triangular and nonsingular. [0,R] is r-by-n (in other words, the 0 is an r-by-n-r zero matrix). The integer r is the rank of , and satisfies .
• is m-by-r, is p-by-r, both are real, nonnegative and diagonal, and . Write and , where and lie in the interval from 0 to 1. The ratios are called the generalized singular values of the pair A, B. If , then the generalized singular value is infinite. and have the following detailed structures, depending on whether or m-r < 0. In the first case, , then Here l is the rank of B, k=r-l, C and S are diagonal matrices satisfying C2 + S2 = I, and S is nonsingular. We may also identify , for , , and for . Thus, the first k generalized singular values are infinite, and the remaining l generalized singular values are finite.

In the second case, when m-r < 0, and Again, l is the rank of B, k=r-l, C and S are diagonal matrices satisfying C2 + S2 = I, S is nonsingular, and we may identify , for , , , for , and . Thus, the first k generalized singular values are infinite, and the remaining l generalized singular values are finite.

Here are some important special cases of the generalized singular value decomposition. First, if B is square and nonsingular, then r=n and the generalized singular value decomposition of A and B is equivalent to the singular value decomposition of AB-1, where the singular values of AB-1 are equal to the generalized singular values of the pair A, B: Second, if the columns of are orthonormal, then r=n, R=I and the generalized singular value decomposition of A and B is equivalent to the CS (Cosine-Sine) decomposition of : Third, the generalized eigenvalues and eigenvectors of can be expressed in terms of the generalized singular value decomposition: Let Then Therefore, the columns of X are the eigenvectors of , and the nontrivial'' eigenvalues are the squares of the generalized singular values (see also section 2.3.5.1). Trivial'' eigenvalues are those corresponding to the leading n-r columns of X, which span the common null space of AT A and BT B. The trivial eigenvalues'' are not well defined2.1.

A single driver routine xGGSVD computes the generalized singular value decomposition of A and B (see Table 2.6). The method is based on the method described in [83,10,8].

 Type of Function and storage scheme Single precision Double precision problem real complex real complex GSEP simple driver SSYGV CHEGV DSYGV ZHEGV divide and conquer driver SSYGVD CHEGVD DSYGVD ZHEGVD expert driver SSYGVX CHEGVX DSYGVX ZHEGVX simple driver (packed storage) SSPGV CHPGV DSPGV ZHPGV divide and conquer driver SSPGVD CHPGVD DSPGVD ZHPGVD expert driver SSPGVX CHPGVX DSPGVX ZHPGVX simple driver (band matrices) SSBGV CHBGV DSBGV ZHBGV divide and conquer driver SSBGVD CHBGVD DSBGV ZHBGVD expert driver SSBGVX CHBGVX DSBGVX ZHBGVX GNEP simple driver for Schur factorization SGGES CGGES DGGES ZGGES expert driver for Schur factorization SGGESX CGGESX DGGESX ZGGESX simple driver for eigenvalues/vectors SGGEV CGGEV DGGEV ZGGEV expert driver for eigenvalues/vectors SGGEVX CGGEVX DGGEVX ZGGEVX GSVD singular values/vectors SGGSVD CGGSVD DGGSVD ZGGSVD     Next: Computational Routines Up: Generalized Eigenvalue and Singular Previous: Generalized Nonsymmetric Eigenproblems (GNEP)   Contents   Index
Susan Blackford
1999-10-01