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references/igraph-1.0.1/vendor/lapack/dlarre.c
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/* -- translated by f2c (version 20240504).
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"
/* Table of constant values */
static integer c__1 = 1;
static integer c__2 = 2;
/* > \brief \b DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each un
reduced block Ti, finds base representations and eigenvalues.
=========== DOCUMENTATION ===========
Online html documentation available at
http://www.netlib.org/lapack/explore-html/
> \htmlonly
> Download DLARRE + dependencies
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarre.
f">
> [TGZ]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarre.
f">
> [ZIP]</a>
> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarre.
f">
> [TXT]</a>
> \endhtmlonly
Definition:
===========
SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
WORK, IWORK, INFO )
CHARACTER RANGE
INTEGER IL, INFO, IU, M, N, NSPLIT
DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ),
$ INDEXW( * )
DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ),
$ W( * ),WERR( * ), WGAP( * ), WORK( * )
> \par Purpose:
=============
>
> \verbatim
>
> To find the desired eigenvalues of a given real symmetric
> tridiagonal matrix T, DLARRE sets any "small" off-diagonal
> elements to zero, and for each unreduced block T_i, it finds
> (a) a suitable shift at one end of the block's spectrum,
> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
> (c) eigenvalues of each L_i D_i L_i^T.
> The representations and eigenvalues found are then used by
> DSTEMR to compute the eigenvectors of T.
> The accuracy varies depending on whether bisection is used to
> find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
> conpute all and then discard any unwanted one.
> As an added benefit, DLARRE also outputs the n
> Gerschgorin intervals for the matrices L_i D_i L_i^T.
> \endverbatim
Arguments:
==========
> \param[in] RANGE
> \verbatim
> RANGE is CHARACTER*1
> = 'A': ("All") all eigenvalues will be found.
> = 'V': ("Value") all eigenvalues in the half-open interval
> (VL, VU] will be found.
> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
> entire matrix) will be found.
> \endverbatim
>
> \param[in] N
> \verbatim
> N is INTEGER
> The order of the matrix. N > 0.
> \endverbatim
>
> \param[in,out] VL
> \verbatim
> VL is DOUBLE PRECISION
> \endverbatim
>
> \param[in,out] VU
> \verbatim
> VU is DOUBLE PRECISION
> If RANGE='V', the lower and upper bounds for the eigenvalues.
> Eigenvalues less than or equal to VL, or greater than VU,
> will not be returned. VL < VU.
> If RANGE='I' or ='A', DLARRE computes bounds on the desired
> part of the spectrum.
> \endverbatim
>
> \param[in] IL
> \verbatim
> IL is INTEGER
> \endverbatim
>
> \param[in] IU
> \verbatim
> IU is INTEGER
> If RANGE='I', the indices (in ascending order) of the
> smallest and largest eigenvalues to be returned.
> 1 <= IL <= IU <= N.
> \endverbatim
>
> \param[in,out] D
> \verbatim
> D is DOUBLE PRECISION array, dimension (N)
> On entry, the N diagonal elements of the tridiagonal
> matrix T.
> On exit, the N diagonal elements of the diagonal
> matrices D_i.
> \endverbatim
>
> \param[in,out] E
> \verbatim
> E is DOUBLE PRECISION array, dimension (N)
> On entry, the first (N-1) entries contain the subdiagonal
> elements of the tridiagonal matrix T; E(N) need not be set.
> On exit, E contains the subdiagonal elements of the unit
> bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
> 1 <= I <= NSPLIT, contain the base points sigma_i on output.
> \endverbatim
>
> \param[in,out] E2
> \verbatim
> E2 is DOUBLE PRECISION array, dimension (N)
> On entry, the first (N-1) entries contain the SQUARES of the
> subdiagonal elements of the tridiagonal matrix T;
> E2(N) need not be set.
> On exit, the entries E2( ISPLIT( I ) ),
> 1 <= I <= NSPLIT, have been set to zero
> \endverbatim
>
> \param[in] RTOL1
> \verbatim
> RTOL1 is DOUBLE PRECISION
> \endverbatim
>
> \param[in] RTOL2
> \verbatim
> RTOL2 is DOUBLE PRECISION
> Parameters for bisection.
