doublereal crzt01_(integer *m, integer *n, complex *a, complex *af, integer *
lda, complex *tau, complex *work, integer *lwork)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2, i__3, i__4;
real ret_val;
/* Local variables */
integer i__, j, info;
real norma;
extern /* Subroutine */ int caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *);
real rwork[1];
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *), slamch_(char *);
extern /* Subroutine */ int claset_(char *, integer *, integer *, complex
*, complex *, complex *, integer *), xerbla_(char *,
integer *), cunmrz_(char *, char *, integer *, integer *,
integer *, integer *, complex *, integer *, complex *, complex *,
integer *, complex *, integer *, integer *);
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CRZT01 returns */
/* || A - R*Q || / ( M * eps * ||A|| ) */
/* for an upper trapezoidal A that was factored with CTZRZF. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrices A and AF. */
/* N (input) INTEGER */
/* The number of columns of the matrices A and AF. */
/* A (input) COMPLEX array, dimension (LDA,N) */
/* The original upper trapezoidal M by N matrix A. */
/* AF (input) COMPLEX array, dimension (LDA,N) */
/* The output of CTZRZF for input matrix A. */
/* The lower triangle is not referenced. */
/* LDA (input) INTEGER */
/* The leading dimension of the arrays A and AF. */
/* TAU (input) COMPLEX array, dimension (M) */
/* Details of the Householder transformations as returned by */
/* CTZRZF. */
/* WORK (workspace) COMPLEX array, dimension (LWORK) */
/* LWORK (input) INTEGER */
/* The length of the array WORK. LWORK >= m*n + m. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
af_dim1 = *lda;
af_offset = 1 + af_dim1;
af -= af_offset;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
--work;
/* Function Body */
ret_val = 0.f;
if (*lwork < *m * *n + *m) {
xerbla_("CRZT01", &c__8);
return ret_val;
}
/* Quick return if possible */
if (*m <= 0 || *n <= 0) {
return ret_val;
}
norma = clange_("One-norm", m, n, &a[a_offset], lda, rwork);
/* Copy upper triangle R */
claset_("Full", m, n, &c_b6, &c_b6, &work[1], m);
i__1 = *m;
for (j = 1; j <= i__1; ++j) {
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = (j - 1) * *m + i__;
i__4 = i__ + j * af_dim1;
work[i__3].r = af[i__4].r, work[i__3].i = af[i__4].i;
/* L10: */
}
/* L20: */
}
/* R = R * P(1) * ... *P(m) */
i__1 = *n - *m;
i__2 = *lwork - *m * *n;
cunmrz_("Right", "No tranpose", m, n, m, &i__1, &af[af_offset], lda, &tau[
1], &work[1], m, &work[*m * *n + 1], &i__2, &info);
/* R = R - A */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
caxpy_(m, &c_b15, &a[i__ * a_dim1 + 1], &c__1, &work[(i__ - 1) * *m +
1], &c__1);
/* L30: */
}
ret_val = clange_("One-norm", m, n, &work[1], m, rwork);
ret_val /= slamch_("Epsilon") * (real) max(*m,*n);
if (norma != 0.f) {
ret_val /= norma;
}
return ret_val;
/* End of CRZT01 */
} /* crzt01_ */
doublereal crzt02_(integer *m, integer *n, complex *af, integer *lda, complex
*tau, complex *work, integer *lwork)
{
/* System generated locals */
integer af_dim1, af_offset, i__1, i__2, i__3;
real ret_val;
complex q__1;
/* Local variables */
integer i__, info;
real rwork[1];
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *), slamch_(char *);
extern /* Subroutine */ int claset_(char *, integer *, integer *, complex
*, complex *, complex *, integer *), xerbla_(char *,
integer *), cunmrz_(char *, char *, integer *, integer *,
integer *, integer *, complex *, integer *, complex *, complex *,
integer *, complex *, integer *, integer *);
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CRZT02 returns */
/* || I - Q'*Q || / ( M * eps) */
/* where the matrix Q is defined by the Householder transformations */
/* generated by CTZRZF. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix AF. */
/* N (input) INTEGER */
