/* dgesvj.f -- translated by f2c (version 20061008).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
#include "blaswrap.h"
/* Table of constant values */
static doublereal c_b17 = 0.;
static doublereal c_b18 = 1.;
static integer c__1 = 1;
static integer c__0 = 0;
static integer c__2 = 2;
/* Subroutine */ int dgesvj_(char *joba, char *jobu, char *jobv, integer *m,
integer *n, doublereal *a, integer *lda, doublereal *sva, integer *mv,
doublereal *v, integer *ldv, doublereal *work, integer *lwork,
integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal), d_sign(doublereal *, doublereal *);
/* Local variables */
doublereal bigtheta;
integer pskipped, i__, p, q;
doublereal t;
integer n2, n4;
doublereal rootsfmin;
integer n34;
doublereal cs, sn;
integer ir1, jbc;
doublereal big;
integer kbl, igl, ibr, jgl, nbl;
doublereal tol;
integer mvl;
doublereal aapp, aapq, aaqq;
extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *,
integer *);
doublereal ctol;
integer ierr;
doublereal aapp0;
extern doublereal dnrm2_(integer *, doublereal *, integer *);
doublereal temp1;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
doublereal scale, large, apoaq, aqoap;
extern logical lsame_(char *, char *);
doublereal theta, small, sfmin;
logical lsvec;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
doublereal fastr[5];
extern /* Subroutine */ int dswap_(integer *, doublereal *, integer *,
doublereal *, integer *);
logical applv, rsvec;
extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *);
logical uctol;
extern /* Subroutine */ int drotm_(integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *);
logical lower, upper, rotok;
extern /* Subroutine */ int dgsvj0_(char *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
doublereal *, integer *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *), dgsvj1_(
char *, integer *, integer *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *);
extern doublereal dlamch_(char *);
extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *);
extern integer idamax_(integer *, doublereal *, integer *);
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *),
xerbla_(char *, integer *);
integer ijblsk, swband, blskip;
doublereal mxaapq;
extern /* Subroutine */ int dlassq_(integer *, doublereal *, integer *,
doublereal *, doublereal *);
doublereal thsign, mxsinj;
integer emptsw, notrot, iswrot, lkahead;
logical goscale, noscale;
doublereal rootbig, epsilon, rooteps;
integer rowskip;
doublereal roottol;
/* -- LAPACK routine (version 3.2) -- */
/* -- Contributed by Zlatko Drmac of the University of Zagreb and -- */
/* -- Kresimir Veselic of the Fernuniversitaet Hagen -- */
/* -- November 2008 -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* This routine is also part of SIGMA (version 1.23, October 23. 2008.) */
/* SIGMA is a library of algorithms for highly accurate algorithms for */
/* computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the */
/* eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0. */
/* -#- Scalar Arguments -#- */
/* -#- Array Arguments -#- */
/* .. */
/* Purpose */
/* ~~~~~~~ */
/* DGESVJ 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^t, [++] = [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. */
/* Further Details */
/* ~~~~~~~~~~~~~~~ */
/* The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane */
/* rotations. The rotations are implemented as fast scaled rotations of */
/* Anda and Park [1]. In the case of underflow of the Jacobi angle, a */
/* modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses */
/* column interchanges of de Rijk [2]. The relative accuracy of the computed */
/* singular values and the accuracy of the computed singular vectors (in */
/* angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. */
/* The condition number that determines the accuracy in the full rank case */
/* is essentially min_{D=diag} kappa(A*D), where kappa(.) is the */
/* spectral condition number. The best performance of this Jacobi SVD */
/* procedure is achieved if used in an accelerated version of Drmac and */
/* Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. */
/* Some tunning parameters (marked with [TP]) are available for the */
/* implementer. */
/* The computational range for the nonzero singular values is the machine */
/* number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even */
/* denormalized singular values can be computed with the corresponding */
/* gradual loss of accurate digits. */
/* Contributors */
/* ~~~~~~~~~~~~ */
/* Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) */
/* References */
/* ~~~~~~~~~~ */
/* [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling. */
/* SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174. */
/* [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the */
/* singular value decomposition on a vector computer. */
/* SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371. */
/* [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR. */
/* [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular */
/* value computation in floating point arithmetic. */
/* SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222. */
/* [5] 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. */
/* [6] 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. */
/* [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, */
/* QSVD, (H,K)-SVD computations. */
/* Department of Mathematics, University of Zagreb, 2008. */