> An interval [LEFT,RIGHT] has converged if
> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
> \endverbatim
>
> \param[in] SPLTOL
> \verbatim
> SPLTOL is DOUBLE PRECISION
> The threshold for splitting.
> \endverbatim
>
> \param[out] NSPLIT
> \verbatim
> NSPLIT is INTEGER
> The number of blocks T splits into. 1 <= NSPLIT <= N.
> \endverbatim
>
> \param[out] ISPLIT
> \verbatim
> ISPLIT is INTEGER array, dimension (N)
> The splitting points, at which T breaks up into blocks.
> The first block consists of rows/columns 1 to ISPLIT(1),
> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
> etc., and the NSPLIT-th consists of rows/columns
> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
> \endverbatim
>
> \param[out] M
> \verbatim
> M is INTEGER
> The total number of eigenvalues (of all L_i D_i L_i^T)
> found.
> \endverbatim
>
> \param[out] W
> \verbatim
> W is DOUBLE PRECISION array, dimension (N)
> The first M elements contain the eigenvalues. The
> eigenvalues of each of the blocks, L_i D_i L_i^T, are
> sorted in ascending order ( DLARRE may use the
> remaining N-M elements as workspace).
> \endverbatim
>
> \param[out] WERR
> \verbatim
> WERR is DOUBLE PRECISION array, dimension (N)
> The error bound on the corresponding eigenvalue in W.
> \endverbatim
>
> \param[out] WGAP
> \verbatim
> WGAP is DOUBLE PRECISION array, dimension (N)
> The separation from the right neighbor eigenvalue in W.
> The gap is only with respect to the eigenvalues of the same block
> as each block has its own representation tree.
> Exception: at the right end of a block we store the left gap
> \endverbatim
>
> \param[out] IBLOCK
> \verbatim
> IBLOCK is INTEGER array, dimension (N)
> The indices of the blocks (submatrices) associated with the
> corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
> W(i) belongs to the first block from the top, =2 if W(i)
> belongs to the second block, etc.
> \endverbatim
>
> \param[out] INDEXW
> \verbatim
> INDEXW is INTEGER array, dimension (N)
> The indices of the eigenvalues within each block (submatrix);
> for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
> i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
> \endverbatim
>
> \param[out] GERS
> \verbatim
> GERS is DOUBLE PRECISION array, dimension (2*N)
> The N Gerschgorin intervals (the i-th Gerschgorin interval
> is (GERS(2*i-1), GERS(2*i)).
> \endverbatim
>
> \param[out] PIVMIN
> \verbatim
> PIVMIN is DOUBLE PRECISION
> The minimum pivot in the Sturm sequence for T.
> \endverbatim
>
> \param[out] WORK
> \verbatim
> WORK is DOUBLE PRECISION array, dimension (6*N)
> Workspace.
> \endverbatim
>
> \param[out] IWORK
> \verbatim
> IWORK is INTEGER array, dimension (5*N)
> Workspace.
> \endverbatim
>
> \param[out] INFO
> \verbatim
> INFO is INTEGER
> = 0: successful exit
> > 0: A problem occured in DLARRE.
> < 0: One of the called subroutines signaled an internal problem.
> Needs inspection of the corresponding parameter IINFO
> for further information.
>
> =-1: Problem in DLARRD.
> = 2: No base representation could be found in MAXTRY iterations.
> Increasing MAXTRY and recompilation might be a remedy.
> =-3: Problem in DLARRB when computing the refined root
> representation for DLASQ2.
> =-4: Problem in DLARRB when preforming bisection on the
> desired part of the spectrum.
> =-5: Problem in DLASQ2.
> =-6: Problem in DLASQ2.
> \endverbatim
Authors:
========
> \author Univ. of Tennessee
> \author Univ. of California Berkeley
> \author Univ. of Colorado Denver
> \author NAG Ltd.
> \date September 2012
> \ingroup auxOTHERauxiliary
> \par Further Details:
=====================
>
> \verbatim
>
> The base representations are required to suffer very little
> element growth and consequently define all their eigenvalues to
> high relative accuracy.