/* The number of columns of the matrix AF. */
/* AF (input) COMPLEX array, dimension (LDA,N) */
/* The output of CTZRZF. */
/* LDA (input) INTEGER */
/* The leading dimension of the array AF. */
/* TAU (input) COMPLEX array, dimension (M) */
/* Details of the Householder transformations as returned by */
/* CTZRZF. */
/* WORK (workspace) COMPLEX array, dimension (LWORK) */
/* LWORK (input) INTEGER */
/* Length of WORK array. LWORK >= N*N+N. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
af_dim1 = *lda;
af_offset = 1 + af_dim1;
af -= af_offset;
--tau;
--work;
/* Function Body */
ret_val = 0.f;
if (*lwork < *n * *n + *n) {
xerbla_("CRZT02", &c__7);
return ret_val;
}
/* Quick return if possible */
if (*m <= 0 || *n <= 0) {
return ret_val;
}
/* Q := I */
claset_("Full", n, n, &c_b5, &c_b6, &work[1], n);
/* Q := P(1) * ... * P(m) * Q */
i__1 = *n - *m;
i__2 = *lwork - *n * *n;
cunmrz_("Left", "No transpose", n, n, m, &i__1, &af[af_offset], lda, &tau[
1], &work[1], n, &work[*n * *n + 1], &i__2, &info);
/* Q := P(m)' * ... * P(1)' * Q */
i__1 = *n - *m;
i__2 = *lwork - *n * *n;
cunmrz_("Left", "Conjugate transpose", n, n, m, &i__1, &af[af_offset],
lda, &tau[1], &work[1], n, &work[*n * *n + 1], &i__2, &info);
/* Q := Q - I */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = (i__ - 1) * *n + i__;
i__3 = (i__ - 1) * *n + i__;
q__1.r = work[i__3].r - 1.f, q__1.i = work[i__3].i;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
/* L10: */
}
ret_val = clange_("One-norm", n, n, &work[1], n, rwork) / (
slamch_("Epsilon") * (real) max(*m,*n));
return ret_val;
/* End of CRZT02 */
} /* crzt02_ */
/* Subroutine */ int cgelsy_(integer *m, integer *n, integer *nrhs, complex *
a, integer *lda, complex *b, integer *ldb, integer *jpvt, real *rcond,
integer *rank, complex *work, integer *lwork, real *rwork, integer *
info)
{
/* -- LAPACK driver 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
=======
CGELSY computes the minimum-norm solution to a complex linear least
squares problem:
minimize || A * X - B ||
using a complete orthogonal factorization of A. A is an M-by-N
matrix which may be rank-deficient.
Several right hand side vectors b and solution vectors x can be
handled in a single call; they are stored as the columns of the
M-by-NRHS right hand side matrix B and the N-by-NRHS solution
matrix X.
The routine first computes a QR factorization with column pivoting:
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/RCOND. The order of R11, RANK,
is the effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated
by unitary transformations from the right, arriving at the
complete orthogonal factorization:
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first RANK columns of Q.
This routine is basically identical to the original xGELSX except
three differences:
o The permutation of matrix B (the right hand side) is faster and
more simple.
o The call to the subroutine xGEQPF has been substituted by the
the call to the subroutine xGEQP3. This subroutine is a Blas-3
version of the QR factorization with column pivoting.
o Matrix B (the right hand side) is updated with Blas-3.
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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of
columns of matrices B and X. NRHS >= 0.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A has been overwritten by details of its
complete orthogonal factorization.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) COMPLEX array, dimension (LDB,NRHS)
On entry, the M-by-NRHS right hand side matrix B.
On exit, the N-by-NRHS solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M,N).
JPVT (input/output) INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
to the front of AP, otherwise column i is a free column.
On exit, if JPVT(i) = k, then the i-th column of A*P
was the k-th column of A.
