realGEsing(3) LAPACK realGEsing(3)

realGEsing

# SYNOPSIS¶

## Functions¶

subroutine sgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
SGEJSV subroutine sgesdd (JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
SGESDD 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 sgesvdx (JOBU, JOBVT, RANGE, M, N, A, LDA, VL, VU, IL, IU, NS, S, U, LDU, VT, LDVT, WORK, LWORK, IWORK, INFO)
SGESVDX computes the singular value decomposition (SVD) for GE matrices subroutine sggsvd3 (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)
SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

# Detailed Description¶

This is the group of real singular value driver functions for GE matrices

# Function Documentation¶

## subroutine sgejsv (character*1 JOBA, character*1 JOBU, character*1 JOBV, character*1 JOBR, character*1 JOBT, character*1 JOBP, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( n ) SVA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( lwork ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

SGEJSV

Purpose:

``` SGEJSV computes the singular value decomposition (SVD) of a real M-by-N
matrix [A], where M >= N. The SVD of [A] is written as
[A] = [U] * [SIGMA] * [V]^t,
where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
[V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
the singular values of [A]. The columns of [U] and [V] are the left and
the right singular vectors of [A], respectively. The matrices [U] and [V]
are computed and stored in the arrays U and V, respectively. The diagonal
of [SIGMA] is computed and stored in the array SVA.
SGEJSV can sometimes compute tiny singular values and their singular vectors much
more accurately than other SVD routines, see below under Further Details.
```

Parameters:

JOBA

```          JOBA is CHARACTER*1
Specifies the level of accuracy:
= 'C': This option works well (high relative accuracy) if A = B * D,
with well-conditioned B and arbitrary diagonal matrix D.
The accuracy cannot be spoiled by COLUMN scaling. The
accuracy of the computed output depends on the condition of
B, and the procedure aims at the best theoretical accuracy.
The relative error max_{i=1:N}|d sigma_i| / sigma_i is
bounded by f(M,N)*epsilon* cond(B), independent of D.
The input matrix is preprocessed with the QRF with column
pivoting. This initial preprocessing and preconditioning by
a rank revealing QR factorization is common for all values of
JOBA. Additional actions are specified as follows:
= 'E': Computation as with 'C' with an additional estimate of the
condition number of B. It provides a realistic error bound.
= 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
D1, D2, and well-conditioned matrix C, this option gives
higher accuracy than the 'C' option. If the structure of the
input matrix is not known, and relative accuracy is
desirable, then this option is advisable. The input matrix A
is preprocessed with QR factorization with FULL (row and
column) pivoting.
= 'G'  Computation as with 'F' with an additional estimate of the
condition number of B, where A=D*B. If A has heavily weighted
rows, then using this condition number gives too pessimistic
error bound.
= 'A': Small singular values are the noise and the matrix is treated
as numerically rank deficient. The error in the computed
singular values is bounded by f(m,n)*epsilon*||A||.
The computed SVD A = U * S * V^t restores A up to
f(m,n)*epsilon*||A||.
This gives the procedure the licence to discard (set to zero)
all singular values below N*epsilon*||A||.
= 'R': Similar as in 'A'. Rank revealing property of the initial
QR factorization is used do reveal (using triangular factor)
a gap sigma_{r+1} < epsilon * sigma_r in which case the
numerical RANK is declared to be r. The SVD is computed with
absolute error bounds, but more accurately than with 'A'.
```

JOBU

```          JOBU is CHARACTER*1
Specifies whether to compute the columns of U:
= 'U': N columns of U are returned in the array U.
= 'F': full set of M left sing. vectors is returned in the array U.
= 'W': U may be used as workspace of length M*N. See the description
of U.
= 'N': U is not computed.
```

JOBV

```          JOBV is CHARACTER*1
Specifies whether to compute the matrix V:
= 'V': N columns of V are returned in the array V; Jacobi rotations
are not explicitly accumulated.
= 'J': N columns of V are returned in the array V, but they are
computed as the product of Jacobi rotations. This option is
allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
= 'W': V may be used as workspace of length N*N. See the description
of V.
= 'N': V is not computed.
```

JOBR

```          JOBR is CHARACTER*1
Specifies the RANGE for the singular values. Issues the licence to
set to zero small positive singular values if they are outside
specified range. If A .NE. 0 is scaled so that the largest singular
value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
the licence to kill columns of A whose norm in c*A is less than
SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
= 'N': Do not kill small columns of c*A. This option assumes that
BLAS and QR factorizations and triangular solvers are
implemented to work in that range. If the condition of A
is greater than BIG, use SGESVJ.
= 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
(roughly, as described above). This option is recommended.
===========================
For computing the singular values in the FULL range [SFMIN,BIG]
use SGESVJ.
```

JOBT

```          JOBT is CHARACTER*1
If the matrix is square then the procedure may determine to use
transposed A if A^t seems to be better with respect to convergence.
If the matrix is not square, JOBT is ignored. This is subject to
changes in the future.
The decision is based on two values of entropy over the adjoint
orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
= 'T': transpose if entropy test indicates possibly faster
convergence of Jacobi process if A^t is taken as input. If A is
replaced with A^t, then the row pivoting is included automatically.
= 'N': do not speculate.
This option can be used to compute only the singular values, or the
full SVD (U, SIGMA and V). For only one set of singular vectors
(U or V), the caller should provide both U and V, as one of the
matrices is used as workspace if the matrix A is transposed.
The implementer can easily remove this constraint and make the
code more complicated. See the descriptions of U and V.
```

JOBP

```          JOBP is CHARACTER*1
Issues the licence to introduce structured perturbations to drown
denormalized numbers. This licence should be active if the
denormals are poorly implemented, causing slow computation,
especially in cases of fast convergence (!). For details see [1,2].
For the sake of simplicity, this perturbations are included only
when the full SVD or only the singular values are requested. The
implementer/user can easily add the perturbation for the cases of
computing one set of singular vectors.
= 'P': introduce perturbation
= 'N': do not perturb
```

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. M >= N >= 0.
```

A

```          A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
```

LDA

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

SVA

```          SVA is REAL array, dimension (N)
On exit,
- For WORK(1)/WORK(2) = ONE: The singular values of A. During the
computation SVA contains Euclidean column norms of the
iterated matrices in the array A.
- For WORK(1) .NE. WORK(2): The singular values of A are
(WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
sigma_max(A) overflows or if small singular values have been
saved from underflow by scaling the input matrix A.
- If JOBR='R' then some of the singular values may be returned
as exact zeros obtained by "set to zero" because they are
below the numerical rank threshold or are denormalized numbers.
```

U

```          U is REAL array, dimension ( LDU, N )
If JOBU = 'U', then U contains on exit the M-by-N matrix of
the left singular vectors.
If JOBU = 'F', then U contains on exit the M-by-M matrix of
the left singular vectors, including an ONB
of the orthogonal complement of the Range(A).
If JOBU = 'W'  .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
then U is used as workspace if the procedure
replaces A with A^t. In that case, [V] is computed
in U as left singular vectors of A^t and then
copied back to the V array. This 'W' option is just
a reminder to the caller that in this case U is
reserved as workspace of length N*N.
If JOBU = 'N'  U is not referenced, unless JOBT='T'.
```

LDU

```          LDU is INTEGER
The leading dimension of the array U,  LDU >= 1.
IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.
```

V

```          V is REAL array, dimension ( LDV, N )
If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
then V is used as workspace if the pprocedure
replaces A with A^t. In that case, [U] is computed
in V as right singular vectors of A^t and then
copied back to the U array. This 'W' option is just
a reminder to the caller that in this case V is
reserved as workspace of length N*N.
If JOBV = 'N'  V is not referenced, unless JOBT='T'.
```

LDV

```          LDV is INTEGER
The leading dimension of the array V,  LDV >= 1.
If JOBV = 'V' or 'J' or 'W', then LDV >= N.
```

WORK

```          WORK is REAL array, dimension (LWORK)
On exit,
WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
that SCALE*SVA(1:N) are the computed singular values
of A. (See the description of SVA().)
WORK(2) = See the description of WORK(1).
WORK(3) = SCONDA is an estimate for the condition number of
column equilibrated A. (If JOBA .EQ. 'E' or 'G')
SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
It is computed using SPOCON. It holds
N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
where R is the triangular factor from the QRF of A.
However, if R is truncated and the numerical rank is
determined to be strictly smaller than N, SCONDA is
returned as -1, thus indicating that the smallest
singular values might be lost.
If full SVD is needed, the following two condition numbers are
useful for the analysis of the algorithm. They are provied for
a developer/implementer who is familiar with the details of
the method.
WORK(4) = an estimate of the scaled condition number of the
triangular factor in the first QR factorization.
WORK(5) = an estimate of the scaled condition number of the
triangular factor in the second QR factorization.
The following two parameters are computed if JOBT .EQ. 'T'.
They are provided for a developer/implementer who is familiar
with the details of the method.
WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
of diag(A^t*A) / Trace(A^t*A) taken as point in the
probability simplex.
WORK(7) = the entropy of A*A^t.
```

LWORK

```          LWORK is INTEGER
Length of WORK to confirm proper allocation of work space.
LWORK depends on the job:
If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
-> .. no scaled condition estimate required (JOBE.EQ.'N'):
LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
block size for DGEQP3 and DGEQRF.
In general, optimal LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
-> .. an estimate of the scaled condition number of A is
required (JOBA='E', 'G'). In this case, LWORK is the maximum
of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
N+N*N+LWORK(DPOCON),7).
If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
-> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
-> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
DORMLQ. In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
If SIGMA and the left singular vectors are needed
-> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
-> For optimal performance:
if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or
M*NB (for JOBU.EQ.'F').
If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
-> if JOBV.EQ.'V'
the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
-> if JOBV.EQ.'J' the minimal requirement is
LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
-> For optimal performance, LWORK should be additionally
larger than N+M*NB, where NB is the optimal block size
for DORMQR.
```

IWORK

```          IWORK is INTEGER array, dimension (M+3*N).
On exit,
IWORK(1) = the numerical rank determined after the initial
QR factorization with pivoting. See the descriptions
of JOBA and JOBR.
IWORK(2) = the number of the computed nonzero singular values
IWORK(3) = if nonzero, a warning message:
If IWORK(3).EQ.1 then some of the column norms of A
were denormalized floats. The requested high accuracy
is not warranted by the data.
```

INFO

```          INFO is INTEGER
< 0  : if INFO = -i, then the i-th argument had an illegal value.
= 0 :  successful exit;
> 0 :  SGEJSV  did not converge in the maximal allowed number
of sweeps. The computed values may be inaccurate.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

June 2016

Further Details:

```  SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an
additional row pivoting can be used as a preprocessor, which in some
cases results in much higher accuracy. An example is matrix A with the
structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
diagonal matrices and C is well-conditioned matrix. In that case, complete
pivoting in the first QR factorizations provides accuracy dependent on the
condition number of C, and independent of D1, D2. Such higher accuracy is
not completely understood theoretically, but it works well in practice.
Further, if A can be written as A = B*D, with well-conditioned B and some
diagonal D, then the high accuracy is guaranteed, both theoretically and
in software, independent of D. For more details see [1], [2].
The computational range for the singular values can be the full range
( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
& LAPACK routines called by SGEJSV are implemented to work in that range.
If that is not the case, then the restriction for safe computation with
the singular values in the range of normalized IEEE numbers is that the
spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
overflow. This code (SGEJSV) is best used in this restricted range,
meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
returned as zeros. See JOBR for details on this.
Further, this implementation is somewhat slower than the one described
in [1,2] due to replacement of some non-LAPACK components, and because
the choice of some tuning parameters in the iterative part (SGESVJ) is
left to the implementer on a particular machine.
The rank revealing QR factorization (in this code: SGEQP3) should be
implemented as in [3]. We have a new version of SGEQP3 under development
that is more robust than the current one in LAPACK, with a cleaner cut in
rank deficient cases. It will be available in the SIGMA library [4].
If M is much larger than N, it is obvious that the initial QRF with
column pivoting can be preprocessed by the QRF without pivoting. That
well known trick is not used in SGEJSV because in some cases heavy row
weighting can be treated with complete pivoting. The overhead in cases
M much larger than N is then only due to pivoting, but the benefits in
terms of accuracy have prevailed. The implementer/user can incorporate
this extra QRF step easily. The implementer can also improve data movement
(matrix transpose, matrix copy, matrix transposed copy) - this
implementation of SGEJSV uses only the simplest, naive data movement.
```

Contributors:

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

References:

``` [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
factorization software - a case study.
ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
LAPACK Working note 176.
[4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008.
```

Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you.

## subroutine sgesdd (character JOBZ, 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, dimension( * ) IWORK, integer INFO)¶

SGESDD

Purpose:

``` SGESDD computes the singular value decomposition (SVD) of a real
M-by-N matrix A, optionally computing the left and right singular
vectors.  If singular vectors are desired, it uses a
divide-and-conquer algorithm.
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 VT = V**T, not V.
The divide and conquer algorithm makes very mild assumptions about
floating point arithmetic. It will work on machines with a guard
digit in add/subtract, or on those binary machines without guard
digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
Cray-2. It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
```

Parameters:

JOBZ

```          JOBZ is CHARACTER*1
Specifies options for computing all or part of the matrix U:
= 'A':  all M columns of U and all N rows of V**T are
returned in the arrays U and VT;
= 'S':  the first min(M,N) columns of U and the first
min(M,N) rows of V**T are returned in the arrays U
and VT;
= 'O':  If M >= N, the first N columns of U are overwritten
on the array A and all rows of V**T are returned in
the array VT;
otherwise, all columns of U are returned in the
array U and the first M rows of V**T are overwritten
in the array A;
= 'N':  no columns of U or rows of V**T are computed.
```

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 JOBZ = 'O',  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) otherwise.
if JOBZ .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)
UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
UCOL = min(M,N) if JOBZ = 'S'.
If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
orthogonal 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.
```

LDU

```          LDU is INTEGER
The leading dimension of the array U.  LDU >= 1; if
JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
```

VT

```          VT is REAL array, dimension (LDVT,N)
If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
N-by-N orthogonal 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.
```

LDVT

```          LDVT is INTEGER
The leading dimension of the array VT.  LDVT >= 1;
if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
if JOBZ = '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;
```

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK >= 1.
If LWORK = -1, a workspace query is assumed.  The optimal
size for the WORK array is calculated and stored in WORK(1),
and no other work except argument checking is performed.
Let mx = max(M,N) and mn = min(M,N).
If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
These are not tight minimums in all cases; see comments inside code.
For good performance, LWORK should generally be larger;
a query is recommended.
```

IWORK

```          IWORK is INTEGER array, dimension (8*min(M,N))
```

INFO

```          INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  SBDSDC did not converge, updating process failed.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

June 2016

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

## 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.

Date:

April 2012

## subroutine sgesvdx (character JOBU, character JOBVT, character RANGE, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real VL, real VU, integer IL, integer IU, integer NS, real, dimension( * ) S, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldvt, * ) VT, integer LDVT, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

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

Purpose:

```  SGESVDX 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.
SGESVDX uses an eigenvalue problem for obtaining the SVD, which
allows for the computation of a subset of singular values and
vectors. See SBDSVDX for details.
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:
= 'V':  the first min(m,n) columns of U (the left singular
vectors) or as specified by RANGE are returned in
the array U;
= '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:
= 'V':  the first min(m,n) rows of V**T (the right singular
vectors) or as specified by RANGE are returned in
the array VT;
= 'N':  no rows of V**T (no right singular vectors) are
computed.
```

RANGE

```          RANGE is CHARACTER*1
= 'A': all singular values will be found.
= 'V': all singular values in the half-open interval (VL,VU]
will be found.
= 'I': the IL-th through IU-th singular values will be found.
```

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, the contents of A are destroyed.
```

LDA

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

VL

```          VL is REAL
If RANGE='V', the lower bound of the interval to
be searched for singular values. VU > VL.
Not referenced if RANGE = 'A' or 'I'.
```

VU

```          VU is REAL
If RANGE='V', the upper bound of the interval to
be searched for singular values. VU > VL.
Not referenced if RANGE = 'A' or 'I'.
```

IL

```          IL is INTEGER
If RANGE='I', the index of the
smallest singular value to be returned.
1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
Not referenced if RANGE = 'A' or 'V'.
```

IU

```          IU is INTEGER
If RANGE='I', the index of the
largest singular value to be returned.
1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
Not referenced if RANGE = 'A' or 'V'.
```

NS

```          NS is INTEGER
The total number of singular values found,
0 <= NS <= min(M,N).
If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.
```

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)
If JOBU = 'V', U contains columns of U (the left singular
vectors, stored columnwise) as specified by RANGE; if
JOBU = 'N', U is not referenced.
Note: The user must ensure that UCOL >= NS; if RANGE = 'V',
the exact value of NS is not known in advance and an upper
bound must be used.
```

LDU

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

VT

```          VT is REAL array, dimension (LDVT,N)
If JOBVT = 'V', VT contains the rows of V**T (the right singular
vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N',
VT is not referenced.
Note: The user must ensure that LDVT >= NS; if RANGE = 'V',
the exact value of NS is not known in advance and an upper
bound must be used.
```

LDVT

```          LDVT is INTEGER
The leading dimension of the array VT.  LDVT >= 1; if
JOBVT = 'V', LDVT >= NS (see above).
```

WORK

```          WORK is REAL 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,MIN(M,N)*(MIN(M,N)+4)) for the paths (see
- PATH 1  (M much larger than N)
- PATH 1t (N much larger than M)
LWORK >= MAX(1,MIN(M,N)*2+MAX(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.
```

IWORK

```          IWORK is INTEGER array, dimension (12*MIN(M,N))
If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0,
then IWORK contains the indices of the eigenvectors that failed
to converge in SBDSVDX/SSTEVX.
```

INFO

```     INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  if INFO = i, then i eigenvectors failed to converge
in SBDSVDX/SSTEVX.
if INFO = N*2 + 1, an internal error occurred in
SBDSVDX
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

June 2016

## subroutine sggsvd3 (character JOBU, character JOBV, character JOBQ, integer M, integer N, integer P, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)¶

SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:

``` SGGSVD3 computes the generalized singular value decomposition (GSVD)
of an M-by-N real matrix A and P-by-N real matrix B:
U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices.
Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and
D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the
following structures, respectively:
If M-K-L >= 0,
K  L
D1 =     K ( I  0 )
L ( 0  C )
M-K-L ( 0  0 )
K  L
D2 =   L ( 0  S )
P-L ( 0  0 )
N-K-L  K    L
( 0 R ) = K (  0   R11  R12 )
L (  0    0   R22 )
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1),  ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 =   K ( I  0    0   )
M-K ( 0  C    0   )
K M-K K+L-M
D2 =   M-K ( 0  S    0  )
K+L-M ( 0  0    I  )
P-L ( 0  0    0  )
N-K-L  K   M-K  K+L-M
( 0 R ) =     K ( 0    R11  R12  R13  )
M-K ( 0     0   R22  R23  )
K+L-M ( 0     0    0   R33  )
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1),  ... , BETA(M) ),
C**2 + S**2 = I.
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
( 0  R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
The routine computes C, S, R, and optionally the orthogonal
transformation matrices U, V and Q.
In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
A and B implicitly gives the SVD of A*inv(B):
A*inv(B) = U*(D1*inv(D2))*V**T.
If ( A**T,B**T)**T  has orthonormal columns, then the GSVD of A and B is
also equal to the CS decomposition of A and B. Furthermore, the GSVD
can be used to derive the solution of the eigenvalue problem:
A**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, D1 and D2 are
``diagonal''.  The former GSVD form can be converted to the latter
form by taking the nonsingular matrix X as
X = Q*( I   0    )
( 0 inv(R) ).
```

Parameters:

JOBU

```          JOBU is CHARACTER*1
= 'U':  Orthogonal matrix U is computed;
= 'N':  U is not computed.
```

JOBV

```          JOBV is CHARACTER*1
= 'V':  Orthogonal matrix V is computed;
= 'N':  V is not computed.
```

JOBQ

```          JOBQ is CHARACTER*1
= 'Q':  Orthogonal matrix Q is computed;
= 'N':  Q is not computed.
```

M

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

N

```          N is INTEGER
The number of columns of the matrices A and B.  N >= 0.
```

P

```          P is INTEGER
The number of rows of the matrix B.  P >= 0.
```

K

```          K is INTEGER
```

L

```          L is INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A**T,B**T)**T.
```

A

```          A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
See Purpose for details.
```

LDA

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

B

```          B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-K-L < 0.
See Purpose for details.
```

LDB

```          LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
```

ALPHA

```          ALPHA is REAL array, dimension (N)
```

BETA

```          BETA is REAL array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
BETA(1:K)  = 0,
and if M-K-L >= 0,
ALPHA(K+1:K+L) = C,
BETA(K+1:K+L)  = S,
or if M-K-L < 0,
ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
BETA(K+1:M) =S, BETA(M+1:K+L) =1
and
ALPHA(K+L+1:N) = 0
BETA(K+L+1:N)  = 0
```

U

```          U is REAL array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M orthogonal matrix U.
If JOBU = 'N', U is not referenced.
```

LDU

```          LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
```

V

```          V is REAL array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P orthogonal matrix V.
If JOBV = 'N', V is not referenced.
```

LDV

```          LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
```

Q

```          Q is REAL array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.
```

LDQ

```          LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
```

WORK

```          WORK is REAL 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.
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.
```

IWORK

```          IWORK is INTEGER array, dimension (N)
On exit, IWORK stores the sorting information. More
precisely, the following loop will sort ALPHA
for I = K+1, min(M,K+L)
swap ALPHA(I) and ALPHA(IWORK(I))
endfor
such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
```

INFO

```          INFO is INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = 1, the Jacobi-type procedure failed to
converge.  For further details, see subroutine STGSJA.
```

Internal Parameters:

```  TOLA    REAL
TOLB    REAL
TOLA and TOLB are the thresholds to determine the effective
rank of (A**T,B**T)**T. Generally, they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

August 2015

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Further Details:

SGGSVD3 replaces the deprecated subroutine SGGSVD.

# Author¶

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