> \endverbatim
> \par Contributors:
==================
>
> Beresford Parlett, University of California, Berkeley, USA \n
> Jim Demmel, University of California, Berkeley, USA \n
> Inderjit Dhillon, University of Texas, Austin, USA \n
> Osni Marques, LBNL/NERSC, USA \n
> Christof Voemel, University of California, Berkeley, USA \n
>
=====================================================================
Subroutine */ int igraphdlarre_(char *range, integer *n, doublereal *vl,
doublereal *vu, integer *il, integer *iu, doublereal *d__, doublereal
*e, doublereal *e2, doublereal *rtol1, doublereal *rtol2, doublereal *
spltol, integer *nsplit, integer *isplit, integer *m, doublereal *w,
doublereal *werr, doublereal *wgap, integer *iblock, integer *indexw,
doublereal *gers, doublereal *pivmin, doublereal *work, integer *
iwork, integer *info)
{
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal), log(doublereal);
/* Local variables */
integer i__, j;
doublereal s1, s2;
integer mb;
doublereal gl;
integer in, mm;
doublereal gu;
integer cnt;
doublereal eps, tau, tmp, rtl;
integer cnt1, cnt2;
doublereal tmp1, eabs;
integer iend, jblk;
doublereal eold;
integer indl;
doublereal dmax__, emax;
integer wend, idum, indu;
doublereal rtol;
integer iseed[4];
doublereal avgap, sigma;
extern logical igraphlsame_(char *, char *);
integer iinfo;
extern /* Subroutine */ int igraphdcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
logical norep;
extern /* Subroutine */ int igraphdlasq2_(integer *, doublereal *, integer *);
extern doublereal igraphdlamch_(char *);
integer ibegin;
logical forceb;
integer irange;
doublereal sgndef;
extern /* Subroutine */ int igraphdlarra_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
integer *), igraphdlarrb_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, integer *, integer *), igraphdlarrc_(char *
, integer *, doublereal *, doublereal *, doublereal *, doublereal
*, doublereal *, integer *, integer *, integer *, integer *);
integer wbegin;
extern /* Subroutine */ int igraphdlarrd_(char *, char *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *
, integer *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
doublereal safmin, spdiam;
extern /* Subroutine */ int igraphdlarrk_(integer *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *);
logical usedqd;
doublereal clwdth, isleft;
extern /* Subroutine */ int igraphdlarnv_(integer *, integer *, integer *,
doublereal *);
doublereal isrght, bsrtol, dpivot;
/* -- LAPACK auxiliary routine (version 3.4.2) --
-- LAPACK is a software package provided by Univ. of Tennessee, --
-- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
September 2012
=====================================================================
Parameter adjustments */
--iwork;
--work;
--gers;
--indexw;
--iblock;
--wgap;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
/* Function Body */
*info = 0;
/* Decode RANGE */
if (igraphlsame_(range, "A")) {
irange = 1;
} else if (igraphlsame_(range, "V")) {
irange = 3;
} else if (igraphlsame_(range, "I")) {
irange = 2;
}
*m = 0;
/* Get machine constants */
safmin = igraphdlamch_("S");
eps = igraphdlamch_("P");
/* Set parameters */
rtl = sqrt(eps);
bsrtol = sqrt(eps);
/* Treat case of 1x1 matrix for quick return */
if (*n == 1) {
if (irange == 1 || irange == 3 && d__[1] > *vl && d__[1] <= *vu ||
irange == 2 && *il == 1 && *iu == 1) {
*m = 1;
w[1] = d__[1];
/* The computation error of the eigenvalue is zero */
werr[1] = 0.;
wgap[1] = 0.;
iblock[1] = 1;
indexw[1] = 1;
gers[1] = d__[1];
gers[2] = d__[1];
}
/* store the shift for the initial RRR, which is zero in this case */
e[1] = 0.;
return 0;
}
/* General case: tridiagonal matrix of order > 1
Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter.