RCOND (input) REAL
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number < 1/RCOND.
RANK (output) INTEGER
The effective rank of A, i.e., the order of the submatrix
R11. This is the same as the order of the submatrix T11
in the complete orthogonal factorization of A.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK.
The unblocked strategy requires that:
LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
where MN = min(M,N).
The block algorithm requires that:
LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
where NB is an upper bound on the blocksize returned
by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR,
and CUNMRZ.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK (workspace) REAL array, dimension (2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details
===============
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
=====================================================================
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {0.f,0.f};
static complex c_b2 = {1.f,0.f};
static integer c__1 = 1;
static integer c_n1 = -1;
static integer c__0 = 0;
static integer c__2 = 2;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4;
real r__1, r__2;
complex q__1;
/* Builtin functions */
double c_abs(complex *);
/* Local variables */
static real anrm, bnrm, smin, smax;
static integer i__, j, iascl, ibscl;
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
static integer ismin, ismax;
static complex c1, c2;
extern /* Subroutine */ int ctrsm_(char *, char *, char *, char *,
integer *, integer *, complex *, complex *, integer *, complex *,
integer *), claic1_(integer *,
integer *, complex *, real *, complex *, complex *, real *,
complex *, complex *);
static real wsize;
static complex s1, s2;
extern /* Subroutine */ int cgeqp3_(integer *, integer *, complex *,
integer *, integer *, complex *, complex *, integer *, real *,
integer *);
static integer nb;
extern /* Subroutine */ int slabad_(real *, real *);
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
static integer mn;
extern /* Subroutine */ int clascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, complex *, integer *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int claset_(char *, integer *, integer *, complex
*, complex *, complex *, integer *), xerbla_(char *,
integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static real bignum;
static integer nb1, nb2, nb3, nb4;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
static real sminpr, smaxpr, smlnum;
extern /* Subroutine */ int cunmrz_(char *, char *, integer *, integer *,
integer *, integer *, complex *, integer *, complex *, complex *,
integer *, complex *, integer *, integer *);
static integer lwkopt;
static logical lquery;
extern /* Subroutine */ int ctzrzf_(integer *, integer *, complex *,
integer *, complex *, complex *, integer *, integer *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--jpvt;
--work;
--rwork;
/* Function Body */
mn = min(*m,*n);
ismin = mn + 1;
ismax = (mn << 1) + 1;
/* Test the input arguments. */
*info = 0;
nb1 = ilaenv_(&c__1, "CGEQRF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
nb2 = ilaenv_(&c__1, "CGERQF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
nb3 = ilaenv_(&c__1, "CUNMQR", " ", m, n, nrhs, &c_n1, (ftnlen)6, (ftnlen)
1);
nb4 = ilaenv_(&c__1, "CUNMRQ", " ", m, n, nrhs, &c_n1, (ftnlen)6, (ftnlen)
1);
/* Computing MAX */
i__1 = max(nb1,nb2), i__1 = max(i__1,nb3);
nb = max(i__1,nb4);
/* Computing MAX */
i__1 = 1, i__2 = mn + (*n << 1) + nb * (*n + 1), i__1 = max(i__1,i__2),
i__2 = (mn << 1) + nb * *nrhs;
lwkopt = max(i__1,i__2);
q__1.r = (real) lwkopt, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
