naginterfaces.library.lapackeig.dgesdd

naginterfaces.library.lapackeig.dgesdd(jobz, a)[source]

dgesdd computes the singular value decomposition (SVD) of a real matrix , optionally computing the left and/or right singular vectors, by using a divide-and-conquer method.

For full information please refer to the NAG Library document for f08kd

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f08/f08kdf.html

Parameters
jobzstr, length 1

Specifies options for computing all or part of the matrix .

All columns of and all rows of are returned in the arrays and .

The first columns of and the first rows of are returned in the arrays and .

If , the first columns of are overwritten on the array and all rows of are returned in the array . Otherwise, all columns of are returned in the array and the first rows of are overwritten in the array .

No columns of or rows of are computed.

afloat, array-like, shape

The matrix .

Returns
afloat, ndarray, shape

If , is overwritten with the first columns of (the left singular vectors, stored column-wise) if ; is overwritten with the first rows of (the right singular vectors, stored row-wise) otherwise.

If , the contents of are destroyed.

sfloat, ndarray, shape

The singular values of , sorted so that .

ufloat, ndarray, shape

If or and , contains the orthogonal matrix .

If , contains the first columns of (the left singular vectors, stored column-wise).

If and , or , is not referenced.

vtfloat, ndarray, shape

If or and , contains the orthogonal matrix .

If , contains the first rows of (the right singular vectors, stored row-wise).

If and , or , is not referenced.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: , , or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

dgesdd did not converge, the updating process failed.

Notes

The SVD is written as

where is an matrix which is zero except for its diagonal elements, is an orthogonal matrix, and is an orthogonal matrix. The diagonal elements of are the singular values of ; they are real and non-negative, and are returned in descending order. The first columns of and are the left and right singular vectors of .

Note that the function returns , not .

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore