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 sgesvdq (JOBA, JOBP, JOBR, JOBU, JOBV, M, N, A, LDA, S, U, LDU, V, LDV, NUMRANK, IWORK, LIWORK, WORK, LWORK, RWORK, LRWORK, INFO)
SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method 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 = '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 = 'V' .AND. JOBT = 'T' .AND. M = 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 = 'U' AND JOBT = 'T' AND M = 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 = '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 = '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 = 'N', JOBV = 'N' ) and

-> .. no scaled condition estimate required (JOBE = '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 = '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 = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),

if JOBU = '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 = 'U') or

M*NB (for JOBU = 'F').

If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and

-> if JOBV = 'V'

the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).

-> if JOBV = '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) = 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 , .

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 . 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 .

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:

```
 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.

 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.

 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.

 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 sgesvdq (character JOBA, character JOBP, character JOBR, character JOBU, character JOBV, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, integer NUMRANK, integer, dimension( * ) IWORK, integer LIWORK, real, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer INFO)¶

SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices

Purpose:

```
SGESVDQ computes the singular value decomposition (SVD) of a real

M-by-N matrix A, where M >= N. The SVD of A is written as

[++]   [xx]   [x0]   [xx]

A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]

[++]   [xx]

where SIGMA is an N-by-N diagonal matrix, U is an M-by-N 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.```

Parameters

JOBA

```
JOBA is CHARACTER*1

Specifies the level of accuracy in the computed SVD

= 'A' The requested accuracy corresponds to having the backward

error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,

where EPS = SLAMCH('Epsilon'). This authorises CGESVDQ to

truncate the computed triangular factor in a rank revealing

QR factorization whenever the truncated part is below the

threshold of the order of EPS * ||A||_F. This is aggressive

truncation level.

= 'M' Similarly as with 'A', but the truncation is more gentle: it

is allowed only when there is a drop on the diagonal of the

triangular factor in the QR factorization. This is medium

truncation level.

= 'H' High accuracy requested. No numerical rank determination based

on the rank revealing QR factorization is attempted.

= 'E' Same as 'H', and in addition the condition number of column

scaled A is estimated and returned in  RWORK(1).

N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)```

JOBP

```
JOBP is CHARACTER*1

= 'P' The rows of A are ordered in decreasing order with respect to

||A(i,:)||_infty. This enhances numerical accuracy at the cost

of extra data movement. Recommended for numerical robustness.

= 'N' No row pivoting.```

JOBR

```
JOBR is CHARACTER*1

= 'T' After the initial pivoted QR factorization, SGESVD is applied to

the transposed R**T of the computed triangular factor R. This involves

some extra data movement (matrix transpositions). Useful for

experiments, research and development.

= 'N' The triangular factor R is given as input to SGESVD. This may be

preferred as it involves less data movement.```

JOBU

```
JOBU is CHARACTER*1

= 'A' All M left singular vectors are computed and returned in the

matrix U. See the description of U.

= 'S' or 'U' N = min(M,N) left singular vectors are computed and returned

in the matrix U. See the description of U.

= 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular

vectors are computed and returned in the matrix U.

= 'F' The N left singular vectors are returned in factored form as the

product of the Q factor from the initial QR factorization and the

N left singular vectors of (R**T , 0)**T. If row pivoting is used,

then the necessary information on the row pivoting is stored in

IWORK(N+1:N+M-1).

= 'N' The left singular vectors are not computed.```

JOBV

```
JOBV is CHARACTER*1

= 'A', 'V' All N right singular vectors are computed and returned in

the matrix V.

= 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular

vectors are computed and returned in the matrix V. This option is

allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.

= 'N' The right singular vectors are not 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.  M >= N >= 0.```

A

```
A is REAL array of dimensions LDA x N

On entry, the input matrix A.

On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains

the Householder vectors as stored by SGEQP3. If JOBU = 'F', these Householder

vectors together with WORK(1:N) can be used to restore the Q factors from

the initial pivoted QR factorization of A. See the description of U.```

LDA

```
LDA is INTEGER.

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

S

```
S is REAL array of dimension N.

The singular values of A, ordered so that S(i) >= S(i+1).```

U

```
U is REAL array, dimension

LDU x M if JOBU = 'A'; see the description of LDU. In this case,

on exit, U contains the M left singular vectors.

LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this

case, U contains the leading N or the leading NUMRANK left singular vectors.

LDU x N if JOBU = 'F' ; see the description of LDU. In this case U

contains N x N orthogonal matrix that can be used to form the left

singular vectors.

If JOBU = 'N', U is not referenced.```

LDU

```
LDU is INTEGER.

The leading dimension of the array U.

If JOBU = 'A', 'S', 'U', 'R',  LDU >= max(1,M).

If JOBU = 'F',                 LDU >= max(1,N).

Otherwise,                     LDU >= 1.```

V

```
V is REAL array, dimension

LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .

If JOBV = 'A', or 'V',  V contains the N-by-N orthogonal matrix  V**T;

If JOBV = 'R', V contains the first NUMRANK rows of V**T (the right

singular vectors, stored rowwise, of the NUMRANK largest singular values).

If JOBV = 'N' and JOBA = 'E', V is used as a workspace.

If JOBV = 'N', and JOBA.NE.'E', V is not referenced.```

LDV

```
LDV is INTEGER

The leading dimension of the array V.

If JOBV = 'A', 'V', 'R',  or JOBA = 'E', LDV >= max(1,N).