lquery = *lwork == -1;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*m)) {
*info = -5;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = max(1,*m);
if (*ldb < max(i__1,*n)) {
*info = -7;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = mn << 1, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = mn +
*nrhs;
if (*lwork < mn + max(i__1,i__2) && ! lquery) {
*info = -12;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGELSY", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible
Computing MIN */
i__1 = min(*m,*n);
if (min(i__1,*nrhs) == 0) {
*rank = 0;
return 0;
}
/* Get machine parameters */
smlnum = slamch_("S") / slamch_("P");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
/* Scale A, B if max entries outside range [SMLNUM,BIGNUM] */
anrm = clange_("M", m, n, &a[a_offset], lda, &rwork[1]);
iascl = 0;
if (anrm > 0.f && anrm < smlnum) {
/* Scale matrix norm up to SMLNUM */
clascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda,
info);
iascl = 1;
} else if (anrm > bignum) {
/* Scale matrix norm down to BIGNUM */
clascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda,
info);
iascl = 2;
} else if (anrm == 0.f) {
/* Matrix all zero. Return zero solution. */
i__1 = max(*m,*n);
claset_("F", &i__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
*rank = 0;
goto L70;
}
bnrm = clange_("M", m, nrhs, &b[b_offset], ldb, &rwork[1]);
ibscl = 0;
if (bnrm > 0.f && bnrm < smlnum) {
/* Scale matrix norm up to SMLNUM */
clascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, nrhs, &b[b_offset], ldb,
info);
ibscl = 1;
} else if (bnrm > bignum) {
/* Scale matrix norm down to BIGNUM */
clascl_("G", &c__0, &c__0, &bnrm, &bignum, m, nrhs, &b[b_offset], ldb,
info);
ibscl = 2;
}
/* Compute QR factorization with column pivoting of A:
A * P = Q * R */
i__1 = *lwork - mn;
cgeqp3_(m, n, &a[a_offset], lda, &jpvt[1], &work[1], &work[mn + 1], &i__1,
&rwork[1], info);
i__1 = mn + 1;
wsize = mn + work[i__1].r;
/* complex workspace: MN+NB*(N+1). real workspace 2*N.
Details of Householder rotations stored in WORK(1:MN).
Determine RANK using incremental condition estimation */
i__1 = ismin;
work[i__1].r = 1.f, work[i__1].i = 0.f;
i__1 = ismax;
work[i__1].r = 1.f, work[i__1].i = 0.f;
smax = c_abs(&a_ref(1, 1));
smin = smax;
if (c_abs(&a_ref(1, 1)) == 0.f) {
*rank = 0;
i__1 = max(*m,*n);
claset_("F", &i__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
goto L70;
} else {
*rank = 1;
}
L10:
if (*rank < mn) {
i__ = *rank + 1;
claic1_(&c__2, rank, &work[ismin], &smin, &a_ref(1, i__), &a_ref(i__,
i__), &sminpr, &s1, &c1);
claic1_(&c__1, rank, &work[ismax], &smax, &a_ref(1, i__), &a_ref(i__,
i__), &smaxpr, &s2, &c2);
if (smaxpr * *rcond <= sminpr) {
i__1 = *rank;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = ismin + i__ - 1;
i__3 = ismin + i__ - 1;
q__1.r = s1.r * work[i__3].r - s1.i * work[i__3].i, q__1.i =
s1.r * work[i__3].i + s1.i * work[i__3].r;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
i__2 = ismax + i__ - 1;
i__3 = ismax + i__ - 1;
q__1.r = s2.r * work[i__3].r - s2.i * work[i__3].i, q__1.i =
s2.r * work[i__3].i + s2.i * work[i__3].r;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
/* L20: */
}
i__1 = ismin + *rank;
work[i__1].r = c1.r, work[i__1].i = c1.i;
i__1 = ismax + *rank;
work[i__1].r = c2.r, work[i__1].i = c2.i;
smin = sminpr;
smax = smaxpr;
++(*rank);
goto L10;
}
}
/* complex workspace: 3*MN.
Logically partition R = [ R11 R12 ]
[ 0 R22 ]
where R11 = R(1:RANK,1:RANK)
[R11,R12] = [ T11, 0 ] * Y */
if (*rank < *n) {
i__1 = *lwork - (mn << 1);
ctzrzf_(rank, n, &a[a_offset], lda, &work[mn + 1], &work[(mn << 1) +
1], &i__1, info);
}
/* complex workspace: 2*MN.
