4.8 Singular Value Decomposition

gesvd( A, S[, jobu='N'[, jobvt='N'[, U=None[, Vt=None]]]])

Singular value decomposition

$\begin{displaymath} A = U \Sigma V^T, \qquad A = U \Sigma V^H \end{displaymath}$

of a real or complex m by n matrix A.

S is a real matrix of length at least min{m, n}. On exit, its first min{m, n} elements are the singular values in descending order.

The argument jobu controls how many left singular vectors are computed. The possible values are 'N', 'A', 'S' and 'O'. If jobu is 'N', no left singular vectors are computed. If jobu is 'A', all left singular vectors are computed and returned as columns of U. If jobu is 'S', the first min{m,n} left singular vectors are computed and returned as columns of U. If jobu is 'O', the first min{m,n} left singular vectors are computed and returned as columns of A. The argument U is None (if jobu is 'N' or 'A') or a matrix of the same type as A.

The argument jobvt controls how many right singular vectors are computed. The possible values are 'N', 'A', 'S' and 'O'. If jobvt is 'N', no right singular vectors are computed. If jobvt is 'A', all right singular vectors are computed and returned as rows of Vt. If jobvt is 'S', the first min{m,n} right singular vectors are computed and their (conjugate) transposes are returned as rows of Vt. If jobvt is 'O', the first min{m, n} right singular vectors are computed and their (conjugate) transposes are returned as rows of A. Note that the (conjugate) transposes of the right singular vectors (i.e., the matrix ) are returned in Vt or A. The argument Vt can be None (if jobvt is 'N' or 'A') or a matrix of the same type as A.

On exit, the contents of A are destroyed.

gesdd( A, S[, jobz='N'[, U=None[, Vt=None]]])

Singular value decomposition of a real or complex m by n matrix A. This function is based on a divide-and-conquer algorithm and is faster than gesvd().

S is a real matrix of length at least min{m, n}. On exit, its first min{m, n} elements are the singular values in descending order.

The argument jobz controls how many singular vectors are computed. The possible values are 'N', 'A', 'S' and 'O'. If jobz is 'N', no singular vectors are computed. If jobz is 'A', all m left singular vectors are computed and returned as columns of U and all n right singular vectors are computed and returned as rows of Vt. If jobz is 'S', the first min{m, n} left and right singular vectors are computed and returned as columns of U and rows of Vt. If jobz is 'O' and m is greater than or equal to n, the first n left singular vectors are returned as columns of A and the n right singular vectors are returned as rows of Vt. If jobz is 'O' and m is less than n, the m left singular vectors are returned as columns of U and the first m right singular vectors are returned as rows of A. Note that the (conjugate) transposes of the right singular vectors are returned in Vt or A.

The argument U can be None (if jobz is 'N' or 'A' of jobz is 'O' and m is greater than or equal to n) or a matrix of the same type as A. The argument Vt can be None (if jobz is 'N' or 'A' or jobz is 'O' and m is less than n) or a matrix of the same type as A.

On exit, the contents of A are destroyed.