/* Subroutine */
int dgerqf_(integer *m, integer *n, doublereal *a, integer * lda, doublereal *tau, doublereal *work, integer *lwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
/* Local variables */
integer i__, k, ib, nb, ki, kk, mu, nu, nx, iws, nbmin, iinfo;
extern /* Subroutine */
int dgerq2_(integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), dlarfb_(char *, char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *), dlarft_(char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *);
integer ldwork, lwkopt;
logical lquery;
/* -- LAPACK computational routine (version 3.4.0) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2011 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
if (*m < 0)
{
*info = -1;
}
else if (*n < 0)
{
*info = -2;
}
else if (*lda < max(1,*m))
{
*info = -4;
}
if (*info == 0)
{
k = min(*m,*n);
if (k == 0)
{
lwkopt = 1;
}
else
{
nb = ilaenv_(&c__1, "DGERQF", " ", m, n, &c_n1, &c_n1);
lwkopt = *m * nb;
}
work[1] = (doublereal) lwkopt;
if (*lwork < max(1,*m) && ! lquery)
{
*info = -7;
}
}
if (*info != 0)
{
i__1 = -(*info);
xerbla_("DGERQF", &i__1);
return 0;
}
else if (lquery)
{
return 0;
}
/* Quick return if possible */
if (k == 0)
{
return 0;
}
nbmin = 2;
nx = 1;
iws = *m;
if (nb > 1 && nb < k)
{
/* Determine when to cross over from blocked to unblocked code. */
/* Computing MAX */
i__1 = 0;
i__2 = ilaenv_(&c__3, "DGERQF", " ", m, n, &c_n1, &c_n1); // , expr subst
nx = max(i__1,i__2);
if (nx < k)
{
/* Determine if workspace is large enough for blocked code. */
ldwork = *m;
iws = ldwork * nb;
if (*lwork < iws)
{
/* Not enough workspace to use optimal NB: reduce NB and */
/* determine the minimum value of NB. */
nb = *lwork / ldwork;
/* Computing MAX */
i__1 = 2;
i__2 = ilaenv_(&c__2, "DGERQF", " ", m, n, &c_n1, & c_n1); // , expr subst
nbmin = max(i__1,i__2);
}
}
}
if (nb >= nbmin && nb < k && nx < k)
{
/* Use blocked code initially. */
/* The last kk rows are handled by the block method. */
ki = (k - nx - 1) / nb * nb;
/* Computing MIN */
i__1 = k;
i__2 = ki + nb; // , expr subst
kk = min(i__1,i__2);
i__1 = k - kk + 1;
i__2 = -nb;
for (i__ = k - kk + ki + 1;
i__2 < 0 ? i__ >= i__1 : i__ <= i__1;
i__ += i__2)
{
/* Computing MIN */
i__3 = k - i__ + 1;
ib = min(i__3,nb);
/* Compute the RQ factorization of the current block */
/* A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1) */
i__3 = *n - k + i__ + ib - 1;
dgerq2_(&ib, &i__3, &a[*m - k + i__ + a_dim1], lda, &tau[i__], & work[1], &iinfo);
if (*m - k + i__ > 1)
{
/* Form the triangular factor of the block reflector */
/* H = H(i+ib-1) . . . H(i+1) H(i) */
i__3 = *n - k + i__ + ib - 1;
dlarft_("Backward", "Rowwise", &i__3, &ib, &a[*m - k + i__ + a_dim1], lda, &tau[i__], &work[1], &ldwork);
/* Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right */
i__3 = *m - k + i__ - 1;
i__4 = *n - k + i__ + ib - 1;
dlarfb_("Right", "No transpose", "Backward", "Rowwise", &i__3, &i__4, &ib, &a[*m - k + i__ + a_dim1], lda, &work[1], &ldwork, &a[a_offset], lda, &work[ib + 1], &ldwork);
}
/* L10: */
}
mu = *m - k + i__ + nb - 1;
nu = *n - k + i__ + nb - 1;
}
else
{
mu = *m;
nu = *n;
}
/* Use unblocked code to factor the last or only block */
if (mu > 0 && nu > 0)
{
dgerq2_(&mu, &nu, &a[a_offset], lda, &tau[1], &work[1], &iinfo);
}
work[1] = (doublereal) iws;
return 0;
/* End of DGERQF */
}
/* Subroutine */ int dgerqf_(integer *m, integer *n, doublereal *a, integer *
lda, doublereal *tau, doublereal *work, integer *lwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
DGERQF computes an RQ factorization of a real M-by-N matrix A:
A = R * Q.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see Further Details).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU (output) DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is
the optimal blocksize.
