# ?gesdd

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

## Syntax

FORTRAN 77:

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)

Fortran 95:

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

## Description

This 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*ΣVH`,

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 VH, not V.

## Input Parameters

jobz

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

  Specifies options for computing all or part of the matrix U.

  If `jobz = 'A'`, all m columns of U and all n rows of VT 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 VT 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 VT 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 VT are overwritten in the array a;

  if `jobz = 'N'`, no columns of U or rows of VT 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 first 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 suggested value of lwork.

rwork

  REAL for cgesdd

  DOUBLE PRECISION for zgesdd

  Workspace array, DIMENSION at least `max(1, 5*min(m,n))` if `jobz = 'N'`.

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

iwork

  INTEGER. Workspace array, DIMENSION 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 VT (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, DIMENSION 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 VT.

  If `jobz = 'S'`, vt contains the first min(m, n) rows of VT (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 required minimal 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 = i`, then ?bdsdc did not converge, updating process failed.

## Fortran 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 Fortran 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,min(m, n)).

vt

  Holds the matrix VT of size (min(m, n),n).

job

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

## Application Notes

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 ≥ 3*(min(m, n))2 + max (max(m, n), 4*(min(m, n))2 + 4*min(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)`;

For good performance, lwork should generally be larger.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

Parent topic: Singular Value Decomposition