# gesvd - Man Page

gesvd: SVD, QR iteration

## Synopsis

### Functions

subroutine cgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info)
CGESVD computes the singular value decomposition (SVD) for GE matrices
subroutine dgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, info)
DGESVD computes the singular value decomposition (SVD) for GE matrices
subroutine sgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, info)
SGESVD computes the singular value decomposition (SVD) for GE matrices
subroutine zgesvd (jobu, jobvt, m, n, a, lda, s, u, ldu, vt, ldvt, work, lwork, rwork, info)
ZGESVD computes the singular value decomposition (SVD) for GE matrices

## Function Documentation

### subroutine cgesvd (character jobu, character jobvt, integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) s, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldvt, * ) vt, integer ldvt, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer info)

CGESVD computes the singular value decomposition (SVD) for GE matrices

Purpose:

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

A = U * SIGMA * conjugate-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 unitary matrix, and
V is an N-by-N unitary 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**H, not V.```
Parameters

JOBU

```          JOBU is 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

```          JOBVT is CHARACTER*1
Specifies options for computing all or part of the matrix
V**H:
= 'A':  all N rows of V**H are returned in the array VT;
= 'S':  the first min(m,n) rows of V**H (the right singular
vectors) are returned in the array VT;
= 'O':  the first min(m,n) rows of V**H (the right singular
vectors) are overwritten on the array A;
= 'N':  no rows of V**H (no right singular vectors) are
computed.

JOBVT and JOBU cannot both be 'O'.```

M

```          M is INTEGER
The number of rows of the input matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the input matrix A.  N >= 0.```

A

```          A is COMPLEX 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**H (the right singular vectors,
stored rowwise);
if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
are destroyed.```

LDA

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

S

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

U

```          U is COMPLEX 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 unitary 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

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

VT

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

LDVT

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

WORK

```          WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >=  MAX(1,2*MIN(M,N)+MAX(M,N)).
For good performance, LWORK should generally be larger.

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 first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

RWORK

```          RWORK is REAL array, dimension (5*min(M,N))
On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the
unconverged superdiagonal elements of an upper bidiagonal
matrix B whose diagonal is in S (not necessarily sorted).
B satisfies A = U * B * VT, so it has the same singular
values as A, and singular vectors related by U and VT.```

INFO

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

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 212 of file cgesvd.f.

### subroutine dgesvd (character jobu, character jobvt, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldvt, * ) vt, integer ldvt, double precision, dimension( * ) work, integer lwork, integer info)

DGESVD computes the singular value decomposition (SVD) for GE matrices

Purpose:

``` DGESVD 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.```
Parameters

JOBU

```          JOBU is 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

```          JOBVT is 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

```          M is INTEGER
The number of rows of the input matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the input matrix A.  N >= 0.```

A

```          A is DOUBLE PRECISION 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

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

S

```          S is DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >= S(i+1).```

U

```          U is DOUBLE PRECISION 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 orthogonal 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

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

VT

```          VT is DOUBLE PRECISION 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

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

WORK

```          WORK is DOUBLE PRECISION array, dimension (MAX(1,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 bidiagonal matrix B
whose diagonal is in S (not necessarily sorted). B
satisfies A = U * B * VT, so it has the same singular values
as A, and singular vectors related by U and VT.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
- PATH 1  (M much larger than N, JOBU='N')
- PATH 1t (N much larger than M, JOBVT='N')
LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(M,N)) for the other paths
For good performance, LWORK should generally be larger.

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 first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if DBDSQR 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.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 209 of file dgesvd.f.

### subroutine sgesvd (character jobu, character jobvt, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldvt, * ) vt, integer ldvt, real, dimension( * ) work, integer lwork, integer info)

SGESVD computes the singular value decomposition (SVD) for GE matrices

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.```
Parameters

JOBU

```          JOBU is 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

```          JOBVT is 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

```          M is INTEGER
The number of rows of the input matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the input matrix A.  N >= 0.```

A

```          A is 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

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

S

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

U

```          U is 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 orthogonal 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

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

VT

```          VT is 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

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

WORK

```          WORK is REAL array, dimension (MAX(1,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 bidiagonal matrix B
whose diagonal is in S (not necessarily sorted). B
satisfies A = U * B * VT, so it has the same singular values
as A, and singular vectors related by U and VT.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
- PATH 1  (M much larger than N, JOBU='N')
- PATH 1t (N much larger than M, JOBVT='N')
LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) for the other paths
For good performance, LWORK should generally be larger.

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 first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

INFO

```          INFO is 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.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 209 of file sgesvd.f.

### subroutine zgesvd (character jobu, character jobvt, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldvt, * ) vt, integer ldvt, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer info)

ZGESVD computes the singular value decomposition (SVD) for GE matrices

Purpose:

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

A = U * SIGMA * conjugate-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 unitary matrix, and
V is an N-by-N unitary 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**H, not V.```
Parameters

JOBU

```          JOBU is 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

```          JOBVT is CHARACTER*1
Specifies options for computing all or part of the matrix
V**H:
= 'A':  all N rows of V**H are returned in the array VT;
= 'S':  the first min(m,n) rows of V**H (the right singular
vectors) are returned in the array VT;
= 'O':  the first min(m,n) rows of V**H (the right singular
vectors) are overwritten on the array A;
= 'N':  no rows of V**H (no right singular vectors) are
computed.

JOBVT and JOBU cannot both be 'O'.```

M

```          M is INTEGER
The number of rows of the input matrix A.  M >= 0.```

N

```          N is INTEGER
The number of columns of the input matrix A.  N >= 0.```

A

```          A is COMPLEX*16 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**H (the right singular vectors,
stored rowwise);
if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
are destroyed.```

LDA

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

S

```          S is DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >= S(i+1).```

U

```          U is COMPLEX*16 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 unitary 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

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

VT

```          VT is COMPLEX*16 array, dimension (LDVT,N)
If JOBVT = 'A', VT contains the N-by-N unitary matrix
V**H;
if JOBVT = 'S', VT contains the first min(m,n) rows of
V**H (the right singular vectors, stored rowwise);
if JOBVT = 'N' or 'O', VT is not referenced.```

LDVT

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

WORK

```          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
LWORK >=  MAX(1,2*MIN(M,N)+MAX(M,N)).
For good performance, LWORK should generally be larger.

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 first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

RWORK

```          RWORK is DOUBLE PRECISION array, dimension (5*min(M,N))
On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the
unconverged superdiagonal elements of an upper bidiagonal
matrix B whose diagonal is in S (not necessarily sorted).
B satisfies A = U * B * VT, so it has the same singular
values as A, and singular vectors related by U and VT.```

INFO

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

Univ. of Tennessee

Univ. of California Berkeley