Compute maximum off-diagonal entry and pivmin. */
gl = d__[1];
gu = d__[1];
eold = 0.;
emax = 0.;
e[*n] = 0.;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
werr[i__] = 0.;
wgap[i__] = 0.;
eabs = (d__1 = e[i__], abs(d__1));
if (eabs >= emax) {
emax = eabs;
}
tmp1 = eabs + eold;
gers[(i__ << 1) - 1] = d__[i__] - tmp1;
/* Computing MIN */
d__1 = gl, d__2 = gers[(i__ << 1) - 1];
gl = min(d__1,d__2);
gers[i__ * 2] = d__[i__] + tmp1;
/* Computing MAX */
d__1 = gu, d__2 = gers[i__ * 2];
gu = max(d__1,d__2);
eold = eabs;
/* L5: */
}
/* The minimum pivot allowed in the Sturm sequence for T
Computing MAX
Computing 2nd power */
d__3 = emax;
d__1 = 1., d__2 = d__3 * d__3;
*pivmin = safmin * max(d__1,d__2);
/* Compute spectral diameter. The Gerschgorin bounds give an
estimate that is wrong by at most a factor of SQRT(2) */
spdiam = gu - gl;
/* Compute splitting points */
igraphdlarra_(n, &d__[1], &e[1], &e2[1], spltol, &spdiam, nsplit, &isplit[1], &
iinfo);
/* Can force use of bisection instead of faster DQDS.
Option left in the code for future multisection work. */
forceb = FALSE_;
/* Initialize USEDQD, DQDS should be used for ALLRNG unless someone
explicitly wants bisection. */
usedqd = irange == 1 && ! forceb;
if (irange == 1 && ! forceb) {
/* Set interval [VL,VU] that contains all eigenvalues */
*vl = gl;
*vu = gu;
} else {
/* We call DLARRD to find crude approximations to the eigenvalues
in the desired range. In case IRANGE = INDRNG, we also obtain the
interval (VL,VU] that contains all the wanted eigenvalues.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT))
DLARRD needs a WORK of size 4*N, IWORK of size 3*N */
igraphdlarrd_(range, "B", n, vl, vu, il, iu, &gers[1], &bsrtol, &d__[1], &e[
1], &e2[1], pivmin, nsplit, &isplit[1], &mm, &w[1], &werr[1],
vl, vu, &iblock[1], &indexw[1], &work[1], &iwork[1], &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0 */
i__1 = *n;
for (i__ = mm + 1; i__ <= i__1; ++i__) {
w[i__] = 0.;
werr[i__] = 0.;
iblock[i__] = 0;
indexw[i__] = 0;
/* L14: */
}
}
/* **
Loop over unreduced blocks */
ibegin = 1;
wbegin = 1;
i__1 = *nsplit;
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = isplit[jblk];
in = iend - ibegin + 1;
/* 1 X 1 block */
if (in == 1) {
if (irange == 1 || irange == 3 && d__[ibegin] > *vl && d__[ibegin]
<= *vu || irange == 2 && iblock[wbegin] == jblk) {
++(*m);
w[*m] = d__[ibegin];
werr[*m] = 0.;
/* The gap for a single block doesn't matter for the later
algorithm and is assigned an arbitrary large value */
wgap[*m] = 0.;
iblock[*m] = jblk;
indexw[*m] = 1;
++wbegin;
}
/* E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.;
ibegin = iend + 1;
goto L170;
}
/* Blocks of size larger than 1x1
E( IEND ) will hold the shift for the initial RRR, for now set it =0 */
e[iend] = 0.;
/* Find local outer bounds GL,GU for the block */
gl = d__[ibegin];
gu = d__[ibegin];
i__2 = iend;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing MIN */
d__1 = gers[(i__ << 1) - 1];
gl = min(d__1,gl);
/* Computing MAX */
d__1 = gers[i__ * 2];
gu = max(d__1,gu);
/* L15: */
}
spdiam = gu - gl;
if (! (irange == 1 && ! forceb)) {
/* Count the number of eigenvalues in the current block. */
mb = 0;
i__2 = mm;
for (i__ = wbegin; i__ <= i__2; ++i__) {
if (iblock[i__] == jblk) {
++mb;
} else {
goto L21;
}
/* L20: */
}
L21:
if (mb == 0) {
/* No eigenvalue in the current block lies in the desired range
E( IEND ) holds the shift for the initial RRR */
e[iend] = 0.;
ibegin = iend + 1;
goto L170;
} else {
/* Decide whether dqds or bisection is more efficient */