/* Bugs, Examples and Comments */
/* ~~~~~~~~~~~~~~~~~~~~~~~~~~~ */
/* Please report all bugs and send interesting test examples and comments to */
/* drmac@math.hr. Thank you. */
/* Arguments */
/* ~~~~~~~~~ */
/* JOBA (input) CHARACTER* 1 */
/* Specifies the structure of A. */
/* = 'L': The input matrix A is lower triangular; */
/* = 'U': The input matrix A is upper triangular; */
/* = 'G': The input matrix A is general M-by-N matrix, M >= N. */
/* JOBU (input) CHARACTER*1 */
/* Specifies whether to compute the left singular vectors */
/* (columns of U): */
/* = 'U': The left singular vectors corresponding to the nonzero */
/* singular values are computed and returned in the leading */
/* columns of A. See more details in the description of A. */
/* The default numerical orthogonality threshold is set to */
/* approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E'). */
/* = 'C': Analogous to JOBU='U', except that user can control the */
/* level of numerical orthogonality of the computed left */
/* singular vectors. TOL can be set to TOL = CTOL*EPS, where */
/* CTOL is given on input in the array WORK. */
/* No CTOL smaller than ONE is allowed. CTOL greater */
/* than 1 / EPS is meaningless. The option 'C' */
/* can be used if M*EPS is satisfactory orthogonality */
/* of the computed left singular vectors, so CTOL=M could */
/* save few sweeps of Jacobi rotations. */
/* See the descriptions of A and WORK(1). */
/* = 'N': The matrix U is not computed. However, see the */
/* description of A. */
/* JOBV (input) CHARACTER*1 */
/* Specifies whether to compute the right singular vectors, that */
/* is, the matrix V: */
/* = 'V' : the matrix V is computed and returned in the array V */
/* = 'A' : the Jacobi rotations are applied to the MV-by-N */
/* array V. In other words, the right singular vector */
/* matrix V is not computed explicitly, instead it is */
/* applied to an MV-by-N matrix initially stored in the */
/* first MV rows of V. */
/* = 'N' : the matrix V is not computed and the array V is not */
/* referenced */
/* M (input) INTEGER */
/* The number of rows of the input matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the input matrix A. */
/* M >= N >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, */
/* If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C': */
/* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ */
/* If INFO .EQ. 0, */
/* ~~~~~~~~~~~~~~~ */
/* RANKA orthonormal columns of U are returned in the */
/* leading RANKA columns of the array A. Here RANKA <= N */
/* is the number of computed singular values of A that are */
/* above the underflow threshold DLAMCH('S'). The singular */
/* vectors corresponding to underflowed or zero singular */
/* values are not computed. The value of RANKA is returned */
/* in the array WORK as RANKA=NINT(WORK(2)). Also see the */
/* descriptions of SVA and WORK. The computed columns of U */
/* are mutually numerically orthogonal up to approximately */
/* TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'), */
/* see the description of JOBU. */
/* If INFO .GT. 0, */
/* ~~~~~~~~~~~~~~~ */
/* the procedure DGESVJ did not converge in the given number */
/* of iterations (sweeps). In that case, the computed */
/* columns of U may not be orthogonal up to TOL. The output */
/* U (stored in A), SIGMA (given by the computed singular */
/* values in SVA(1:N)) and V is still a decomposition of the */
/* input matrix A in the sense that the residual */
/* ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small. */
/* If JOBU .EQ. 'N': */
/* ~~~~~~~~~~~~~~~~~ */
/* If INFO .EQ. 0 */
/* ~~~~~~~~~~~~~~ */
/* Note that the left singular vectors are 'for free' in the */
/* one-sided Jacobi SVD algorithm. However, if only the */
/* singular values are needed, the level of numerical */
/* orthogonality of U is not an issue and iterations are */
/* stopped when the columns of the iterated matrix are */
/* numerically orthogonal up to approximately M*EPS. Thus, */
/* on exit, A contains the columns of U scaled with the */
/* corresponding singular values. */
/* If INFO .GT. 0, */
/* ~~~~~~~~~~~~~~~ */
/* the procedure DGESVJ did not converge in the given number */
/* of iterations (sweeps). */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* SVA (workspace/output) REAL array, dimension (N) */
/* On exit, */
/* If INFO .EQ. 0, */
/* ~~~~~~~~~~~~~~~ */
/* depending on the value SCALE = WORK(1), we have: */
/* If SCALE .EQ. ONE: */
/* ~~~~~~~~~~~~~~~~~~ */
/* SVA(1:N) contains the computed singular values of A. */
/* During the computation SVA contains the Euclidean column */
/* norms of the iterated matrices in the array A. */
/* If SCALE .NE. ONE: */
/* ~~~~~~~~~~~~~~~~~~ */
/* The singular values of A are SCALE*SVA(1:N), and this */
/* factored representation is due to the fact that some of the */
/* singular values of A might underflow or overflow. */
/* If INFO .GT. 0, */
/* ~~~~~~~~~~~~~~~ */
/* the procedure DGESVJ did not converge in the given number of */
/* iterations (sweeps) and SCALE*SVA(1:N) may not be accurate. */
/* MV (input) INTEGER */
/* If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ */
/* is applied to the first MV rows of V. See the description of JOBV. */
/* V (input/output) REAL array, dimension (LDV,N) */
/* If JOBV = 'V', then V contains on exit the N-by-N matrix of */
/* the right singular vectors; */
/* If JOBV = 'A', then V contains the product of the computed right */