Details of Householder rotations stored in WORK(MN+1:2*MN)
B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) */
i__1 = *lwork - (mn << 1);
cunmqr_("Left", "Conjugate transpose", m, nrhs, &mn, &a[a_offset], lda, &
work[1], &b[b_offset], ldb, &work[(mn << 1) + 1], &i__1, info);
/* Computing MAX */
i__1 = (mn << 1) + 1;
r__1 = wsize, r__2 = (mn << 1) + work[i__1].r;
wsize = dmax(r__1,r__2);
/* complex workspace: 2*MN+NB*NRHS.
B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) */
ctrsm_("Left", "Upper", "No transpose", "Non-unit", rank, nrhs, &c_b2, &a[
a_offset], lda, &b[b_offset], ldb);
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = *rank + 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
b[i__3].r = 0.f, b[i__3].i = 0.f;
/* L30: */
}
/* L40: */
}
/* B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS) */
if (*rank < *n) {
i__1 = *n - *rank;
i__2 = *lwork - (mn << 1);
cunmrz_("Left", "Conjugate transpose", n, nrhs, rank, &i__1, &a[
a_offset], lda, &work[mn + 1], &b[b_offset], ldb, &work[(mn <<
1) + 1], &i__2, info);
}
/* complex workspace: 2*MN+NRHS.
B(1:N,1:NRHS) := P * B(1:N,1:NRHS) */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = jpvt[i__];
i__4 = b_subscr(i__, j);
work[i__3].r = b[i__4].r, work[i__3].i = b[i__4].i;
/* L50: */
}
ccopy_(n, &work[1], &c__1, &b_ref(1, j), &c__1);
/* L60: */
}
/* complex workspace: N.
Undo scaling */
if (iascl == 1) {
clascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb,
info);
clascl_("U", &c__0, &c__0, &smlnum, &anrm, rank, rank, &a[a_offset],
lda, info);
} else if (iascl == 2) {
clascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb,
info);
clascl_("U", &c__0, &c__0, &bignum, &anrm, rank, rank, &a[a_offset],
lda, info);
}
if (ibscl == 1) {
clascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb,
info);
} else if (ibscl == 2) {
clascl_("G", &c__0, &c__0, &bignum, &bnrm, n, nrhs, &b[b_offset], ldb,
info);
}
L70:
q__1.r = (real) lwkopt, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
return 0;
/* End of CGELSY */
} /* cgelsy_ */
int cgelsy_(int *m, int *n, int *nrhs, complex *
a, int *lda, complex *b, int *ldb, int *jpvt, float *rcond,
int *rank, complex *work, int *lwork, float *rwork, int *
info)
{
/* System generated locals */
int a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4;
float r__1, r__2;
complex q__1;
/* Builtin functions */
double c_abs(complex *);
/* Local variables */
int i__, j;
complex c1, c2, s1, s2;
int nb, mn, nb1, nb2, nb3, nb4;
float anrm, bnrm, smin, smax;
int iascl, ibscl;
extern int ccopy_(int *, complex *, int *,
complex *, int *);
int ismin, ismax;
extern int ctrsm_(char *, char *, char *, char *,
int *, int *, complex *, complex *, int *, complex *,
int *), claic1_(int *,
int *, complex *, float *, complex *, complex *, float *,
complex *, complex *);
float wsize;
extern int cgeqp3_(int *, int *, complex *,
int *, int *, complex *, complex *, int *, float *,
int *), slabad_(float *, float *);
extern double clange_(char *, int *, int *, complex *,
int *, float *);
extern int clascl_(char *, int *, int *, float *,
float *, int *, int *, complex *, int *, int *);
extern double slamch_(char *);
extern int claset_(char *, int *, int *, complex
*, complex *, complex *, int *), xerbla_(char *,
int *);
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
float bignum;
extern int cunmqr_(char *, char *, int *, int *,
int *, complex *, int *, complex *, complex *, int *,
complex *, int *, int *);
float sminpr, smaxpr, smlnum;