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 (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details
===============
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i).
=====================================================================
Test the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static integer c__3 = 3;
static integer c__2 = 2;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
/* Local variables */
static integer i__, k, nbmin, iinfo;
extern /* Subroutine */ int dgerq2_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
static integer ib, nb, ki, kk;
extern /* Subroutine */ int dlarfb_(char *, char *, char *, char *,
integer *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
integer *);
static integer mu, nu, nx;
extern /* Subroutine */ int dlarft_(char *, char *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ldwork, lwkopt;
static logical lquery;
static integer iws;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
nb = ilaenv_(&c__1, "DGERQF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (ftnlen)
1);
lwkopt = *m * nb;
work[1] = (doublereal) lwkopt;
lquery = *lwork == -1;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
} else if (*lwork < max(1,*m) && ! lquery) {
*info = -7;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGERQF", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
k = min(*m,*n);
if (k == 0) {
work[1] = 1.;
return 0;
}
nbmin = 2;
nx = 1;
iws = *m;
if (nb > 1 && nb < k) {
/* Determine when to cross over from blocked to unblocked code.
Computing MAX */
i__1 = 0, i__2 = ilaenv_(&c__3, "DGERQF", " ", m, n, &c_n1, &c_n1, (
ftnlen)6, (ftnlen)1);
nx = max(i__1,i__2);
if (nx < k) {
/* Determine if workspace is large enough for blocked code. */
ldwork = *m;
iws = ldwork * nb;
if (*lwork < iws) {
/* Not enough workspace to use optimal NB: reduce NB and
determine the minimum value of NB. */
nb = *lwork / ldwork;
/* Computing MAX */
i__1 = 2, i__2 = ilaenv_(&c__2, "DGERQF", " ", m, n, &c_n1, &
c_n1, (ftnlen)6, (ftnlen)1);
nbmin = max(i__1,i__2);
}
}
}
if (nb >= nbmin && nb < k && nx < k) {
/* Use blocked code initially.
The last kk rows are handled by the block method. */
ki = (k - nx - 1) / nb * nb;
/* Computing MIN */
i__1 = k, i__2 = ki + nb;
kk = min(i__1,i__2);
i__1 = k - kk + 1;
i__2 = -nb;
for (i__ = k - kk + ki + 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__
+= i__2) {
/* Computing MIN */
i__3 = k - i__ + 1;
ib = min(i__3,nb);
/* Compute the RQ factorization of the current block
A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1) */
i__3 = *n - k + i__ + ib - 1;
dgerq2_(&ib, &i__3, &a_ref(*m - k + i__, 1), lda, &tau[i__], &
work[1], &iinfo);
if (*m - k + i__ > 1) {
/* Form the triangular factor of the block reflector
H = H(i+ib-1) . . . H(i+1) H(i) */
i__3 = *n - k + i__ + ib - 1;
dlarft_("Backward", "Rowwise", &i__3, &ib, &a_ref(*m - k +
i__, 1), lda, &tau[i__], &work[1], &ldwork);
/* Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right */
i__3 = *m - k + i__ - 1;
i__4 = *n - k + i__ + ib - 1;
dlarfb_("Right", "No transpose", "Backward", "Rowwise", &i__3,
&i__4, &ib, &a_ref(*m - k + i__, 1), lda, &work[1], &
ldwork, &a[a_offset], lda, &work[ib + 1], &ldwork);
}
/* L10: */
}
mu = *m - k + i__ + nb - 1;
nu = *n - k + i__ + nb - 1;
} else {
mu = *m;
nu = *n;
}
/* Use unblocked code to factor the last or only block */