/* singular vector matrix and the initial matrix in */
/* the array V. */
/* If JOBV = 'N', then V is not referenced. */
/* LDV (input) INTEGER */
/* The leading dimension of the array V, LDV .GE. 1. */
/* If JOBV .EQ. 'V', then LDV .GE. max(1,N). */
/* If JOBV .EQ. 'A', then LDV .GE. max(1,MV) . */
/* WORK (input/workspace/output) REAL array, dimension max(4,M+N). */
/* On entry, */
/* If JOBU .EQ. 'C', */
/* ~~~~~~~~~~~~~~~~~ */
/* WORK(1) = CTOL, where CTOL defines the threshold for convergence. */
/* The process stops if all columns of A are mutually */
/* orthogonal up to CTOL*EPS, EPS=DLAMCH('E'). */
/* It is required that CTOL >= ONE, i.e. it is not */
/* allowed to force the routine to obtain orthogonality */
/* below EPSILON. */
/* On exit, */
/* WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N) */
/* are the computed singular vcalues of A. */
/* (See description of SVA().) */
/* WORK(2) = NINT(WORK(2)) is the number of the computed nonzero */
/* singular values. */
/* WORK(3) = NINT(WORK(3)) is the number of the computed singular */
/* values that are larger than the underflow threshold. */
/* WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi */
/* rotations needed for numerical convergence. */
/* WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep. */
/* This is useful information in cases when DGESVJ did */
/* not converge, as it can be used to estimate whether */
/* the output is stil useful and for post festum analysis. */
/* WORK(6) = the largest absolute value over all sines of the */
/* Jacobi rotation angles in the last sweep. It can be */
/* useful for a post festum analysis. */
/* LWORK length of WORK, WORK >= MAX(6,M+N) */
/* INFO (output) INTEGER */
/* = 0 : successful exit. */
/* < 0 : if INFO = -i, then the i-th argument had an illegal value */
/* > 0 : DGESVJ did not converge in the maximal allowed number (30) */
/* of sweeps. The output may still be useful. See the */
/* description of WORK. */
/* Local Parameters */
/* Local Scalars */
/* Local Arrays */
/* Intrinsic Functions */
/* External Functions */
/* .. from BLAS */
/* .. from LAPACK */
/* External Subroutines */
/* .. from BLAS */
/* .. from LAPACK */
/* Test the input arguments */
/* Parameter adjustments */
--sva;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
--work;
/* Function Body */
lsvec = lsame_(jobu, "U");
uctol = lsame_(jobu, "C");
rsvec = lsame_(jobv, "V");
applv = lsame_(jobv, "A");
upper = lsame_(joba, "U");
lower = lsame_(joba, "L");
if (! (upper || lower || lsame_(joba, "G"))) {
*info = -1;
} else if (! (lsvec || uctol || lsame_(jobu, "N")))
{
*info = -2;
} else if (! (rsvec || applv || lsame_(jobv, "N")))
{
*info = -3;
} else if (*m < 0) {
*info = -4;
} else if (*n < 0 || *n > *m) {
*info = -5;
} else if (*lda < *m) {
*info = -7;
} else if (*mv < 0) {
*info = -9;
} else if (rsvec && *ldv < *n || applv && *ldv < *mv) {
*info = -11;
} else if (uctol && work[1] <= 1.) {
*info = -12;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = *m + *n;
if (*lwork < max(i__1,6)) {
*info = -13;
} else {
*info = 0;
}
}
/* #:( */
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGESVJ", &i__1);
return 0;
}
/* #:) Quick return for void matrix */
if (*m == 0 || *n == 0) {
return 0;
}
/* Set numerical parameters */
/* The stopping criterion for Jacobi rotations is */
/* max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS */
/* where EPS is the round-off and CTOL is defined as follows: */
if (uctol) {
/* ... user controlled */
ctol = work[1];
} else {
/* ... default */
if (lsvec || rsvec || applv) {
ctol = sqrt((doublereal) (*m));
} else {
ctol = (doublereal) (*m);
}
}
/* ... and the machine dependent parameters are */
/* [!] (Make sure that DLAMCH() works properly on the target machine.) */
epsilon = dlamch_("Epsilon");
rooteps = sqrt(epsilon);
sfmin = dlamch_("SafeMinimum");
rootsfmin = sqrt(sfmin);
small = sfmin / epsilon;
big = dlamch_("Overflow");
/* BIG = ONE / SFMIN */
rootbig = 1. / rootsfmin;
large = big / sqrt((doublereal) (*m * *n));
bigtheta = 1. / rooteps;
tol = ctol * epsilon;
roottol = sqrt(tol);
if ((doublereal) (*m) * epsilon >= 1.) {
*info = -5;
i__1 = -(*info);
xerbla_("DGESVJ", &i__1);
return 0;
}
/* Initialize the right singular vector matrix. */
if (rsvec) {
mvl = *n;
dlaset_("A", &mvl, n, &c_b17, &c_b18, &v[v_offset], ldv);
} else if (applv) {
mvl = *mv;
}
rsvec = rsvec || applv;
/* Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N ) */
/* (!) If necessary, scale A to protect the largest singular value */
/* from overflow. It is possible that saving the largest singular */
/* value destroys the information about the small ones. */
/* This initial scaling is almost minimal in the sense that the */
/* goal is to make sure that no column norm overflows, and that */
/* DSQRT(N)*max_i SVA(i) does not overflow. If INFinite entries */
/* in A are detected, the procedure returns with INFO=-6. */
scale = 1. / sqrt((doublereal) (*m) * (doublereal) (*n));
noscale = TRUE_;
goscale = TRUE_;
if (lower) {
/* the input matrix is M-by-N lower triangular (trapezoidal) */
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
aapp = 0.;
aaqq = 0.;
i__2 = *m - p + 1;
dlassq_(&i__2, &a[p + p * a_dim1], &c__1, &aapp, &aaqq);
if (aapp > big) {
*info = -6;
i__2 = -(*info);
xerbla_("DGESVJ", &i__2);
return 0;
}
aaqq = sqrt(aaqq);
if (aapp < big / aaqq && noscale) {
sva[p] = aapp * aaqq;
} else {
noscale = FALSE_;
sva[p] = aapp * (aaqq * scale);
if (goscale) {
goscale = FALSE_;
i__2 = p - 1;
for (q = 1; q <= i__2; ++q) {
sva[q] *= scale;
/* L1873: */
}
}
}
/* L1874: */
}
} else if (upper) {
/* the input matrix is M-by-N upper triangular (trapezoidal) */
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
aapp = 0.;
aaqq = 0.;
dlassq_(&p, &a[p * a_dim1 + 1], &c__1, &aapp, &aaqq);
if (aapp > big) {
*info = -6;
i__2 = -(*info);
xerbla_("DGESVJ", &i__2);
return 0;
}
aaqq = sqrt(aaqq);
if (aapp < big / aaqq && noscale) {
sva[p] = aapp * aaqq;
} else {
noscale = FALSE_;
sva[p] = aapp * (aaqq * scale);
if (goscale) {
goscale = FALSE_;
i__2 = p - 1;
for (q = 1; q <= i__2; ++q) {
sva[q] *= scale;
/* L2873: */
}
}
}
/* L2874: */
}
} else {
/* the input matrix is M-by-N general dense */
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
aapp = 0.;
aaqq = 0.;
dlassq_(m, &a[p * a_dim1 + 1], &c__1, &aapp, &aaqq);
if (aapp > big) {
*info = -6;
i__2 = -(*info);
xerbla_("DGESVJ", &i__2);
return 0;
}
aaqq = sqrt(aaqq);
if (aapp < big / aaqq && noscale) {
sva[p] = aapp * aaqq;
} else {
noscale = FALSE_;
sva[p] = aapp * (aaqq * scale);
if (goscale) {
goscale = FALSE_;
i__2 = p - 1;
for (q = 1; q <= i__2; ++q) {
sva[q] *= scale;
/* L3873: */
}
}
}
/* L3874: */
}
}
if (noscale) {
scale = 1.;
}
/* Move the smaller part of the spectrum from the underflow threshold */
/* (!) Start by determining the position of the nonzero entries of the */
/* array SVA() relative to ( SFMIN, BIG ). */
aapp = 0.;
aaqq = big;
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
if (sva[p] != 0.) {
/* Computing MIN */
d__1 = aaqq, d__2 = sva[p];
aaqq = min(d__1,d__2);
}
/* Computing MAX */
d__1 = aapp, d__2 = sva[p];
aapp = max(d__1,d__2);
/* L4781: */
}
/* #:) Quick return for zero matrix */
if (aapp == 0.) {
if (lsvec) {
dlaset_("G", m, n, &c_b17, &c_b18, &a[a_offset], lda);
}
work[1] = 1.;
work[2] = 0.;
work[3] = 0.;
work[4] = 0.;
work[5] = 0.;
work[6] = 0.;
return 0;
}
/* #:) Quick return for one-column matrix */
if (*n == 1) {
if (lsvec) {
dlascl_("G", &c__0, &c__0, &sva[1], &scale, m, &c__1, &a[a_dim1 +
1], lda, &ierr);
}
work[1] = 1. / scale;
if (sva[1] >= sfmin) {
work[2] = 1.;
} else {
work[2] = 0.;
}
work[3] = 0.;
work[4] = 0.;
work[5] = 0.;
work[6] = 0.;
return 0;
}
/* Protect small singular values from underflow, and try to */
/* avoid underflows/overflows in computing Jacobi rotations. */
sn = sqrt(sfmin / epsilon);
temp1 = sqrt(big / (doublereal) (*n));
if (aapp <= sn || aaqq >= temp1 || sn <= aaqq && aapp <= temp1) {
/* Computing MIN */
d__1 = big, d__2 = temp1 / aapp;
temp1 = min(d__1,d__2);
/* AAQQ = AAQQ*TEMP1 */
/* AAPP = AAPP*TEMP1 */
} else if (aaqq <= sn && aapp <= temp1) {
/* Computing MIN */
d__1 = sn / aaqq, d__2 = big / (aapp * sqrt((doublereal) (*n)));
temp1 = min(d__1,d__2);
/* AAQQ = AAQQ*TEMP1 */
/* AAPP = AAPP*TEMP1 */
} else if (aaqq >= sn && aapp >= temp1) {
/* Computing MAX */
d__1 = sn / aaqq, d__2 = temp1 / aapp;
temp1 = max(d__1,d__2);
/* AAQQ = AAQQ*TEMP1 */
/* AAPP = AAPP*TEMP1 */
} else if (aaqq <= sn && aapp >= temp1) {
/* Computing MIN */
d__1 = sn / aaqq, d__2 = big / (sqrt((doublereal) (*n)) * aapp);
temp1 = min(d__1,d__2);
/* AAQQ = AAQQ*TEMP1 */
/* AAPP = AAPP*TEMP1 */
} else {
temp1 = 1.;
}
/* Scale, if necessary */
if (temp1 != 1.) {
dlascl_("G", &c__0, &c__0, &c_b18, &temp1, n, &c__1, &sva[1], n, &
ierr);
}
scale = temp1 * scale;
if (scale != 1.) {
dlascl_(joba, &c__0, &c__0, &c_b18, &scale, m, n, &a[a_offset], lda, &
ierr);
scale = 1. / scale;
}
/* Row-cyclic Jacobi SVD algorithm with column pivoting */
emptsw = *n * (*n - 1) / 2;
notrot = 0;
fastr[0] = 0.;
/* A is represented in factored form A = A * diag(WORK), where diag(WORK) */
/* is initialized to identity. WORK is updated during fast scaled */
/* rotations. */
i__1 = *n;
for (q = 1; q <= i__1; ++q) {
work[q] = 1.;
/* L1868: */
}
swband = 3;
/* [TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective */
/* if DGESVJ is used as a computational routine in the preconditioned */
/* Jacobi SVD algorithm DGESVJ. For sweeps i=1:SWBAND the procedure */
/* works on pivots inside a band-like region around the diagonal. */
/* The boundaries are determined dynamically, based on the number of */
/* pivots above a threshold. */
kbl = min(8,*n);
/* [TP] KBL is a tuning parameter that defines the tile size in the */
/* tiling of the p-q loops of pivot pairs. In general, an optimal */
/* value of KBL depends on the matrix dimensions and on the */
/* parameters of the computer's memory. */
nbl = *n / kbl;
if (nbl * kbl != *n) {
++nbl;
}
/* Computing 2nd power */
i__1 = kbl;
blskip = i__1 * i__1;
/* [TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. */
rowskip = min(5,kbl);
/* [TP] ROWSKIP is a tuning parameter. */
lkahead = 1;
/* [TP] LKAHEAD is a tuning parameter. */
/* Quasi block transformations, using the lower (upper) triangular */
/* structure of the input matrix. The quasi-block-cycling usually */
/* invokes cubic convergence. Big part of this cycle is done inside */
/* canonical subspaces of dimensions less than M. */
/* Computing MAX */
i__1 = 64, i__2 = kbl << 2;
if ((lower || upper) && *n > max(i__1,i__2)) {
/* [TP] The number of partition levels and the actual partition are */
/* tuning parameters. */
n4 = *n / 4;
n2 = *n / 2;
n34 = n4 * 3;
if (applv) {
q = 0;
} else {
q = 1;
}
if (lower) {
/* This works very well on lower triangular matrices, in particular */
/* in the framework of the preconditioned Jacobi SVD (xGEJSV). */
/* The idea is simple: */