extern int cunmrz_(char *, char *, int *, int *,
int *, int *, complex *, int *, complex *, complex *,
int *, complex *, int *, int *);
int lwkopt;
int lquery;
extern int ctzrzf_(int *, int *, complex *,
int *, complex *, complex *, int *, int *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CGELSY computes the minimum-norm solution to a complex linear least */
/* squares problem: */
/* minimize || A * X - B || */
/* using a complete orthogonal factorization of A. A is an M-by-N */
/* matrix which may be rank-deficient. */
/* Several right hand side vectors b and solution vectors x can be */
/* handled in a single call; they are stored as the columns of the */
/* M-by-NRHS right hand side matrix B and the N-by-NRHS solution */
/* matrix X. */
/* The routine first computes a QR factorization with column pivoting: */
/* A * P = Q * [ R11 R12 ] */
/* [ 0 R22 ] */
/* with R11 defined as the largest leading submatrix whose estimated */
/* condition number is less than 1/RCOND. The order of R11, RANK, */
/* is the effective rank of A. */
/* Then, R22 is considered to be negligible, and R12 is annihilated */
/* by unitary transformations from the right, arriving at the */
/* complete orthogonal factorization: */
/* A * P = Q * [ T11 0 ] * Z */
/* [ 0 0 ] */
/* The minimum-norm solution is then */
/* X = P * Z' [ inv(T11)*Q1'*B ] */
/* [ 0 ] */
/* where Q1 consists of the first RANK columns of Q. */
/* This routine is basically identical to the original xGELSX except */
/* three differences: */
/* o The permutation of matrix B (the right hand side) is faster and */
/* more simple. */
/* o The call to the subroutine xGEQPF has been substituted by the */
/* the call to the subroutine xGEQP3. This subroutine is a Blas-3 */
/* version of the QR factorization with column pivoting. */
/* o Matrix B (the right hand side) is updated with Blas-3. */
/* 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. */
/* NRHS (input) INTEGER */
/* The number of right hand sides, i.e., the number of */
/* columns of matrices B and X. NRHS >= 0. */
/* A (input/output) COMPLEX array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, A has been overwritten by details of its */
/* complete orthogonal factorization. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,M). */
/* B (input/output) COMPLEX array, dimension (LDB,NRHS) */
/* On entry, the M-by-NRHS right hand side matrix B. */
/* On exit, the N-by-NRHS solution matrix X. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= MAX(1,M,N). */
/* JPVT (input/output) INTEGER array, dimension (N) */
/* On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted */
/* to the front of AP, otherwise column i is a free column. */
/* On exit, if JPVT(i) = k, then the i-th column of A*P */
/* was the k-th column of A. */
/* RCOND (input) REAL */
/* RCOND is used to determine the effective rank of A, which */
/* is defined as the order of the largest leading triangular */
/* submatrix R11 in the QR factorization with pivoting of A, */
/* whose estimated condition number < 1/RCOND. */
/* RANK (output) INTEGER */
/* The effective rank of A, i.e., the order of the submatrix */
/* R11. This is the same as the order of the submatrix T11 */
/* in the complete orthogonal factorization of A. */
/* WORK (workspace/output) COMPLEX 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. */
/* The unblocked strategy requires that: */
/* LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) */
/* where MN = MIN(M,N). */
