doublereal srzt01_(integer *m, integer *n, real *a, real *af, integer *lda,
real *tau, real *work, integer *lwork)
{
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, i__1, i__2;
real ret_val;
/* Local variables */
integer i__, j, info;
real norma, rwork[1];
extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *,
real *, integer *);
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
extern /* Subroutine */ int xerbla_(char *, integer *), slaset_(
char *, integer *, integer *, real *, real *, real *, integer *), sormrz_(char *, char *, integer *, integer *, integer *,
integer *, real *, integer *, real *, real *, integer *, real *,
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 */
/* ======= */
/* SRZT01 returns */
/* || A - R*Q || / ( M * eps * ||A|| ) */
/* for an upper trapezoidal A that was factored with STZRZF. */
/* 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) REAL array, dimension (LDA,N) */
/* The original upper trapezoidal M by N matrix A. */
/* AF (input) REAL array, dimension (LDA,N) */
/* The output of STZRZF for input matrix A. */
/* The lower triangle is not referenced. */
/* LDA (input) INTEGER */
/* The leading dimension of the arrays A and AF. */
/* TAU (input) REAL array, dimension (M) */
/* Details of the Householder transformations as returned by */
/* STZRZF. */
/* WORK (workspace) REAL array, dimension (LWORK) */
/* LWORK (input) INTEGER */
/* The length of the array WORK. LWORK >= m*n + m*nb. */
/* ===================================================================== */
/* .. 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_("SRZT01", &c__8);
return ret_val;
}
/* Quick return if possible */
if (*m <= 0 || *n <= 0) {
return ret_val;
}
norma = slange_("One-norm", m, n, &a[a_offset], lda, rwork);
/* Copy upper triangle R */
slaset_("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__) {
work[(j - 1) * *m + i__] = af[i__ + j * af_dim1];
/* L10: */
}
/* L20: */
}
/* R = R * P(1) * ... *P(m) */
i__1 = *n - *m;
i__2 = *lwork - *m * *n;
sormrz_("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__) {
saxpy_(m, &c_b13, &a[i__ * a_dim1 + 1], &c__1, &work[(i__ - 1) * *m +
1], &c__1);
/* L30: */
}
ret_val = slange_("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 SRZT01 */
} /* srzt01_ */
/* Subroutine */ int sgelsy_(integer *m, integer *n, integer *nrhs, real *a,
integer *lda, real *b, integer *ldb, integer *jpvt, real *rcond,
integer *rank, real *work, integer *lwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2;
real r__1, r__2;
/* Local variables */
integer i__, j;
real c1, c2, s1, s2;
integer nb, mn, nb1, nb2, nb3, nb4;
real anrm, bnrm, smin, smax;
integer iascl, ibscl, ismin, ismax;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
real wsize;
extern /* Subroutine */ int strsm_(char *, char *, char *, char *,
integer *, integer *, real *, real *, integer *, real *, integer *
), slaic1_(integer *, integer *,
real *, real *, real *, real *, real *, real *, real *), sgeqp3_(
integer *, integer *, real *, integer *, integer *, real *, real *
, integer *, integer *), slabad_(real *, real *);
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
real bignum;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *), slaset_(char *, integer *, integer *, real *, real *,
real *, integer *);
integer lwkmin;
real sminpr, smaxpr, smlnum;
integer lwkopt;
logical lquery;
extern /* Subroutine */ int sormqr_(char *, char *, integer *, integer *,
integer *, real *, integer *, real *, real *, integer *, real *,
integer *, integer *), sormrz_(char *, char *,
integer *, integer *, integer *, integer *, real *, integer *,
real *, real *, integer *, real *, integer *, integer *), stzrzf_(integer *, integer *, real *, integer *, real *,
real *, integer *, integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGELSY computes the minimum-norm solution to a real 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 orthogonal 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 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. */
/* o The permutation of matrix B (the right hand side) is faster and */
/* more simple. */
/* 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) REAL 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) REAL 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 AP */
/* 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) REAL 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 >= MAX( MN+3*N+1, 2*MN+NRHS ), */
/* where MN = min( M, N ). */
/* The block algorithm requires that: */
/* LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), */
/* where NB is an upper bound on the blocksize returned */