/* [+ 0 0 0] Note that Jacobi transformations of [0 0] */
/* [+ + 0 0] [0 0] */
/* [+ + x 0] actually work on [x 0] [x 0] */
/* [+ + x x] [x x]. [x x] */
i__1 = *m - n34;
i__2 = *n - n34;
i__3 = *lwork - *n;
dgsvj0_(jobv, &i__1, &i__2, &a[n34 + 1 + (n34 + 1) * a_dim1], lda,
&work[n34 + 1], &sva[n34 + 1], &mvl, &v[n34 * q + 1 + (
n34 + 1) * v_dim1], ldv, &epsilon, &sfmin, &tol, &c__2, &
work[*n + 1], &i__3, &ierr);
i__1 = *m - n2;
i__2 = n34 - n2;
i__3 = *lwork - *n;
dgsvj0_(jobv, &i__1, &i__2, &a[n2 + 1 + (n2 + 1) * a_dim1], lda, &
work[n2 + 1], &sva[n2 + 1], &mvl, &v[n2 * q + 1 + (n2 + 1)
* v_dim1], ldv, &epsilon, &sfmin, &tol, &c__2, &work[*n
+ 1], &i__3, &ierr);
i__1 = *m - n2;
i__2 = *n - n2;
i__3 = *lwork - *n;
dgsvj1_(jobv, &i__1, &i__2, &n4, &a[n2 + 1 + (n2 + 1) * a_dim1],
lda, &work[n2 + 1], &sva[n2 + 1], &mvl, &v[n2 * q + 1 + (
n2 + 1) * v_dim1], ldv, &epsilon, &sfmin, &tol, &c__1, &
work[*n + 1], &i__3, &ierr);
i__1 = *m - n4;
i__2 = n2 - n4;
i__3 = *lwork - *n;
dgsvj0_(jobv, &i__1, &i__2, &a[n4 + 1 + (n4 + 1) * a_dim1], lda, &
work[n4 + 1], &sva[n4 + 1], &mvl, &v[n4 * q + 1 + (n4 + 1)
* v_dim1], ldv, &epsilon, &sfmin, &tol, &c__1, &work[*n
+ 1], &i__3, &ierr);
i__1 = *lwork - *n;
dgsvj0_(jobv, m, &n4, &a[a_offset], lda, &work[1], &sva[1], &mvl,
&v[v_offset], ldv, &epsilon, &sfmin, &tol, &c__1, &work[*
n + 1], &i__1, &ierr);
i__1 = *lwork - *n;
dgsvj1_(jobv, m, &n2, &n4, &a[a_offset], lda, &work[1], &sva[1], &
mvl, &v[v_offset], ldv, &epsilon, &sfmin, &tol, &c__1, &
work[*n + 1], &i__1, &ierr);
} else if (upper) {
i__1 = *lwork - *n;
dgsvj0_(jobv, &n4, &n4, &a[a_offset], lda, &work[1], &sva[1], &
mvl, &v[v_offset], ldv, &epsilon, &sfmin, &tol, &c__2, &
work[*n + 1], &i__1, &ierr);
i__1 = *lwork - *n;
dgsvj0_(jobv, &n2, &n4, &a[(n4 + 1) * a_dim1 + 1], lda, &work[n4
+ 1], &sva[n4 + 1], &mvl, &v[n4 * q + 1 + (n4 + 1) *
v_dim1], ldv, &epsilon, &sfmin, &tol, &c__1, &work[*n + 1]
, &i__1, &ierr);
i__1 = *lwork - *n;
dgsvj1_(jobv, &n2, &n2, &n4, &a[a_offset], lda, &work[1], &sva[1],
&mvl, &v[v_offset], ldv, &epsilon, &sfmin, &tol, &c__1, &
work[*n + 1], &i__1, &ierr);
i__1 = n2 + n4;
i__2 = *lwork - *n;
dgsvj0_(jobv, &i__1, &n4, &a[(n2 + 1) * a_dim1 + 1], lda, &work[
n2 + 1], &sva[n2 + 1], &mvl, &v[n2 * q + 1 + (n2 + 1) *
v_dim1], ldv, &epsilon, &sfmin, &tol, &c__1, &work[*n + 1]
, &i__2, &ierr);
}
}
/* -#- Row-cyclic pivot strategy with de Rijk's pivoting -#- */
for (i__ = 1; i__ <= 30; ++i__) {
/* .. go go go ... */
mxaapq = 0.;
mxsinj = 0.;
iswrot = 0;
notrot = 0;
pskipped = 0;
/* Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs */
/* 1 <= p < q <= N. This is the first step toward a blocked implementation */
/* of the rotations. New implementation, based on block transformations, */
/* is under development. */
i__1 = nbl;
for (ibr = 1; ibr <= i__1; ++ibr) {
igl = (ibr - 1) * kbl + 1;
/* Computing MIN */
i__3 = lkahead, i__4 = nbl - ibr;
i__2 = min(i__3,i__4);
for (ir1 = 0; ir1 <= i__2; ++ir1) {
igl += ir1 * kbl;
/* Computing MIN */
i__4 = igl + kbl - 1, i__5 = *n - 1;
i__3 = min(i__4,i__5);
for (p = igl; p <= i__3; ++p) {
/* .. de Rijk's pivoting */
i__4 = *n - p + 1;
q = idamax_(&i__4, &sva[p], &c__1) + p - 1;
if (p != q) {
dswap_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 +
1], &c__1);
if (rsvec) {
dswap_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q *
v_dim1 + 1], &c__1);
}
temp1 = sva[p];
sva[p] = sva[q];
sva[q] = temp1;
temp1 = work[p];
work[p] = work[q];
work[q] = temp1;
}
if (ir1 == 0) {
/* Column norms are periodically updated by explicit */
/* norm computation. */
/* Caveat: */
/* Unfortunately, some BLAS implementations compute DNRM2(M,A(1,p),1) */
/* as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to */
/* overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and to */
/* underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold). */
/* Hence, DNRM2 cannot be trusted, not even in the case when */
/* the true norm is far from the under(over)flow boundaries. */
/* If properly implemented DNRM2 is available, the IF-THEN-ELSE */
/* below should read "AAPP = DNRM2( M, A(1,p), 1 ) * WORK(p)". */
if (sva[p] < rootbig && sva[p] > rootsfmin) {
sva[p] = dnrm2_(m, &a[p * a_dim1 + 1], &c__1) *
work[p];
} else {
temp1 = 0.;
aapp = 0.;
dlassq_(m, &a[p * a_dim1 + 1], &c__1, &temp1, &
aapp);
sva[p] = temp1 * sqrt(aapp) * work[p];
}
aapp = sva[p];
} else {
aapp = sva[p];
}
if (aapp > 0.) {
pskipped = 0;
/* Computing MIN */
i__5 = igl + kbl - 1;
i__4 = min(i__5,*n);
for (q = p + 1; q <= i__4; ++q) {
aaqq = sva[q];
if (aaqq > 0.) {
aapp0 = aapp;
if (aaqq >= 1.) {
rotok = small * aapp <= aaqq;
if (aapp < big / aaqq) {
aapq = ddot_(m, &a[p * a_dim1 + 1], &
c__1, &a[q * a_dim1 + 1], &
c__1) * work[p] * work[q] /
aaqq / aapp;
} else {
dcopy_(m, &a[p * a_dim1 + 1], &c__1, &
work[*n + 1], &c__1);
dlascl_("G", &c__0, &c__0, &aapp, &
work[p], m, &c__1, &work[*n +
1], lda, &ierr);
aapq = ddot_(m, &work[*n + 1], &c__1,
&a[q * a_dim1 + 1], &c__1) *
work[q] / aaqq;
}
} else {
rotok = aapp <= aaqq / small;
if (aapp > small / aaqq) {
aapq = ddot_(m, &a[p * a_dim1 + 1], &
c__1, &a[q * a_dim1 + 1], &
c__1) * work[p] * work[q] /
aaqq / aapp;
} else {
dcopy_(m, &a[q * a_dim1 + 1], &c__1, &
work[*n + 1], &c__1);
dlascl_("G", &c__0, &c__0, &aaqq, &
work[q], m, &c__1, &work[*n +
1], lda, &ierr);
aapq = ddot_(m, &work[*n + 1], &c__1,
&a[p * a_dim1 + 1], &c__1) *
work[p] / aapp;
}
}
/* Computing MAX */
d__1 = mxaapq, d__2 = abs(aapq);
mxaapq = max(d__1,d__2);