usedqd = (doublereal) mb > in * .5 && ! forceb;
wend = wbegin + mb - 1;
/* Calculate gaps for the current block
In later stages, when representations for individual
eigenvalues are different, we use SIGMA = E( IEND ). */
sigma = 0.;
i__2 = wend - 1;
for (i__ = wbegin; i__ <= i__2; ++i__) {
/* Computing MAX */
d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
werr[i__]);
wgap[i__] = max(d__1,d__2);
/* L30: */
}
/* Computing MAX */
d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = max(d__1,d__2);
/* Find local index of the first and last desired evalue. */
indl = indexw[wbegin];
indu = indexw[wend];
}
}
if (irange == 1 && ! forceb || usedqd) {
/* Case of DQDS
Find approximations to the extremal eigenvalues of the block */
igraphdlarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Computing MAX */
d__2 = gl, d__3 = tmp - tmp1 - eps * 100. * (d__1 = tmp - tmp1,
abs(d__1));
isleft = max(d__2,d__3);
igraphdlarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
rtl, &tmp, &tmp1, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* Computing MIN */
d__2 = gu, d__3 = tmp + tmp1 + eps * 100. * (d__1 = tmp + tmp1,
abs(d__1));
isrght = min(d__2,d__3);
/* Improve the estimate of the spectral diameter */
spdiam = isrght - isleft;
} else {
/* Case of bisection
Find approximations to the wanted extremal eigenvalues
Computing MAX */
d__2 = gl, d__3 = w[wbegin] - werr[wbegin] - eps * 100. * (d__1 =
w[wbegin] - werr[wbegin], abs(d__1));
isleft = max(d__2,d__3);
/* Computing MIN */
d__2 = gu, d__3 = w[wend] + werr[wend] + eps * 100. * (d__1 = w[
wend] + werr[wend], abs(d__1));
isrght = min(d__2,d__3);
}
/* Decide whether the base representation for the current block
L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I
should be on the left or the right end of the current block.
The strategy is to shift to the end which is "more populated"
Furthermore, decide whether to use DQDS for the computation of
the eigenvalue approximations at the end of DLARRE or bisection.
dqds is chosen if all eigenvalues are desired or the number of
eigenvalues to be computed is large compared to the blocksize. */
if (irange == 1 && ! forceb) {
/* If all the eigenvalues have to be computed, we use dqd */
usedqd = TRUE_;
/* INDL is the local index of the first eigenvalue to compute */
indl = 1;
indu = in;
/* MB = number of eigenvalues to compute */
mb = in;
wend = wbegin + mb - 1;
/* Define 1/4 and 3/4 points of the spectrum */
s1 = isleft + spdiam * .25;
s2 = isrght - spdiam * .25;
} else {
/* DLARRD has computed IBLOCK and INDEXW for each eigenvalue
approximation.
choose sigma */
if (usedqd) {
s1 = isleft + spdiam * .25;
s2 = isrght - spdiam * .25;
} else {
tmp = min(isrght,*vu) - max(isleft,*vl);
s1 = max(isleft,*vl) + tmp * .25;
s2 = min(isrght,*vu) - tmp * .25;
}
}
/* Compute the negcount at the 1/4 and 3/4 points */
if (mb > 1) {
igraphdlarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
cnt, &cnt1, &cnt2, &iinfo);
}
if (mb == 1) {
sigma = gl;
sgndef = 1.;
} else if (cnt1 - indl >= indu - cnt2) {
if (irange == 1 && ! forceb) {
sigma = max(isleft,gl);
} else if (usedqd) {
/* use Gerschgorin bound as shift to get pos def matrix
for dqds */
sigma = isleft;
} else {
/* use approximation of the first desired eigenvalue of the
block as shift */
sigma = max(isleft,*vl);
}
sgndef = 1.;
} else {
if (irange == 1 && ! forceb) {
sigma = min(isrght,gu);
} else if (usedqd) {
/* use Gerschgorin bound as shift to get neg def matrix
for dqds */
sigma = isrght;
} else {
/* use approximation of the first desired eigenvalue of the