if (mu > 0 && nu > 0) {
dgerq2_(&mu, &nu, &a[a_offset], lda, &tau[1], &work[1], &iinfo);
}
work[1] = (doublereal) iws;
return 0;
/* End of DGERQF */
} /* dgerqf_ */
int dgerqf_(int *m, int *n, double *a, int *
lda, double *tau, double *work, int *lwork, int *info)
{
/* System generated locals */
int a_dim1, a_offset, i__1, i__2, i__3, i__4;
/* Local variables */
int i__, k, ib, nb, ki, kk, mu, nu, nx, iws, nbmin, iinfo;
extern int dgerq2_(int *, int *, double *,
int *, double *, double *, int *), dlarfb_(char *,
char *, char *, char *, int *, int *, int *,
double *, int *, double *, int *, double *,
int *, double *, int *), dlarft_(char *, char *, int *, int *, double
*, int *, double *, double *, int *), xerbla_(char *, int *);
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
int ldwork, lwkopt;
int lquery;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DGERQF computes an RQ factorization of a float M-by-N matrix A: */
/* A = R * Q. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0. */
/* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, */
/* if m <= n, the upper triangle of the subarray */
/* A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R; */
/* if m >= n, the elements on and above the (m-n)-th subdiagonal */
/* contain the M-by-N upper trapezoidal matrix R; */
/* the remaining elements, with the array TAU, represent the */
/* orthogonal matrix Q as a product of MIN(m,n) elementary */
/* reflectors (see Further Details). */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,M). */
/* TAU (output) DOUBLE PRECISION array, dimension (MIN(M,N)) */
/* The scalar factors of the elementary reflectors (see Further */
/* Details). */
/* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= MAX(1,M). */
/* For optimum performance LWORK >= M*NB, where NB is */
/* the optimal blocksize. */
/* 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 (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* The matrix Q is represented as a product of elementary reflectors */
/* Q = H(1) H(2) . . . H(k), where k = MIN(m,n). */
/* Each H(i) has the form */
/* H(i) = I - tau * v * v' */
/* where tau is a float scalar, and v is a float vector with */
/* v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in */
/* A(m-k+i,1:n-k+i-1), and tau in TAU(i). */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < MAX(1,*m)) {
*info = -4;
}
if (*info == 0) {
k = MIN(*m,*n);
if (k == 0) {
lwkopt = 1;
} else {
nb = ilaenv_(&c__1, "DGERQF", " ", m, n, &c_n1, &c_n1);
lwkopt = *m * nb;
}
work[1] = (double) lwkopt;
if (*lwork < MAX(1,*m) && ! lquery) {
*info = -7;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGERQF", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (k == 0) {
return 0;
}
nbmin = 2;
nx = 1;
iws = *m;
if (nb > 1 && nb < k) {
/* Determine when to cross over from blocked to unblocked code. */
/* Computing MAX */
i__1 = 0, i__2 = ilaenv_(&c__3, "DGERQF", " ", m, n, &c_n1, &c_n1);
nx = MAX(i__1,i__2);
if (nx < k) {
/* Determine if workspace is large enough for blocked code. */
ldwork = *m;
iws = ldwork * nb;
if (*lwork < iws) {
/* Not enough workspace to use optimal NB: reduce NB and */
/* determine the minimum value of NB. */
nb = *lwork / ldwork;
/* Computing MAX */
i__1 = 2, i__2 = ilaenv_(&c__2, "DGERQF", " ", m, n, &c_n1, &
c_n1);
nbmin = MAX(i__1,i__2);
}
}
}