Otherwise,                               LDV >= 1.```

NUMRANK

```
NUMRANK is INTEGER

NUMRANK is the numerical rank first determined after the rank

revealing QR factorization, following the strategy specified by the

value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK

leading singular values and vectors are then requested in the call

of SGESVD. The final value of NUMRANK might be further reduced if

some singular values are computed as zeros.```

IWORK

```
IWORK is INTEGER array, dimension (max(1, LIWORK)).

On exit, IWORK(1:N) contains column pivoting permutation of the

rank revealing QR factorization.

If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence

of row swaps used in row pivoting. These can be used to restore the

left singular vectors in the case JOBU = 'F'.

If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,

LIWORK(1) returns the minimal LIWORK.```

LIWORK

```
LIWORK is INTEGER

The dimension of the array IWORK.

LIWORK >= N + M - 1,     if JOBP = 'P' and JOBA .NE. 'E';

LIWORK >= N              if JOBP = 'N' and JOBA .NE. 'E';

LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E';

LIWORK >= N + N          if JOBP = 'N' and JOBA = 'E'.
If LIWORK = -1, then a workspace query is assumed; the routine

only calculates and returns the optimal and minimal sizes

for the WORK, IWORK, and RWORK arrays, and no error

message related to LWORK is issued by XERBLA.```

WORK

```
WORK is REAL array, dimension (max(2, LWORK)), used as a workspace.

On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters

needed to recover the Q factor from the QR factorization computed by

SGEQP3.

If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,

WORK(1) returns the optimal LWORK, and

WORK(2) returns the minimal LWORK.```

LWORK

```
LWORK is INTEGER

The dimension of the array WORK. It is determined as follows:

Let  LWQP3 = 3*N+1,  LWCON = 3*N, and let

LWORQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'

{ MAX( M, 1 ),  if JOBU = 'A'

LWSVD = MAX( 5*N, 1 )

LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ),

LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 )

Then the minimal value of LWORK is:

= MAX( N + LWQP3, LWSVD )        if only the singular values are needed;

= MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,

and a scaled condition estimate requested;

= N + MAX( LWQP3, LWSVD, LWORQ ) if the singular values and the left

singular vectors are requested;

= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the singular values and the left

singular vectors are requested, and also

a scaled condition estimate requested;

= N + MAX( LWQP3, LWSVD )        if the singular values and the right

singular vectors are requested;

= N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right

singular vectors are requested, and also

a scaled condition etimate requested;

= N + MAX( LWQP3, LWSVD, LWORQ ) if the full SVD is requested with JOBV = 'R';

independent of JOBR;

= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested,

JOBV = 'R' and, also a scaled condition

estimate requested; independent of JOBR;

= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),

N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the

full SVD is requested with JOBV = 'A' or 'V', and

JOBR ='N'

= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),

N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) )

if the full SVD is requested with JOBV = 'A' or 'V', and

JOBR ='N', and also a scaled condition number estimate

requested.

= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),

N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the

full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'

= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),

N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) )

if the full SVD is requested with JOBV = 'A' or 'V', and

JOBR ='T', and also a scaled condition number estimate

requested.

Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ).

If LWORK = -1, then a workspace query is assumed; the routine

only calculates and returns the optimal and minimal sizes

for the WORK, IWORK, and RWORK arrays, and no error

message related to LWORK is issued by XERBLA.```

RWORK

```
RWORK is REAL array, dimension (max(1, LRWORK)).

On exit,

1. If JOBA = 'E', RWORK(1) contains an estimate of the condition

number of column scaled A. If A = C * D where D is diagonal and C

has unit columns in the Euclidean norm, then, assuming full column rank,

N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).

Otherwise, RWORK(1) = -1.

2. RWORK(2) contains the number of singular values computed as

exact zeros in SGESVD applied to the upper triangular or trapeziodal

R (from the initial QR factorization). In case of early exit (no call to

SGESVD, such as in the case of zero matrix) RWORK(2) = -1.

If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,

RWORK(1) returns the minimal LRWORK.```

LRWORK

```
LRWORK is INTEGER.

The dimension of the array RWORK.

If JOBP ='P', then LRWORK >= MAX(2, M).

Otherwise, LRWORK >= 2
If LRWORK = -1, then a workspace query is assumed; the routine

only calculates and returns the optimal and minimal sizes

for the WORK, IWORK, and RWORK arrays, 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 (computed in SGESVD) did not

converge to zero.```

Further Details:

```
1. The data movement (matrix transpose) is coded using simple nested

DO-loops because BLAS and LAPACK do not provide corresponding subroutines.

Those DO-loops are easily identified in this source code - by the CONTINUE

statements labeled with 11**. In an optimized version of this code, the

nested DO loops should be replaced with calls to an optimized subroutine.

2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause

column norm overflow. This is the minial precaution and it is left to the

SVD routine (CGESVD) to do its own preemptive scaling if potential over-

or underflows are detected. To avoid repeated scanning of the array A,

an optimal implementation would do all necessary scaling before calling

CGESVD and the scaling in CGESVD can be switched off.

3. Other comments related to code optimization are given in comments in the

code, enlosed in [[double brackets]].```

```

drmac@math.hr. Thank you.```

References

```
 Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for

Computing the SVD with High Accuracy. ACM Trans. Math. Softw.

44(1): 11:1-11:30 (2017)

SIGMA library, xGESVDQ section updated February 2016.

Developed and coded by Zlatko Drmac, Department of Mathematics

University of Zagreb, Croatia, drmac@math.hr```

Contributors:

```
Developed and coded by Zlatko Drmac, Department of Mathematics

University of Zagreb, Croatia, drmac@math.hr```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

November 2018

## 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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