/* The block algorithm requires that: */
/* LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) */
/* where NB is an upper bound on the blocksize returned */
/* by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR, */
/* and CUNMRZ. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* RWORK (workspace) REAL array, dimension (2*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA */
/* E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */
/* G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--jpvt;
--work;
--rwork;
/* Function Body */
mn = MIN(*m,*n);
ismin = mn + 1;
ismax = (mn << 1) + 1;
/* Test the input arguments. */
*info = 0;
nb1 = ilaenv_(&c__1, "CGEQRF", " ", m, n, &c_n1, &c_n1);
nb2 = ilaenv_(&c__1, "CGERQF", " ", m, n, &c_n1, &c_n1);
nb3 = ilaenv_(&c__1, "CUNMQR", " ", m, n, nrhs, &c_n1);
nb4 = ilaenv_(&c__1, "CUNMRQ", " ", m, n, nrhs, &c_n1);
/* Computing MAX */
i__1 = MAX(nb1,nb2), i__1 = MAX(i__1,nb3);
nb = MAX(i__1,nb4);
/* Computing MAX */
i__1 = 1, i__2 = mn + (*n << 1) + nb * (*n + 1), i__1 = MAX(i__1,i__2),
i__2 = (mn << 1) + nb * *nrhs;
lwkopt = MAX(i__1,i__2);
q__1.r = (float) lwkopt, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
lquery = *lwork == -1;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < MAX(1,*m)) {
*info = -5;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = MAX(1,*m);
if (*ldb < MAX(i__1,*n)) {
*info = -7;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = mn << 1, i__2 = *n + 1, i__1 = MAX(i__1,i__2), i__2 = mn +
*nrhs;
if (*lwork < mn + MAX(i__1,i__2) && ! lquery) {
*info = -12;
}
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGELSY", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
/* Computing MIN */
i__1 = MIN(*m,*n);
if (MIN(i__1,*nrhs) == 0) {
*rank = 0;
return 0;
}
/* Get machine parameters */
smlnum = slamch_("S") / slamch_("P");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
/* Scale A, B if max entries outside range [SMLNUM,BIGNUM] */
anrm = clange_("M", m, n, &a[a_offset], lda, &rwork[1]);
iascl = 0;
if (anrm > 0.f && anrm < smlnum) {
/* Scale matrix norm up to SMLNUM */
clascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda,
info);
iascl = 1;
} else if (anrm > bignum) {
/* Scale matrix norm down to BIGNUM */
clascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda,
info);
iascl = 2;
} else if (anrm == 0.f) {
/* Matrix all zero. Return zero solution. */
i__1 = MAX(*m,*n);
claset_("F", &i__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
*rank = 0;
goto L70;
}
bnrm = clange_("M", m, nrhs, &b[b_offset], ldb, &rwork[1]);
ibscl = 0;
if (bnrm > 0.f && bnrm < smlnum) {
/* Scale matrix norm up to SMLNUM */
clascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, nrhs, &b[b_offset], ldb,
info);
ibscl = 1;
} else if (bnrm > bignum) {
/* Scale matrix norm down to BIGNUM */
clascl_("G", &c__0, &c__0, &bnrm, &bignum, m, nrhs, &b[b_offset], ldb,
info);
ibscl = 2;
}
/* Compute QR factorization with column pivoting of A: */
/* A * P = Q * R */
i__1 = *lwork - mn;
cgeqp3_(m, n, &a[a_offset], lda, &jpvt[1], &work[1], &work[mn + 1], &i__1,
&rwork[1], info);
i__1 = mn + 1;
wsize = mn + work[i__1].r;
/* complex workspace: MN+NB*(N+1). float workspace 2*N. */
/* Details of Householder rotations stored in WORK(1:MN). */
/* Determine RANK using incremental condition estimation */
i__1 = ismin;
work[i__1].r = 1.f, work[i__1].i = 0.f;
i__1 = ismax;
work[i__1].r = 1.f, work[i__1].i = 0.f;
smax = c_abs(&a[a_dim1 + 1]);
smin = smax;
if (c_abs(&a[a_dim1 + 1]) == 0.f) {