if (nb >= nbmin && nb < k && nx < k) {
/* Use blocked code initially. */
/* The last kk rows are handled by the block method. */
ki = (k - nx - 1) / nb * nb;
/* Computing MIN */
i__1 = k, i__2 = ki + nb;
kk = MIN(i__1,i__2);
i__1 = k - kk + 1;
i__2 = -nb;
for (i__ = k - kk + ki + 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__
+= i__2) {
/* Computing MIN */
i__3 = k - i__ + 1;
ib = MIN(i__3,nb);
/* Compute the RQ factorization of the current block */
/* A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1) */
i__3 = *n - k + i__ + ib - 1;
dgerq2_(&ib, &i__3, &a[*m - k + i__ + a_dim1], lda, &tau[i__], &
work[1], &iinfo);
if (*m - k + i__ > 1) {
/* Form the triangular factor of the block reflector */
/* H = H(i+ib-1) . . . H(i+1) H(i) */
i__3 = *n - k + i__ + ib - 1;
dlarft_("Backward", "Rowwise", &i__3, &ib, &a[*m - k + i__ +
a_dim1], lda, &tau[i__], &work[1], &ldwork);
/* Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right */
i__3 = *m - k + i__ - 1;
i__4 = *n - k + i__ + ib - 1;
dlarfb_("Right", "No transpose", "Backward", "Rowwise", &i__3,
&i__4, &ib, &a[*m - k + i__ + a_dim1], lda, &work[1],
&ldwork, &a[a_offset], lda, &work[ib + 1], &ldwork);
}
/* L10: */
}
mu = *m - k + i__ + nb - 1;
nu = *n - k + i__ + nb - 1;
} else {
mu = *m;
nu = *n;
}
/* Use unblocked code to factor the last or only block */
if (mu > 0 && nu > 0) {
dgerq2_(&mu, &nu, &a[a_offset], lda, &tau[1], &work[1], &iinfo);
}
work[1] = (double) iws;
return 0;
/* End of DGERQF */
} /* dgerqf_ */
/* Subroutine */ int dggsvp_(char *jobu, char *jobv, char *jobq, integer *m,
integer *p, integer *n, doublereal *a, integer *lda, doublereal *b,
integer *ldb, doublereal *tola, doublereal *tolb, integer *k, integer
*l, doublereal *u, integer *ldu, doublereal *v, integer *ldv,
doublereal *q, integer *ldq, integer *iwork, doublereal *tau,
doublereal *work, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
DGGSVP computes orthogonal matrices U, V and Q such that
N-K-L K L
U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V'*B*Q = L ( 0 0 B13 )
P-L ( 0 0 0 )
where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
otherwise A23 is (M-K)-by-L upper trapezoidal. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the
transpose of Z.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
DGGSVD.
Arguments
=========
JOBU (input) CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
JOBV (input) CHARACTER*1
= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.
JOBQ (input) CHARACTER*1
= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix described in
the Purpose section.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
TOLA (input) DOUBLE PRECISION
TOLB (input) DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
numerical rank of matrix B and a subblock of A. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MAZHEPS,
TOLB = MAX(P,N)*norm(B)*MAZHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
K (output) INTEGER
L (output) INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose.
K + L = effective numerical rank of (A',B')'.
U (output) DOUBLE PRECISION array, dimension (LDU,M)
If JOBU = 'U', U contains the orthogonal matrix U.
If JOBU = 'N', U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V (output) DOUBLE PRECISION array, dimension (LDV,M)
If JOBV = 'V', V contains the orthogonal matrix V.
If JOBV = 'N', V is not referenced.
LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
IWORK (workspace) INTEGER array, dimension (N)
TAU (workspace) DOUBLE PRECISION array, dimension (N)
WORK (workspace) DOUBLE PRECISION array, dimension (max(3*N,M,P))
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
The subroutine uses LAPACK subroutine DGEQPF for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static doublereal c_b12 = 0.;
static doublereal c_b22 = 1.;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1,
u_offset, v_dim1, v_offset, i__1, i__2, i__3;
doublereal d__1;
/* Local variables */
static integer i__, j;
extern logical lsame_(char *, char *);
static logical wantq, wantu, wantv;
extern /* Subroutine */ int dgeqr2_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *), dgerq2_(
integer *, integer *, doublereal *, integer *, doublereal *,
doublereal *, integer *), dorg2r_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *),
dorm2r_(char *, char *, integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *), dormr2_(char *, char *,
integer *, integer *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *), dgeqpf_(integer *, integer *, doublereal *,
integer *, integer *, doublereal *, doublereal *, integer *),
dlacpy_(char *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *), dlaset_(char *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *), dlapmt_(logical *,
integer *, integer *, doublereal *, integer *, integer *);
static logical forwrd;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define v_ref(a_1,a_2) v[(a_2)*v_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--iwork;
--tau;
--work;
/* Function Body */
wantu = lsame_(jobu, "U");
wantv = lsame_(jobv, "V");
wantq = lsame_(jobq, "Q");
forwrd = TRUE_;
*info = 0;
if (! (wantu || lsame_(jobu, "N"))) {
*info = -1;
} else if (! (wantv || lsame_(jobv, "N"))) {
*info = -2;
} else if (! (wantq || lsame_(jobq, "N"))) {
*info = -3;
} else if (*m < 0) {
*info = -4;
} else if (*p < 0) {
*info = -5;
} else if (*n < 0) {
*info = -6;
} else if (*lda < max(1,*m)) {
*info = -8;
} else if (*ldb < max(1,*p)) {
*info = -10;
} else if (*ldu < 1 || wantu && *ldu < *m) {
*info = -16;
} else if (*ldv < 1 || wantv && *ldv < *p) {
*info = -18;
} else if (*ldq < 1 || wantq && *ldq < *n) {
*info = -20;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGGSVP", &i__1);
return 0;
}
/* QR with column pivoting of B: B*P = V*( S11 S12 )
( 0 0 ) */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
iwork[i__] = 0;
/* L10: */
}
dgeqpf_(p, n, &b[b_offset], ldb, &iwork[1], &tau[1], &work[1], info);
/* Update A := A*P */
dlapmt_(&forwrd, m, n, &a[a_offset], lda, &iwork[1]);
/* Determine the effective rank of matrix B. */
*l = 0;
i__1 = min(*p,*n);
for (i__ = 1; i__ <= i__1; ++i__) {
if ((d__1 = b_ref(i__, i__), abs(d__1)) > *tolb) {
++(*l);
}
/* L20: */
}
if (wantv) {
/* Copy the details of V, and form V. */
dlaset_("Full", p, p, &c_b12, &c_b12, &v[v_offset], ldv);
if (*p > 1) {
i__1 = *p - 1;
dlacpy_("Lower", &i__1, n, &b_ref(2, 1), ldb, &v_ref(2, 1), ldv);
}
i__1 = min(*p,*n);
dorg2r_(p, p, &i__1, &v[v_offset], ldv, &tau[1], &work[1], info);
}
/* Clean up B */
i__1 = *l - 1;
for (j = 1; j <= i__1; ++j) {
i__2 = *l;
for (i__ = j + 1; i__ <= i__2; ++i__) {
b_ref(i__, j) = 0.;
/* L30: */
}
/* L40: */
}
if (*p > *l) {
i__1 = *p - *l;
dlaset_("Full", &i__1, n, &c_b12, &c_b12, &b_ref(*l + 1, 1), ldb);
}
if (wantq) {
/* Set Q = I and Update Q := Q*P */
dlaset_("Full", n, n, &c_b12, &c_b22, &q[q_offset], ldq);
dlapmt_(&forwrd, n, n, &q[q_offset], ldq, &iwork[1]);
}
if (*p >= *l && *n != *l) {
