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## 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 , , with . If A is complex, 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 3.10, 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 and lower bidiagonal when m < n, so that B is nonzero only on the main diagonal and either on the first superdiagonal (if ) 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 PxGEBRD and the SVD of B is computed using the LAPACK routine xBDSQR.

The routine PxGEBRD represents and in factored form as products of elementary reflectors, as described in section 3.4. If A is real, the matrices and may be multiplied by other matrices without forming and using routine PxORMBR . If A is complex, one instead uses PxUNMBR .

If , it may be more efficient to first perform a QR factorization of A, using the routine PxGEQRF , 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 , 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 PxGESVD.

The SVD may be used to find a minimum norm solution to a (possibly) rank-deficient linear least squares problem (3.1). 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 by using PxORMBR.

Table 3.10: Computational routines for the singular value decomposition

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