block as shift */
sigma = min(isrght,*vu);
}
sgndef = -1.;
}
/* An initial SIGMA has been chosen that will be used for computing
T - SIGMA I = L D L^T
Define the increment TAU of the shift in case the initial shift
needs to be refined to obtain a factorization with not too much
element growth. */
if (usedqd) {
/* The initial SIGMA was to the outer end of the spectrum
the matrix is definite and we need not retreat. */
tau = spdiam * eps * *n + *pivmin * 2.;
/* Computing MAX */
d__1 = tau, d__2 = eps * 2. * abs(sigma);
tau = max(d__1,d__2);
} else {
if (mb > 1) {
clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
avgap = (d__1 = clwdth / (doublereal) (wend - wbegin), abs(
d__1));
if (sgndef == 1.) {
/* Computing MAX */
d__1 = wgap[wbegin];
tau = max(d__1,avgap) * .5;
/* Computing MAX */
d__1 = tau, d__2 = werr[wbegin];
tau = max(d__1,d__2);
} else {
/* Computing MAX */
d__1 = wgap[wend - 1];
tau = max(d__1,avgap) * .5;
/* Computing MAX */
d__1 = tau, d__2 = werr[wend];
tau = max(d__1,d__2);
}
} else {
tau = werr[wbegin];
}
}
for (idum = 1; idum <= 6; ++idum) {
/* Compute L D L^T factorization of tridiagonal matrix T - sigma I.
Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of
pivots in WORK(2*IN+1:3*IN) */
dpivot = d__[ibegin] - sigma;
work[1] = dpivot;
dmax__ = abs(work[1]);
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(in << 1) + i__] = 1. / work[i__];
tmp = e[j] * work[(in << 1) + i__];
work[in + i__] = tmp;
dpivot = d__[j + 1] - sigma - tmp * e[j];
work[i__ + 1] = dpivot;
/* Computing MAX */
d__1 = dmax__, d__2 = abs(dpivot);
dmax__ = max(d__1,d__2);
++j;
/* L70: */
}
/* check for element growth */
if (dmax__ > spdiam * 64.) {
norep = TRUE_;
} else {
norep = FALSE_;
}
if (usedqd && ! norep) {
/* Ensure the definiteness of the representation
All entries of D (of L D L^T) must have the same sign */
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
tmp = sgndef * work[i__];
if (tmp < 0.) {
norep = TRUE_;
}
/* L71: */
}
}
if (norep) {
/* Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin
shift which makes the matrix definite. So we should end up
here really only in the case of IRANGE = VALRNG or INDRNG. */
if (idum == 5) {
if (sgndef == 1.) {
/* The fudged Gerschgorin shift should succeed */
sigma = gl - spdiam * 2. * eps * *n - *pivmin * 4.;
} else {
sigma = gu + spdiam * 2. * eps * *n + *pivmin * 4.;
}
} else {
sigma -= sgndef * tau;
tau *= 2.;
}
} else {
/* an initial RRR is found */
goto L83;
}
/* L80: */
}
/* if the program reaches this point, no base representation could be
found in MAXTRY iterations. */
*info = 2;
return 0;
L83:
/* At this point, we have found an initial base representation
T - SIGMA I = L D L^T with not too much element growth.
Store the shift. */
e[iend] = sigma;
/* Store D and L. */
igraphdcopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
igraphdcopy_(&i__2, &work[in + 1], &c__1, &e[ibegin], &c__1);
if (mb > 1) {
/* Perturb each entry of the base representation by a small
(but random) relative amount to overcome difficulties with
glued matrices. */
for (i__ = 1; i__ <= 4; ++i__) {
iseed[i__ - 1] = 1;
/* L122: */
}
i__2 = (in << 1) - 1;
igraphdlarnv_(&c__2, iseed, &i__2, &work[1]);
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
d__[ibegin + i__ - 1] *= eps * 8. * work[i__] + 1.;
e[ibegin + i__ - 1] *= eps * 8. * work[in + i__] + 1.;
/* L125: */
}
d__[iend] *= eps * 4. * work[in] + 1.;
}
/* Don't update the Gerschgorin intervals because keeping track
of the updates would be too much work in DLARRV.