/* by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR, */
/* and SORMRZ. */
/* 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 */
/* =============== */
/* 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 Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. 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;
/* Function Body */
mn = min(*m,*n);
ismin = mn + 1;
ismax = (mn << 1) + 1;
/* Test the input arguments. */
*info = 0;
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;
}
}
/* Figure out optimal block size */
if (*info == 0) {
if (mn == 0 || *nrhs == 0) {
lwkmin = 1;
lwkopt = 1;
} else {
nb1 = ilaenv_(&c__1, "SGEQRF", " ", m, n, &c_n1, &c_n1);
nb2 = ilaenv_(&c__1, "SGERQF", " ", m, n, &c_n1, &c_n1);
nb3 = ilaenv_(&c__1, "SORMQR", " ", m, n, nrhs, &c_n1);
nb4 = ilaenv_(&c__1, "SORMRQ", " ", 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 = mn << 1, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = mn +
*nrhs;
lwkmin = mn + max(i__1,i__2);
/* Computing MAX */
i__1 = lwkmin, 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);
}
work[1] = (real) lwkopt;
if (*lwork < lwkmin && ! lquery) {
*info = -12;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGELSY", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (mn == 0 || *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 = slange_("M", m, n, &a[a_offset], lda, &work[1]);
iascl = 0;
if (anrm > 0.f && anrm < smlnum) {
/* Scale matrix norm up to SMLNUM */
slascl_("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 */
slascl_("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);
slaset_("F", &i__1, nrhs, &c_b31, &c_b31, &b[b_offset], ldb);
*rank = 0;
goto L70;
}
bnrm = slange_("M", m, nrhs, &b[b_offset], ldb, &work[1]);
ibscl = 0;
if (bnrm > 0.f && bnrm < smlnum) {
/* Scale matrix norm up to SMLNUM */
slascl_("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 */
slascl_("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;
sgeqp3_(m, n, &a[a_offset], lda, &jpvt[1], &work[1], &work[mn + 1], &i__1,
info);
wsize = mn + work[mn + 1];
/* workspace: MN+2*N+NB*(N+1). */
/* Details of Householder rotations stored in WORK(1:MN). */
/* Determine RANK using incremental condition estimation */
work[ismin] = 1.f;
work[ismax] = 1.f;
smax = (r__1 = a[a_dim1 + 1], dabs(r__1));
smin = smax;
if ((r__1 = a[a_dim1 + 1], dabs(r__1)) == 0.f) {
*rank = 0;
i__1 = max(*m,*n);
slaset_("F", &i__1, nrhs, &c_b31, &c_b31, &b[b_offset], ldb);
goto L70;
} else {
*rank = 1;
}
L10:
if (*rank < mn) {
i__ = *rank + 1;
slaic1_(&c__2, rank, &work[ismin], &smin, &a[i__ * a_dim1 + 1], &a[
i__ + i__ * a_dim1], &sminpr, &s1, &c1);
slaic1_(&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__) {
work[ismin + i__ - 1] = s1 * work[ismin + i__ - 1];
work[ismax + i__ - 1] = s2 * work[ismax + i__ - 1];
/* L20: */
}
work[ismin + *rank] = c1;
work[ismax + *rank] = c2;
smin = sminpr;
smax = smaxpr;
++(*rank);
goto L10;
}
}
/* 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);
stzrzf_(rank, n, &a[a_offset], lda, &work[mn + 1], &work[(mn << 1) +
1], &i__1, info);
}
/* 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);
sormqr_("Left", "Transpose", m, nrhs, &mn, &a[a_offset], lda, &work[1], &
b[b_offset], ldb, &work[(mn << 1) + 1], &i__1, info);
/* Computing MAX */
r__1 = wsize, r__2 = (mn << 1) + work[(mn << 1) + 1];
wsize = dmax(r__1,r__2);
/* workspace: 2*MN+NB*NRHS. */
/* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) */
strsm_("Left", "Upper", "No transpose", "Non-unit", rank, nrhs, &c_b54, &
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__) {
b[i__ + j * b_dim1] = 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);
sormrz_("Left", "Transpose", n, nrhs, rank, &i__1, &a[a_offset], lda,
&work[mn + 1], &b[b_offset], ldb, &work[(mn << 1) + 1], &i__2,
info);
}
/* 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__) {
work[jpvt[i__]] = b[i__ + j * b_dim1];
/* L50: */
}
scopy_(n, &work[1], &c__1, &b[j * b_dim1 + 1], &c__1);
/* L60: */
}
/* workspace: N. */
/* Undo scaling */
if (iascl == 1) {
slascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb,
info);
slascl_("U", &c__0, &c__0, &smlnum, &anrm, rank, rank, &a[a_offset],
lda, info);
} else if (iascl == 2) {
slascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb,
info);
slascl_("U", &c__0, &c__0, &bignum, &anrm, rank, rank, &a[a_offset],
lda, info);
}
if (ibscl == 1) {
slascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb,
info);
} else if (ibscl == 2) {
slascl_("G", &c__0, &c__0, &bignum, &bnrm, n, nrhs, &b[b_offset], ldb,
info);
}
L70:
work[1] = (real) lwkopt;
return 0;
/* End of SGELSY */
} /* sgelsy_ */