/* TO rotate or NOT to rotate, THAT is the question ... */
if (abs(aapq) > tol) {
/* .. rotate */
/* [RTD] ROTATED = ROTATED + ONE */
if (ir1 == 0) {
notrot = 0;
pskipped = 0;
++iswrot;
}
if (rotok) {
aqoap = aaqq / aapp;
apoaq = aapp / aaqq;
theta = (d__1 = aqoap - apoaq, abs(
d__1)) * -.5 / aapq;
if (abs(theta) > bigtheta) {
t = .5 / theta;
fastr[2] = t * work[p] / work[q];
fastr[3] = -t * work[q] / work[p];
drotm_(m, &a[p * a_dim1 + 1], &
c__1, &a[q * a_dim1 + 1],
&c__1, fastr);
if (rsvec) {
drotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q *
v_dim1 + 1], &c__1, fastr);
}
/* Computing MAX */
d__1 = 0., d__2 = t * apoaq *
aapq + 1.;
sva[q] = aaqq * sqrt((max(d__1,
d__2)));
aapp *= sqrt(1. - t * aqoap *
aapq);
/* Computing MAX */
d__1 = mxsinj, d__2 = abs(t);
mxsinj = max(d__1,d__2);
} else {
/* .. choose correct signum for THETA and rotate */
thsign = -d_sign(&c_b18, &aapq);
t = 1. / (theta + thsign * sqrt(
theta * theta + 1.));
cs = sqrt(1. / (t * t + 1.));
sn = t * cs;
/* Computing MAX */
d__1 = mxsinj, d__2 = abs(sn);
mxsinj = max(d__1,d__2);
/* Computing MAX */
d__1 = 0., d__2 = t * apoaq *
aapq + 1.;
sva[q] = aaqq * sqrt((max(d__1,
d__2)));
/* Computing MAX */
d__1 = 0., d__2 = 1. - t * aqoap *
aapq;
aapp *= sqrt((max(d__1,d__2)));
apoaq = work[p] / work[q];
aqoap = work[q] / work[p];
if (work[p] >= 1.) {
if (work[q] >= 1.) {
fastr[2] = t * apoaq;
fastr[3] = -t * aqoap;
work[p] *= cs;
work[q] *= cs;
drotm_(m, &a[p * a_dim1 + 1], &c__1, &a[q *
a_dim1 + 1], &c__1, fastr);
if (rsvec) {
drotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[
q * v_dim1 + 1], &c__1, fastr);
}
} else {
d__1 = -t * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1, &a[
p * a_dim1 + 1], &c__1);
d__1 = cs * sn * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1, &a[
q * a_dim1 + 1], &c__1);
work[p] *= cs;
work[q] /= cs;
if (rsvec) {
d__1 = -t * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1], &
c__1, &v[p * v_dim1 + 1], &c__1);
d__1 = cs * sn * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1], &
c__1, &v[q * v_dim1 + 1], &c__1);
}
}
} else {
if (work[q] >= 1.) {
d__1 = t * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1, &a[
q * a_dim1 + 1], &c__1);
d__1 = -cs * sn * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1, &a[
p * a_dim1 + 1], &c__1);
work[p] /= cs;
work[q] *= cs;
if (rsvec) {
d__1 = t * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1], &
c__1, &v[q * v_dim1 + 1], &c__1);
d__1 = -cs * sn * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1], &
c__1, &v[p * v_dim1 + 1], &c__1);
}
} else {
if (work[p] >= work[q]) {
d__1 = -t * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1,
&a[p * a_dim1 + 1], &c__1);
d__1 = cs * sn * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1,
&a[q * a_dim1 + 1], &c__1);
work[p] *= cs;
work[q] /= cs;
if (rsvec) {
d__1 = -t * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1],
&c__1, &v[p * v_dim1 + 1], &
c__1);
d__1 = cs * sn * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1],
&c__1, &v[q * v_dim1 + 1], &
c__1);
}
} else {
d__1 = t * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1,
&a[q * a_dim1 + 1], &c__1);
d__1 = -cs * sn * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1,
&a[p * a_dim1 + 1], &c__1);
work[p] /= cs;
work[q] *= cs;
if (rsvec) {
d__1 = t * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1],
&c__1, &v[q * v_dim1 + 1], &
c__1);
d__1 = -cs * sn * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1],
&c__1, &v[p * v_dim1 + 1], &
c__1);
}
}
}
}
}
} else {
/* .. have to use modified Gram-Schmidt like transformation */
dcopy_(m, &a[p * a_dim1 + 1], &c__1, &
work[*n + 1], &c__1);
dlascl_("G", &c__0, &c__0, &aapp, &
c_b18, m, &c__1, &work[*n + 1]
, lda, &ierr);
dlascl_("G", &c__0, &c__0, &aaqq, &
c_b18, m, &c__1, &a[q *
a_dim1 + 1], lda, &ierr);
temp1 = -aapq * work[p] / work[q];
daxpy_(m, &temp1, &work[*n + 1], &
c__1, &a[q * a_dim1 + 1], &
c__1);
dlascl_("G", &c__0, &c__0, &c_b18, &
aaqq, m, &c__1, &a[q * a_dim1
+ 1], lda, &ierr);
/* Computing MAX */
d__1 = 0., d__2 = 1. - aapq * aapq;
sva[q] = aaqq * sqrt((max(d__1,d__2)))
;
mxsinj = max(mxsinj,sfmin);
}
/* END IF ROTOK THEN ... ELSE */
/* In the case of cancellation in updating SVA(q), SVA(p) */
/* recompute SVA(q), SVA(p). */
/* Computing 2nd power */
d__1 = sva[q] / aaqq;
if (d__1 * d__1 <= rooteps) {
if (aaqq < rootbig && aaqq >
rootsfmin) {
sva[q] = dnrm2_(m, &a[q * a_dim1
+ 1], &c__1) * work[q];
} else {
t = 0.;
aaqq = 0.;
dlassq_(m, &a[q * a_dim1 + 1], &
c__1, &t, &aaqq);
sva[q] = t * sqrt(aaqq) * work[q];
}
}
if (aapp / aapp0 <= rooteps) {
if (aapp < rootbig && aapp >
rootsfmin) {
aapp = dnrm2_(m, &a[p * a_dim1 +
1], &c__1) * work[p];
} else {
t = 0.;
aapp = 0.;
dlassq_(m, &a[p * a_dim1 + 1], &
c__1, &t, &aapp);
aapp = t * sqrt(aapp) * work[p];
}
sva[p] = aapp;
}
} else {
/* A(:,p) and A(:,q) already numerically orthogonal */
if (ir1 == 0) {
++notrot;
}
/* [RTD] SKIPPED = SKIPPED + 1 */
++pskipped;
}
} else {
/* A(:,q) is zero column */
if (ir1 == 0) {
++notrot;
}
++pskipped;
}
if (i__ <= swband && pskipped > rowskip) {
if (ir1 == 0) {
aapp = -aapp;
}
notrot = 0;
goto L2103;
}
/* L2002: */
}
/* END q-LOOP */
L2103:
/* bailed out of q-loop */
sva[p] = aapp;
} else {
sva[p] = aapp;
if (ir1 == 0 && aapp == 0.) {
/* Computing MIN */
i__4 = igl + kbl - 1;
notrot = notrot + min(i__4,*n) - p;
}
}
/* L2001: */
}
/* end of the p-loop */
/* end of doing the block ( ibr, ibr ) */
/* L1002: */
}
/* end of ir1-loop */
