?gesdd

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 11/07/2023

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?gesdd

Computes the singular value decomposition of a general rectangular matrix using a divide and conquer method.

Syntax

call sgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, info)

call dgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, iwork, info)

call cgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)

call zgesdd(jobz, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, iwork, info)

call gesdd(a, s [,u] [,vt] [,jobz] [,info])

Include Files

mkl.fi, lapack.f90

Description

The routine computes the singular value decomposition (SVD) of a real/complex m-by-n matrix A, optionally computing the left and/or right singular vectors.

If singular vectors are desired, it uses a divide-and-conquer algorithm. The SVD is written

A = U*Σ*V^T for real routines,

A = U*Σ*V^H for complex routines,

where Σ is an m-by-n matrix which is zero except for its min(m,n) diagonal elements, U is an m-by-m orthogonal/unitary matrix, and V is an n-by-n orthogonal/unitary matrix. The diagonal elements of Σ are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m, n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns vt = V^T (for real flavors) or vt =V^H (for complex flavors), not V.

Input Parameters

jobz

CHARACTER*1. Must be 'A', 'S', 'O', or 'N'.

Specifies options for computing all or part of the matrices U and V.

If jobz = 'A', all m columns of U and all n rows of V^T or V^H are returned in the arrays u and vt;

if jobz = 'S', the first min(m, n) columns of U and the first min(m, n) rows of V^T or V^H are returned in the arrays u and vt;

if jobz = 'O', then

if m≥ n, the first n columns of U are overwritten in the array a and all rows of V^T or V^H are returned in the array vt;

if m<n, all columns of U are returned in the array u and the first m rows of V^T or V^H are overwritten in the array a;

if jobz = 'N', no columns of U or rows of V^T or V^H are computed.

m

INTEGER. The number of rows of the matrix A (m≥ 0).

n

INTEGER. The number of columns in A (n≥ 0).

a, work

REAL for sgesdd

DOUBLE PRECISION for dgesdd

COMPLEX for cgesdd

DOUBLE COMPLEX for zgesdd.

Arrays:

a(lda,*) is an array containing the m-by-n matrix A.

The second dimension of a must be at least max(1, n).

work is a workspace array, its dimension max(1, lwork).

lda

INTEGER. The leading dimension of the array a. Must be at least max(1, m).

ldu, ldvt

INTEGER. The leading dimensions of the output arrays u and vt, respectively.

Constraints:

ldu≥ 1; ldvt≥ 1.

If jobz = 'S' or 'A', or jobz = 'O' and m < n,

then ldu≥m;

If jobz = 'A' or jobz = 'O' and m≥ n,

then ldvt≥n;

If jobz = 'S', ldvt≥ min(m, n).

lwork

INTEGER.

The dimension of the array work; lwork≥ 1.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the work(1), and no error message related to lwork is issued by xerbla.

See Application Notes for the theoretical minimum value of lwork when a workspace query is not performed.

rwork

REAL for cgesdd

DOUBLE PRECISION for zgesdd

Workspace array, size at least max(1, 7*min(m,n)) if jobz = 'N'.

Otherwise, the dimension of rwork must be at least max(1,min(m,n)*max(5*min(m,n)+7,2*max(m,n)+2*min(m,n)+1)). This array is used in complex flavors only.

iwork

INTEGER. Workspace array, size at least max(1, 8 *min(m, n)).

Output Parameters

a

On exit:

If jobz = 'O', then if m≥ n, a is overwritten with the first n columns of U (the left singular vectors, stored columnwise). If m < n, a is overwritten with the first m rows of V^T (the right singular vectors, stored rowwise);

If jobz≠'O', the contents of a are destroyed.

s

REAL for single precision flavors DOUBLE PRECISION for double precision flavors.

Array, size at least max(1, min(m,n)). Contains the singular values of A sorted so that s(i) ≥ s(i+1).

u, vt

REAL for sgesdd

DOUBLE PRECISION for dgesdd

COMPLEX for cgesdd

DOUBLE COMPLEX for zgesdd.

Arrays:

u(ldu,*); the second dimension of u must be at least max(1, m) if jobz = 'A' or jobz = 'O' and m < n.

If jobz = 'S', the second dimension of u must be at least max(1, min(m, n)).

If jobz = 'A'or jobz = 'O' and m < n, u contains the m-by-m orthogonal/unitary matrix U.

If jobz = 'S', u contains the first min(m, n) columns of U (the left singular vectors, stored columnwise).

If jobz = 'O' and m≥n, or jobz = 'N', u is not referenced.

vt(ldvt,*); the second dimension of vt must be at least max(1, n).

If jobz = 'A'or jobz = 'O' and m≥n, vt contains the n-by-n orthogonal/unitary matrix V^T.

If jobz = 'S', vt contains the first min(m, n) rows of V^T (the right singular vectors, stored rowwise).

If jobz = 'O' and m < n, or jobz = 'N', vt is not referenced.

work(1)

On exit, if info = 0, then work(1) returns the optimal size of lwork.

info

INTEGER.

If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = -4, A had a NAN entry.

If info = i, then ?bdsdc did not converge, updating process failed.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gesdd interface are the following:

a

Holds the matrix A of size (m, n).

s

Holds the vector of length min(m, n).

u

Holds the matrix U of size

(m,m) if jobz='A' or jobz='O' and m < n
(m,min(m, n)) if jobz='S'

u is not referenced if jobz is not supplied or if jobz='N' or jobz='O' and m≥n.

vt

Holds the matrix VT of size

(n,n) if jobz='A' or jobz='O' and m≥n
(min(m, n), n) if jobz='S'

vt is not referenced if jobz is not supplied or if jobz='N' or jobz='O' and m < n.

job

Must be 'N', 'A', 'S', or 'O'. The default value is 'N'.

Application Notes

The theoretical minimum value for lwork depends on the flavor of the routine.

For real flavors:

If jobz = 'N', lwork= 3*min(m, n) + max (max(m,n), 6*min(m, n));

If jobz = 'O', lwork= 3*(min(m, n))² + max (max(m, n), 5*(min(m, n))² + 4*min(m, n));

If jobz = 'S' or 'A', lwork= min(m, n)*(6 + 4*min(m, n)) + max(m, n);

For complex flavors:

If jobz = 'N', lwork= 2*min(m, n) + max(m, n);

If jobz = 'O', lwork= 2*(min(m, n))² + max(m, n) + 2*min(m, n);

If jobz = 'S' or 'A', lwork= (min(m, n))² + max(m, n) + 2*min(m, n);

The optimal value of lwork returned by a workspace query generally provides better performance than the theoretical minimum value.The value of lwork returned by a workspace query is generally larger than the theoretical minimum value, but for very small matrices it can be smaller. The absolute minimum value of lwork is the minimum of the workspace query result and the theoretical minimum.

If you set lwork to a value less than the absolute minimum value and not equal to -1, the routine returns immediately with an error exit and does not provide information on the recommended workspace size.

Parent topic: Singular Value Decomposition: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gesdd

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

Application Notes