We update W instead and use it to locate the proper Gerschgorin
intervals.
Compute the required eigenvalues of L D L' by bisection or dqds */
if (! usedqd) {
/* If DLARRD has been used, shift the eigenvalue approximations
according to their representation. This is necessary for
a uniform DLARRV since dqds computes eigenvalues of the
shifted representation. In DLARRV, W will always hold the
UNshifted eigenvalue approximation. */
i__2 = wend;
for (j = wbegin; j <= i__2; ++j) {
w[j] -= sigma;
werr[j] += (d__1 = w[j], abs(d__1)) * eps;
/* L134: */
}
/* call DLARRB to reduce eigenvalue error of the approximations
from DLARRD */
i__2 = iend - 1;
for (i__ = ibegin; i__ <= i__2; ++i__) {
/* Computing 2nd power */
d__1 = e[i__];
work[i__] = d__[i__] * (d__1 * d__1);
/* L135: */
}
/* use bisection to find EV from INDL to INDU */
i__2 = indl - 1;
igraphdlarrb_(&in, &d__[ibegin], &work[ibegin], &indl, &indu, rtol1,
rtol2, &i__2, &w[wbegin], &wgap[wbegin], &werr[wbegin], &
work[(*n << 1) + 1], &iwork[1], pivmin, &spdiam, &in, &
iinfo);
if (iinfo != 0) {
*info = -4;
return 0;
}
/* DLARRB computes all gaps correctly except for the last one
Record distance to VU/GU
Computing MAX */
d__1 = 0., d__2 = *vu - sigma - (w[wend] + werr[wend]);
wgap[wend] = max(d__1,d__2);
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
iblock[*m] = jblk;
indexw[*m] = i__;
/* L138: */
}
} else {
/* Call dqds to get all eigs (and then possibly delete unwanted
eigenvalues).
Note that dqds finds the eigenvalues of the L D L^T representation
of T to high relative accuracy. High relative accuracy
might be lost when the shift of the RRR is subtracted to obtain
the eigenvalues of T. However, T is not guaranteed to define its
eigenvalues to high relative accuracy anyway.
Set RTOL to the order of the tolerance used in DLASQ2
This is an ESTIMATED error, the worst case bound is 4*N*EPS
which is usually too large and requires unnecessary work to be
done by bisection when computing the eigenvectors */
rtol = log((doublereal) in) * 4. * eps;
j = ibegin;
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[(i__ << 1) - 1] = (d__1 = d__[j], abs(d__1));
work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
++j;
/* L140: */
}
work[(in << 1) - 1] = (d__1 = d__[iend], abs(d__1));
work[in * 2] = 0.;
igraphdlasq2_(&in, &work[1], &iinfo);
if (iinfo != 0) {
/* If IINFO = -5 then an index is part of a tight cluster
and should be changed. The index is in IWORK(1) and the
gap is in WORK(N+1) */
*info = -5;
return 0;
} else {
/* Test that all eigenvalues are positive as expected */
i__2 = in;
for (i__ = 1; i__ <= i__2; ++i__) {
if (work[i__] < 0.) {
*info = -6;
return 0;
}
/* L149: */
}
}
if (sgndef > 0.) {
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
w[*m] = work[in - i__ + 1];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L150: */
}
} else {
i__2 = indu;
for (i__ = indl; i__ <= i__2; ++i__) {
++(*m);
w[*m] = -work[i__];
iblock[*m] = jblk;
indexw[*m] = i__;
/* L160: */
}
}
i__2 = *m;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* the value of RTOL below should be the tolerance in DLASQ2 */
werr[i__] = rtol * (d__1 = w[i__], abs(d__1));
/* L165: */
}
i__2 = *m - 1;
for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
/* compute the right gap between the intervals
Computing MAX */
d__1 = 0., d__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] + werr[
i__]);
wgap[i__] = max(d__1,d__2);
/* L166: */
}
/* Computing MAX */
d__1 = 0., d__2 = *vu - sigma - (w[*m] + werr[*m]);
wgap[*m] = max(d__1,d__2);
}
/* proceed with next block */
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return 0;
/* end of DLARRE */
} /* igraphdlarre_ */