/* ... go to the off diagonal blocks */
igl = (ibr - 1) * kbl + 1;
i__2 = nbl;
for (jbc = ibr + 1; jbc <= i__2; ++jbc) {
jgl = (jbc - 1) * kbl + 1;
/* doing the block at ( ibr, jbc ) */
ijblsk = 0;
/* Computing MIN */
i__4 = igl + kbl - 1;
i__3 = min(i__4,*n);
for (p = igl; p <= i__3; ++p) {
aapp = sva[p];
if (aapp > 0.) {
pskipped = 0;
/* Computing MIN */
i__5 = jgl + kbl - 1;
i__4 = min(i__5,*n);
for (q = jgl; q <= i__4; ++q) {
aaqq = sva[q];
if (aaqq > 0.) {
aapp0 = aapp;
/* -#- M x 2 Jacobi SVD -#- */
/* Safe Gram matrix computation */
if (aaqq >= 1.) {
if (aapp >= aaqq) {
rotok = small * aapp <= aaqq;
} else {
rotok = small * aaqq <= aapp;
}
if (aapp < big / aaqq) {
aapq = ddot_(m, &a[p * a_dim1 + 1], &
c__1, &a[q * a_dim1 + 1], &
c__1) * work[p] * work[q] /
aaqq / aapp;
} else {
dcopy_(m, &a[p * a_dim1 + 1], &c__1, &
work[*n + 1], &c__1);
dlascl_("G", &c__0, &c__0, &aapp, &
work[p], m, &c__1, &work[*n +
1], lda, &ierr);
aapq = ddot_(m, &work[*n + 1], &c__1,
&a[q * a_dim1 + 1], &c__1) *
work[q] / aaqq;
}
} else {
if (aapp >= aaqq) {
rotok = aapp <= aaqq / small;
} else {
rotok = aaqq <= aapp / small;
}
if (aapp > small / aaqq) {
aapq = ddot_(m, &a[p * a_dim1 + 1], &
c__1, &a[q * a_dim1 + 1], &
c__1) * work[p] * work[q] /
aaqq / aapp;
} else {
dcopy_(m, &a[q * a_dim1 + 1], &c__1, &
work[*n + 1], &c__1);
dlascl_("G", &c__0, &c__0, &aaqq, &
work[q], m, &c__1, &work[*n +
1], lda, &ierr);
aapq = ddot_(m, &work[*n + 1], &c__1,
&a[p * a_dim1 + 1], &c__1) *
work[p] / aapp;
}
}
/* Computing MAX */
d__1 = mxaapq, d__2 = abs(aapq);
mxaapq = max(d__1,d__2);
/* TO rotate or NOT to rotate, THAT is the question ... */
if (abs(aapq) > tol) {
notrot = 0;
/* [RTD] ROTATED = ROTATED + 1 */
pskipped = 0;
++iswrot;
if (rotok) {
aqoap = aaqq / aapp;
apoaq = aapp / aaqq;
theta = (d__1 = aqoap - apoaq, abs(
d__1)) * -.5 / aapq;
if (aaqq > aapp0) {
theta = -theta;
}
if (abs(theta) > bigtheta) {
t = .5 / theta;
fastr[2] = t * work[p] / work[q];
fastr[3] = -t * work[q] / work[p];
drotm_(m, &a[p * a_dim1 + 1], &
c__1, &a[q * a_dim1 + 1],
&c__1, fastr);
if (rsvec) {
drotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q *
v_dim1 + 1], &c__1, fastr);
}
/* Computing MAX */
d__1 = 0., d__2 = t * apoaq *
aapq + 1.;
sva[q] = aaqq * sqrt((max(d__1,
d__2)));
/* Computing MAX */
d__1 = 0., d__2 = 1. - t * aqoap *
aapq;
aapp *= sqrt((max(d__1,d__2)));
/* Computing MAX */
d__1 = mxsinj, d__2 = abs(t);
mxsinj = max(d__1,d__2);
} else {
/* .. choose correct signum for THETA and rotate */
thsign = -d_sign(&c_b18, &aapq);
if (aaqq > aapp0) {
thsign = -thsign;
}
t = 1. / (theta + thsign * sqrt(
theta * theta + 1.));
cs = sqrt(1. / (t * t + 1.));
sn = t * cs;
/* Computing MAX */
d__1 = mxsinj, d__2 = abs(sn);
mxsinj = max(d__1,d__2);
/* Computing MAX */
d__1 = 0., d__2 = t * apoaq *
aapq + 1.;
sva[q] = aaqq * sqrt((max(d__1,
d__2)));
aapp *= sqrt(1. - t * aqoap *
aapq);
apoaq = work[p] / work[q];
aqoap = work[q] / work[p];
if (work[p] >= 1.) {
if (work[q] >= 1.) {
fastr[2] = t * apoaq;
fastr[3] = -t * aqoap;
work[p] *= cs;
work[q] *= cs;
drotm_(m, &a[p * a_dim1 + 1], &c__1, &a[q *
a_dim1 + 1], &c__1, fastr);
if (rsvec) {
drotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[
q * v_dim1 + 1], &c__1, fastr);
}
} else {
d__1 = -t * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1, &a[
p * a_dim1 + 1], &c__1);
d__1 = cs * sn * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1, &a[
q * a_dim1 + 1], &c__1);
if (rsvec) {
d__1 = -t * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1], &
c__1, &v[p * v_dim1 + 1], &c__1);
d__1 = cs * sn * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1], &
c__1, &v[q * v_dim1 + 1], &c__1);
}
work[p] *= cs;
work[q] /= cs;
}
} else {
if (work[q] >= 1.) {
d__1 = t * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1, &a[
q * a_dim1 + 1], &c__1);
d__1 = -cs * sn * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1, &a[
p * a_dim1 + 1], &c__1);
if (rsvec) {
d__1 = t * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1], &
c__1, &v[q * v_dim1 + 1], &c__1);
d__1 = -cs * sn * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1], &
c__1, &v[p * v_dim1 + 1], &c__1);
}
work[p] /= cs;
work[q] *= cs;
} else {
if (work[p] >= work[q]) {
d__1 = -t * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1,
&a[p * a_dim1 + 1], &c__1);
d__1 = cs * sn * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1,
&a[q * a_dim1 + 1], &c__1);
work[p] *= cs;
work[q] /= cs;
if (rsvec) {
d__1 = -t * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1],
&c__1, &v[p * v_dim1 + 1], &
c__1);
d__1 = cs * sn * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1],
&c__1, &v[q * v_dim1 + 1], &
c__1);
}
} else {
d__1 = t * apoaq;
daxpy_(m, &d__1, &a[p * a_dim1 + 1], &c__1,
&a[q * a_dim1 + 1], &c__1);
d__1 = -cs * sn * aqoap;
daxpy_(m, &d__1, &a[q * a_dim1 + 1], &c__1,
&a[p * a_dim1 + 1], &c__1);
work[p] /= cs;
work[q] *= cs;
if (rsvec) {
d__1 = t * apoaq;
daxpy_(&mvl, &d__1, &v[p * v_dim1 + 1],
&c__1, &v[q * v_dim1 + 1], &
c__1);
d__1 = -cs * sn * aqoap;
daxpy_(&mvl, &d__1, &v[q * v_dim1 + 1],
&c__1, &v[p * v_dim1 + 1], &
c__1);
}
}
}
}
}
} else {
if (aapp > aaqq) {
dcopy_(m, &a[p * a_dim1 + 1], &
c__1, &work[*n + 1], &
c__1);
dlascl_("G", &c__0, &c__0, &aapp,
&c_b18, m, &c__1, &work[*
n + 1], lda, &ierr);
dlascl_("G", &c__0, &c__0, &aaqq,
&c_b18, m, &c__1, &a[q *
a_dim1 + 1], lda, &ierr);
temp1 = -aapq * work[p] / work[q];
daxpy_(m, &temp1, &work[*n + 1], &
c__1, &a[q * a_dim1 + 1],
&c__1);
