Singular Value Decomposition     Next: Generalized Symmetric Definite Up: Computational Routines Previous: Invariant Subspaces and

## Singular Value Decomposition

Let A be a general real m-by-n matrix. The singular value decomposition (SVD) of A is the factorization , where U and V are orthogonal, and , r = min(m , n), with . If A is complex, then its SVD is where U and V are unitary, and is as before with real diagonal elements. The are called the singular values , the first r columns of V the right singular vectors  and the first r columns of U the left singular vectors .

The routines described in this section, and listed in Table 2.12, are used to compute this decomposition. The computation proceeds in the following stages:

1. The matrix A is reduced to bidiagonal  form: if A is real ( if A is complex), where and are orthogonal (unitary if A is complex), and B is real and upper-bidiagonal when m > = n and lower bidiagonal when m < n, so that B is nonzero only on the main diagonal and either on the first superdiagonal (if m > = n) or the first subdiagonal (if m < n).

2. The SVD of the bidiagonal matrix B is computed: , where and are orthogonal and is diagonal as described above. The singular vectors of A are then and .

The reduction to bidiagonal form is performed by the subroutine xGEBRD,        or by xGBBRD      for a band matrix.

The routine xGEBRD represents and in factored form as products of elementary reflectors,     as described in section 5.4. If A is real, the matrices and may be computed explicitly using routine xORGBR,    or multiplied by other matrices without forming and using routine xORMBR  . If A is complex, one instead uses xUNGBR   and xUNMBR  , respectively.

If A is banded and xGBBRD is used to reduce it to bidiagonal form, and are determined as products of Givens rotations  , rather than as products of elementary reflectors. If or is required, it must be formed explicitly by xGBBRD. xGBBRD uses a vectorizable algorithm, similar to that used by xSBTRD (see Kaufman ). xGBBRD may be much faster than xGEBRD when the bandwidth is narrow.

The SVD of the bidiagonal matrix is computed by the subroutine xBDSQR.      xBDSQR is more accurate than its counterparts in LINPACK and EISPACK: barring underflow and overflow, it computes all the singular values of A to nearly full relative precision, independent of their magnitudes. It also computes the singular vectors much more accurately. See section 4.9 and  for details.

If m >> n, it may be more efficient to first perform a QR factorization of A, using the routine xGEQRF     , and then to compute the SVD of the n-by-n matrix R, since if A = QR and , then the SVD of A is given by . Similarly, if m << n, it may be more efficient to first perform an LQ factorization of A, using xGELQF. These preliminary QR and LQ      factorizations are performed by the driver xGESVD.

The SVD may be used to find a minimum norm solution  to a (possibly) rank-deficient linear least squares   problem (2.1). The effective rank, k, of A can be determined as the number of singular values which exceed a suitable threshold. Let be the leading k-by-k submatrix of , and be the matrix consisting of the first k columns of V. Then the solution is given by: where consists of the first k elements of . can be computed using xORMBR, and    xBDSQR has an option to multiply a vector by .

```-----------------------------------------------------------------------------
Type of matrix                             Single precision  Double precision
and storage scheme  Operation              real     complex  real     complex
-----------------------------------------------------------------------------
general             bidiagonal reduction   SGEBRD   CGEBRD   DGEBRD   ZGEBRD
-----------------------------------------------------------------------------
general band        bidiagonal reduction   SGBBRD   CGBBRD   DGBBRD   ZGBBRD
-----------------------------------------------------------------------------
orthogonal/unitary  generate matrix after  SORGBR   CUNGBR   DORGBR   ZUNGBR
bidiagonal reduction
multiply matrix after  SORMBR   CUNMBR   DORMBR   ZUNMBR
bidiagonal reduction
-----------------------------------------------------------------------------
bidiagonal          singular values/       SBDSQR   CBDSQR   DBDSQR   ZBDSQR
singular vectors
-----------------------------------------------------------------------------
```
Table 2.12: Computational routines for the singular value decomposition     Next: Generalized Symmetric Definite Up: Computational Routines Previous: Invariant Subspaces and

Tue Nov 29 14:03:33 EST 1994