/* RQ factorization of (S11 S12): ( S11 S12 ) = ( 0 S12 )*Z */
dgerq2_(l, n, &b[b_offset], ldb, &tau[1], &work[1], info);
/* Update A := A*Z' */
dormr2_("Right", "Transpose", m, n, l, &b[b_offset], ldb, &tau[1], &a[
a_offset], lda, &work[1], info);
if (wantq) {
/* Update Q := Q*Z' */
dormr2_("Right", "Transpose", n, n, l, &b[b_offset], ldb, &tau[1],
&q[q_offset], ldq, &work[1], info);
}
/* Clean up B */
i__1 = *n - *l;
dlaset_("Full", l, &i__1, &c_b12, &c_b12, &b[b_offset], ldb);
i__1 = *n;
for (j = *n - *l + 1; j <= i__1; ++j) {
i__2 = *l;
for (i__ = j - *n + *l + 1; i__ <= i__2; ++i__) {
b_ref(i__, j) = 0.;
/* L50: */
}
/* L60: */
}
}
/* Let N-L L
A = ( A11 A12 ) M,
then the following does the complete QR decomposition of A11:
A11 = U*( 0 T12 )*P1'
( 0 0 ) */
i__1 = *n - *l;
for (i__ = 1; i__ <= i__1; ++i__) {
iwork[i__] = 0;
/* L70: */
}
i__1 = *n - *l;
dgeqpf_(m, &i__1, &a[a_offset], lda, &iwork[1], &tau[1], &work[1], info);
/* Determine the effective rank of A11 */
*k = 0;
/* Computing MIN */
i__2 = *m, i__3 = *n - *l;
i__1 = min(i__2,i__3);
for (i__ = 1; i__ <= i__1; ++i__) {
if ((d__1 = a_ref(i__, i__), abs(d__1)) > *tola) {
++(*k);
}
/* L80: */
}
/* Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N )
Computing MIN */
i__2 = *m, i__3 = *n - *l;
i__1 = min(i__2,i__3);
dorm2r_("Left", "Transpose", m, l, &i__1, &a[a_offset], lda, &tau[1], &
a_ref(1, *n - *l + 1), lda, &work[1], info);
if (wantu) {
/* Copy the details of U, and form U */
dlaset_("Full", m, m, &c_b12, &c_b12, &u[u_offset], ldu);
if (*m > 1) {
i__1 = *m - 1;
i__2 = *n - *l;
dlacpy_("Lower", &i__1, &i__2, &a_ref(2, 1), lda, &u_ref(2, 1),
ldu);
}
/* Computing MIN */
i__2 = *m, i__3 = *n - *l;
i__1 = min(i__2,i__3);
dorg2r_(m, m, &i__1, &u[u_offset], ldu, &tau[1], &work[1], info);
}
if (wantq) {
/* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 */
i__1 = *n - *l;
dlapmt_(&forwrd, n, &i__1, &q[q_offset], ldq, &iwork[1]);
}
/* Clean up A: set the strictly lower triangular part of
A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. */
i__1 = *k - 1;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
a_ref(i__, j) = 0.;
/* L90: */
}
/* L100: */
}
if (*m > *k) {
i__1 = *m - *k;
i__2 = *n - *l;
dlaset_("Full", &i__1, &i__2, &c_b12, &c_b12, &a_ref(*k + 1, 1), lda);
}
if (*n - *l > *k) {
/* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 */
i__1 = *n - *l;
dgerq2_(k, &i__1, &a[a_offset], lda, &tau[1], &work[1], info);
if (wantq) {
/* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' */
i__1 = *n - *l;
dormr2_("Right", "Transpose", n, &i__1, k, &a[a_offset], lda, &
tau[1], &q[q_offset], ldq, &work[1], info);
}
/* Clean up A */
i__1 = *n - *l - *k;
dlaset_("Full", k, &i__1, &c_b12, &c_b12, &a[a_offset], lda);
i__1 = *n - *l;
for (j = *n - *l - *k + 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = j - *n + *l + *k + 1; i__ <= i__2; ++i__) {
a_ref(i__, j) = 0.;
/* L110: */
}
/* L120: */
}
}
if (*m > *k) {
/* QR factorization of A( K+1:M,N-L+1:N ) */
i__1 = *m - *k;
dgeqr2_(&i__1, l, &a_ref(*k + 1, *n - *l + 1), lda, &tau[1], &work[1],
info);
if (wantu) {
/* Update U(:,K+1:M) := U(:,K+1:M)*U1 */
i__1 = *m - *k;
/* Computing MIN */
i__3 = *m - *k;
i__2 = min(i__3,*l);
dorm2r_("Right", "No transpose", m, &i__1, &i__2, &a_ref(*k + 1, *
n - *l + 1), lda, &tau[1], &u_ref(1, *k + 1), ldu, &work[
1], info);
}
/* Clean up */
i__1 = *n;
for (j = *n - *l + 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = j - *n + *k + *l + 1; i__ <= i__2; ++i__) {
a_ref(i__, j) = 0.;
/* L130: */
}
/* L140: */
}
}
return 0;
/* End of DGGSVP */
} /* dggsvp_ */