dlascl_("G", &c__0, &c__0, &c_b18,
&aaqq, m, &c__1, &a[q *
a_dim1 + 1], lda, &ierr);
/* Computing MAX */
d__1 = 0., d__2 = 1. - aapq *
aapq;
sva[q] = aaqq * sqrt((max(d__1,
d__2)));
mxsinj = max(mxsinj,sfmin);
} else {
dcopy_(m, &a[q * a_dim1 + 1], &
c__1, &work[*n + 1], &
c__1);
dlascl_("G", &c__0, &c__0, &aaqq,
&c_b18, m, &c__1, &work[*
n + 1], lda, &ierr);
dlascl_("G", &c__0, &c__0, &aapp,
&c_b18, m, &c__1, &a[p *
a_dim1 + 1], lda, &ierr);
temp1 = -aapq * work[q] / work[p];
daxpy_(m, &temp1, &work[*n + 1], &
c__1, &a[p * a_dim1 + 1],
&c__1);
dlascl_("G", &c__0, &c__0, &c_b18,
&aapp, m, &c__1, &a[p *
a_dim1 + 1], lda, &ierr);
/* Computing MAX */
d__1 = 0., d__2 = 1. - aapq *
aapq;
sva[p] = aapp * sqrt((max(d__1,
d__2)));
mxsinj = max(mxsinj,sfmin);
}
}
/* END IF ROTOK THEN ... ELSE */
/* In the case of cancellation in updating SVA(q) */
/* .. recompute SVA(q) */
/* Computing 2nd power */
d__1 = sva[q] / aaqq;
if (d__1 * d__1 <= rooteps) {
if (aaqq < rootbig && aaqq >
rootsfmin) {
sva[q] = dnrm2_(m, &a[q * a_dim1
+ 1], &c__1) * work[q];
} else {
t = 0.;
aaqq = 0.;
dlassq_(m, &a[q * a_dim1 + 1], &
c__1, &t, &aaqq);
sva[q] = t * sqrt(aaqq) * work[q];
}
}
/* Computing 2nd power */
d__1 = aapp / aapp0;
if (d__1 * d__1 <= rooteps) {
if (aapp < rootbig && aapp >
rootsfmin) {
aapp = dnrm2_(m, &a[p * a_dim1 +
1], &c__1) * work[p];
} else {
t = 0.;
aapp = 0.;
dlassq_(m, &a[p * a_dim1 + 1], &
c__1, &t, &aapp);
aapp = t * sqrt(aapp) * work[p];
}
sva[p] = aapp;
}
/* end of OK rotation */
} else {
++notrot;
/* [RTD] SKIPPED = SKIPPED + 1 */
++pskipped;
++ijblsk;
}
} else {
++notrot;
++pskipped;
++ijblsk;
}
if (i__ <= swband && ijblsk >= blskip) {
sva[p] = aapp;
notrot = 0;
goto L2011;
}
if (i__ <= swband && pskipped > rowskip) {
aapp = -aapp;
notrot = 0;
goto L2203;
}
/* L2200: */
}
/* end of the q-loop */
L2203:
sva[p] = aapp;
} else {
if (aapp == 0.) {
/* Computing MIN */
i__4 = jgl + kbl - 1;
notrot = notrot + min(i__4,*n) - jgl + 1;
}
if (aapp < 0.) {
notrot = 0;
}
}
/* L2100: */
}
/* end of the p-loop */
/* L2010: */
}
/* end of the jbc-loop */
L2011:
/* 2011 bailed out of the jbc-loop */
/* Computing MIN */
i__3 = igl + kbl - 1;
i__2 = min(i__3,*n);
for (p = igl; p <= i__2; ++p) {
sva[p] = (d__1 = sva[p], abs(d__1));
/* L2012: */
}
/* ** */
/* L2000: */
}
/* 2000 :: end of the ibr-loop */
/* .. update SVA(N) */
if (sva[*n] < rootbig && sva[*n] > rootsfmin) {
sva[*n] = dnrm2_(m, &a[*n * a_dim1 + 1], &c__1) * work[*n];
} else {
t = 0.;
aapp = 0.;
dlassq_(m, &a[*n * a_dim1 + 1], &c__1, &t, &aapp);
sva[*n] = t * sqrt(aapp) * work[*n];
}
/* Additional steering devices */
if (i__ < swband && (mxaapq <= roottol || iswrot <= *n)) {
swband = i__;
}
if (i__ > swband + 1 && mxaapq < sqrt((doublereal) (*n)) * tol && (
doublereal) (*n) * mxaapq * mxsinj < tol) {
goto L1994;
}
if (notrot >= emptsw) {
goto L1994;
}
/* L1993: */
}
/* end i=1:NSWEEP loop */
/* #:( Reaching this point means that the procedure has not converged. */
*info = 29;
goto L1995;
L1994:
/* #:) Reaching this point means numerical convergence after the i-th */
/* sweep. */
*info = 0;
/* #:) INFO = 0 confirms successful iterations. */
L1995:
/* Sort the singular values and find how many are above */
/* the underflow threshold. */
n2 = 0;
n4 = 0;
i__1 = *n - 1;
for (p = 1; p <= i__1; ++p) {
i__2 = *n - p + 1;
q = idamax_(&i__2, &sva[p], &c__1) + p - 1;
if (p != q) {
temp1 = sva[p];
sva[p] = sva[q];
sva[q] = temp1;
temp1 = work[p];
work[p] = work[q];
work[q] = temp1;
dswap_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1);
if (rsvec) {
dswap_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], &
c__1);
}
}
if (sva[p] != 0.) {
++n4;
if (sva[p] * scale > sfmin) {
++n2;
}
}
/* L5991: */
}
if (sva[*n] != 0.) {
++n4;
if (sva[*n] * scale > sfmin) {
++n2;
}
}
/* Normalize the left singular vectors. */
if (lsvec || uctol) {
i__1 = n2;
for (p = 1; p <= i__1; ++p) {
d__1 = work[p] / sva[p];
dscal_(m, &d__1, &a[p * a_dim1 + 1], &c__1);
/* L1998: */
}
}
/* Scale the product of Jacobi rotations (assemble the fast rotations). */
if (rsvec) {
if (applv) {
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
dscal_(&mvl, &work[p], &v[p * v_dim1 + 1], &c__1);
/* L2398: */
}
} else {
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
temp1 = 1. / dnrm2_(&mvl, &v[p * v_dim1 + 1], &c__1);
dscal_(&mvl, &temp1, &v[p * v_dim1 + 1], &c__1);
/* L2399: */
}
}
}
/* Undo scaling, if necessary (and possible). */
if (scale > 1. && sva[1] < big / scale || scale < 1. && sva[n2] > sfmin /
scale) {
i__1 = *n;
for (p = 1; p <= i__1; ++p) {
sva[p] = scale * sva[p];
/* L2400: */
}
scale = 1.;
}
work[1] = scale;
/* The singular values of A are SCALE*SVA(1:N). If SCALE.NE.ONE */
/* then some of the singular values may overflow or underflow and */
/* the spectrum is given in this factored representation. */
work[2] = (doublereal) n4;
/* N4 is the number of computed nonzero singular values of A. */
work[3] = (doublereal) n2;
/* N2 is the number of singular values of A greater than SFMIN. */
/* If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers */
/* that may carry some information. */
work[4] = (doublereal) i__;
/* i is the index of the last sweep before declaring convergence. */
work[5] = mxaapq;
/* MXAAPQ is the largest absolute value of scaled pivots in the */
/* last sweep */
work[6] = mxsinj;
/* MXSINJ is the largest absolute value of the sines of Jacobi angles */
/* in the last sweep */
return 0;
/* .. */
/* .. END OF DGESVJ */
/* .. */
} /* dgesvj_ */
