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# sgesvd

```
NAME
SGESVD - compute the singular value decomposition (SVD) of a
real M-by-N matrix A, optionally computing the left and/or
right singular vectors

SYNOPSIS
SUBROUTINE SGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT,
LDVT, WORK, LWORK, INFO )

CHARACTER      JOBU, JOBVT

INTEGER        INFO, LDA, LDU, LDVT, LWORK, M, N

REAL           A( LDA, * ), S( * ), U( LDU, * ), VT(
LDVT, * ), WORK( * )

PURPOSE
SGESVD computes the singular value decomposition (SVD) of a
real M-by-N matrix A, optionally computing the left and/or
right singular vectors. The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its
min(m,n) diagonal elements, U is an M-by-M orthogonal
matrix, and V is an N-by-N orthogonal matrix.  The diagonal
elements of SIGMA 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 V**T, not V.

ARGUMENTS
JOBU    (input) CHARACTER*1
Specifies options for computing all or part of the
matrix U:
= 'A':  all M columns of U are returned in array U:
= 'S':  the first min(m,n) columns of U (the left
singular vectors) are returned in the array U; =
'O':  the first min(m,n) columns of U (the left
singular vectors) are overwritten on the array A; =
'N':  no columns of U (no left singular vectors) are
computed.

JOBVT   (input) CHARACTER*1
Specifies options for computing all or part of the
matrix V**T:
= 'A':  all N rows of V**T are returned in the array
VT;
= 'S':  the first min(m,n) rows of V**T (the right

singular vectors) are returned in the array VT; =
'O':  the first min(m,n) rows of V**T (the right
singular vectors) are overwritten on the array A; =
'N':  no rows of V**T (no right singular vectors)
are computed.

JOBVT and JOBU cannot both be 'O'.

M       (input) INTEGER
The number of rows of the input matrix A.  M >= 0.

N       (input) INTEGER
The number of columns of the input matrix A.  N >=
0.

A       (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.  On exit, if JOBU =
'O',  A is overwritten with the first min(m,n)
columns of U (the left singular vectors, stored
columnwise); if JOBVT = 'O', A is overwritten with
the first min(m,n) rows of V**T (the right singular
vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT
.ne. 'O', the contents of A are destroyed.

LDA     (input) INTEGER
The leading dimension of the array A.  LDA >=
max(1,M).

S       (output) REAL array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >=
S(i+1).

U       (output) REAL array, dimension (LDU,UCOL)
(LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU =
'S'.  If JOBU = 'A', U contains the M-by-M orthogo-
nal matrix U; if JOBU = 'S', U contains the first
min(m,n) columns of U (the left singular vectors,
stored columnwise); if JOBU = 'N' or 'O', U is not
referenced.

LDU     (input) INTEGER
The leading dimension of the array U.  LDU >= 1; if
JOBU = 'S' or 'A', LDU >= M.

VT      (output) REAL array, dimension (LDVT,N)
If JOBVT = 'A', VT contains the N-by-N orthogonal
matrix V**T; if JOBVT = 'S', VT contains the first
min(m,n) rows of V**T (the right singular vectors,
stored rowwise); if JOBVT = 'N' or 'O', VT is not
referenced.

LDVT    (input) INTEGER

The leading dimension of the array VT.  LDVT >= 1;
if JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >=
min(M,N).

WORK    (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK; if INFO > 0, WORK(2:MIN(M,N)) contains the
unconverged superdiagonal elements of an upper bidi-
agonal matrix B whose diagonal is in S (not neces-
sarily sorted). B satisfies A = U * B * VT, so it
has the same singular values as A, and singular vec-
tors related by U and VT.

LWORK   (input) INTEGER
The dimension of the array WORK. LWORK >= 1.  LWORK
>= MAX(3*MIN(M,N)+MAX(M,N),5*MIN(M,N)-4).  For good
performance, LWORK should generally be larger.

INFO    (output) INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal
value.
> 0:  if SBDSQR did not converge, INFO specifies how
many superdiagonals of an intermediate bidiagonal
form B did not converge to zero. See the description
of WORK above for details.
```