*rank = 0;
i__1 = MAX(*m,*n);
claset_("F", &i__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
goto L70;
} else {
*rank = 1;
}
L10:
if (*rank < mn) {
i__ = *rank + 1;
claic1_(&c__2, rank, &work[ismin], &smin, &a[i__ * a_dim1 + 1], &a[
i__ + i__ * a_dim1], &sminpr, &s1, &c1);
claic1_(&c__1, rank, &work[ismax], &smax, &a[i__ * a_dim1 + 1], &a[
i__ + i__ * a_dim1], &smaxpr, &s2, &c2);
if (smaxpr * *rcond <= sminpr) {
i__1 = *rank;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = ismin + i__ - 1;
i__3 = ismin + i__ - 1;
q__1.r = s1.r * work[i__3].r - s1.i * work[i__3].i, q__1.i =
s1.r * work[i__3].i + s1.i * work[i__3].r;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
i__2 = ismax + i__ - 1;
i__3 = ismax + i__ - 1;
q__1.r = s2.r * work[i__3].r - s2.i * work[i__3].i, q__1.i =
s2.r * work[i__3].i + s2.i * work[i__3].r;
work[i__2].r = q__1.r, work[i__2].i = q__1.i;
/* L20: */
}
i__1 = ismin + *rank;
work[i__1].r = c1.r, work[i__1].i = c1.i;
i__1 = ismax + *rank;
work[i__1].r = c2.r, work[i__1].i = c2.i;
smin = sminpr;
smax = smaxpr;
++(*rank);
goto L10;
}
}
/* complex workspace: 3*MN. */
/* Logically partition R = [ R11 R12 ] */
/* [ 0 R22 ] */
/* where R11 = R(1:RANK,1:RANK) */
/* [R11,R12] = [ T11, 0 ] * Y */
if (*rank < *n) {
i__1 = *lwork - (mn << 1);
ctzrzf_(rank, n, &a[a_offset], lda, &work[mn + 1], &work[(mn << 1) +
1], &i__1, info);
}
/* complex workspace: 2*MN. */
/* Details of Householder rotations stored in WORK(MN+1:2*MN) */
/* B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) */
i__1 = *lwork - (mn << 1);
cunmqr_("Left", "Conjugate transpose", m, nrhs, &mn, &a[a_offset], lda, &
work[1], &b[b_offset], ldb, &work[(mn << 1) + 1], &i__1, info);
/* Computing MAX */
i__1 = (mn << 1) + 1;
r__1 = wsize, r__2 = (mn << 1) + work[i__1].r;
wsize = MAX(r__1,r__2);
/* complex workspace: 2*MN+NB*NRHS. */
/* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) */
ctrsm_("Left", "Upper", "No transpose", "Non-unit", rank, nrhs, &c_b2, &a[
a_offset], lda, &b[b_offset], ldb);
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = *rank + 1; i__ <= i__2; ++i__) {
i__3 = i__ + j * b_dim1;
b[i__3].r = 0.f, b[i__3].i = 0.f;
/* L30: */
}
/* L40: */
}
/* B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS) */
if (*rank < *n) {
i__1 = *n - *rank;
i__2 = *lwork - (mn << 1);
cunmrz_("Left", "Conjugate transpose", n, nrhs, rank, &i__1, &a[
a_offset], lda, &work[mn + 1], &b[b_offset], ldb, &work[(mn <<
1) + 1], &i__2, info);
}
/* complex workspace: 2*MN+NRHS. */
/* B(1:N,1:NRHS) := P * B(1:N,1:NRHS) */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = jpvt[i__];
i__4 = i__ + j * b_dim1;
work[i__3].r = b[i__4].r, work[i__3].i = b[i__4].i;
/* L50: */
}
ccopy_(n, &work[1], &c__1, &b[j * b_dim1 + 1], &c__1);
/* L60: */
}
/* complex workspace: N. */
/* Undo scaling */
if (iascl == 1) {
clascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb,
info);
clascl_("U", &c__0, &c__0, &smlnum, &anrm, rank, rank, &a[a_offset],
lda, info);
} else if (iascl == 2) {
clascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb,
info);
clascl_("U", &c__0, &c__0, &bignum, &anrm, rank, rank, &a[a_offset],
lda, info);
}
if (ibscl == 1) {
clascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb,
info);
} else if (ibscl == 2) {
clascl_("G", &c__0, &c__0, &bignum, &bnrm, n, nrhs, &b[b_offset], ldb,
info);
}
L70:
q__1.r = (float) lwkopt, q__1.i = 0.f;
work[1].r = q__1.r, work[1].i = q__1.i;
return 0;
/* End of CGELSY */
} /* cgelsy_ */