#include "blaswrap.h"
/* dlarrv.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"
/* Table of constant values */
static doublereal c_b5 = 0.;
static integer c__1 = 1;
static integer c__2 = 2;
/* Subroutine */ int dlarrv_(integer *n, doublereal *vl, doublereal *vu,
doublereal *d__, doublereal *l, doublereal *pivmin, integer *isplit,
integer *m, integer *dol, integer *dou, doublereal *minrgp,
doublereal *rtol1, doublereal *rtol2, doublereal *w, doublereal *werr,
doublereal *wgap, integer *iblock, integer *indexw, doublereal *gers,
doublereal *z__, integer *ldz, integer *isuppz, doublereal *work,
integer *iwork, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2;
logical L__1;
/* Builtin functions */
double log(doublereal);
/* Local variables */
static integer minwsize, i__, j, k, p, q, miniwsize, ii;
static doublereal gl;
static integer im, in;
static doublereal gu, gap, eps, tau, tol, tmp;
static integer zto;
static doublereal ztz;
static integer iend, jblk;
static doublereal lgap;
static integer done;
static doublereal rgap, left;
static integer wend, iter;
static doublereal bstw;
static integer itmp1;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
static integer indld;
static doublereal fudge;
static integer idone;
static doublereal sigma;
static integer iinfo, iindr;
static doublereal resid;
static logical eskip;
static doublereal right;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static integer nclus, zfrom;
static doublereal rqtol;
static integer iindc1, iindc2;
extern /* Subroutine */ int dlar1v_(integer *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *, logical *,
integer *, doublereal *, doublereal *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *);
static logical stp2ii;
static doublereal lambda;
extern doublereal dlamch_(char *);
static integer ibegin, indeig;
static logical needbs;
static integer indlld;
static doublereal sgndef, mingma;
extern /* Subroutine */ int dlarrb_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, integer *, integer *);
static integer oldien, oldncl, wbegin;
static doublereal spdiam;
static integer negcnt;
extern /* Subroutine */ int dlarrf_(integer *, doublereal *, doublereal *,
doublereal *, integer *, integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *);
static integer oldcls;
static doublereal savgap;
static integer ndepth;
static doublereal ssigma;
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *);
static logical usedbs;
static integer iindwk, offset;
static doublereal gaptol;
static integer newcls, oldfst, indwrk, windex, oldlst;
static logical usedrq;
static integer newfst, newftt, parity, windmn, windpl, isupmn, newlst,
zusedl;
static doublereal bstres;
static integer newsiz, zusedu, zusedw;
static doublereal nrminv, rqcorr;
static logical tryrqc;
static integer isupmx;
/* -- LAPACK auxiliary routine (version 3.1.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
Purpose
=======
DLARRV computes the eigenvectors of the tridiagonal matrix
T = L D L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L^T.
The input eigenvalues should have been computed by DLARRE.
Arguments
=========
N (input) INTEGER
The order of the matrix. N >= 0.
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION
Lower and upper bounds of the interval that contains the desired
eigenvalues. VL < VU. Needed to compute gaps on the left or right
end of the extremal eigenvalues in the desired RANGE.
D (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the N diagonal elements of the diagonal matrix D.
On exit, D may be overwritten.
L (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the (N-1) subdiagonal elements of the unit
bidiagonal matrix L are in elements 1 to N-1 of L
(if the matrix is not splitted.) At the end of each block
is stored the corresponding shift as given by DLARRE.
On exit, L is overwritten.
PIVMIN (in) DOUBLE PRECISION
The minimum pivot allowed in the Sturm sequence.
ISPLIT (input) 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.
M (input) INTEGER
The total number of input eigenvalues. 0 <= M <= N.
DOL (input) INTEGER
DOU (input) INTEGER
If the user wants to compute only selected eigenvectors from all
the eigenvalues supplied, he can specify an index range DOL:DOU.
Or else the setting DOL=1, DOU=M should be applied.
Note that DOL and DOU refer to the order in which the eigenvalues
are stored in W.
If the user wants to compute only selected eigenpairs, then
the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
computed eigenvectors. All other columns of Z are set to zero.
MINRGP (input) DOUBLE PRECISION
RTOL1 (input) DOUBLE PRECISION
RTOL2 (input) DOUBLE PRECISION
Parameters for bisection.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
W (input/output) DOUBLE PRECISION array, dimension (N)
The first M elements of W contain the APPROXIMATE eigenvalues for
which eigenvectors are to be computed. The eigenvalues
should be grouped by split-off block and ordered from
smallest to largest within the block ( The output array
W from DLARRE is expected here ). Furthermore, they are with
respect to the shift of the corresponding root representation
for their block. On exit, W holds the eigenvalues of the
UNshifted matrix.
WERR (input/output) DOUBLE PRECISION array, dimension (N)
The first M elements contain the semiwidth of the uncertainty
interval of the corresponding eigenvalue in W
WGAP (input/output) DOUBLE PRECISION array, dimension (N)
The separation from the right neighbor eigenvalue in W.
IBLOCK (input) 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.
INDEXW (input) 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 the second block.
GERS (input) DOUBLE PRECISION array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval
is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
be computed from the original UNshifted matrix.
Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
If INFO = 0, the first M columns of Z contain the
orthonormal eigenvectors of the matrix T
corresponding to the input eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The I-th eigenvector
is nonzero only in elements ISUPPZ( 2*I-1 ) through
ISUPPZ( 2*I ).
WORK (workspace) DOUBLE PRECISION array, dimension (12*N)
IWORK (workspace) INTEGER array, dimension (7*N)
INFO (output) INTEGER
= 0: successful exit
> 0: A problem occured in DLARRV.
< 0: One of the called subroutines signaled an internal problem.
Needs inspection of the corresponding parameter IINFO
for further information.
=-1: Problem in DLARRB when refining a child's eigenvalues.
=-2: Problem in DLARRF when computing the RRR of a child.
When a child is inside a tight cluster, it can be difficult
to find an RRR. A partial remedy from the user's point of
view is to make the parameter MINRGP smaller and recompile.
However, as the orthogonality of the computed vectors is
proportional to 1/MINRGP, the user should be aware that
he might be trading in precision when he decreases MINRGP.
=-3: Problem in DLARRB when refining a single eigenvalue
after the Rayleigh correction was rejected.
= 5: The Rayleigh Quotient Iteration failed to converge to
full accuracy in MAXITR steps.
Further Details
===============
Based on contributions by
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
=====================================================================
The first N entries of WORK are reserved for the eigenvalues
Parameter adjustments */
--d__;
--l;
--isplit;
--w;
--werr;
--wgap;
--iblock;
--indexw;
--gers;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--isuppz;
--work;
--iwork;
/* Function Body */
indld = *n + 1;
indlld = (*n << 1) + 1;
indwrk = *n * 3 + 1;
minwsize = *n * 12;
i__1 = minwsize;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] = 0.;
/* L5: */
}
/* IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
factorization used to compute the FP vector */
iindr = 0;
/* IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
layer and the one above. */
iindc1 = *n;
iindc2 = *n << 1;
iindwk = *n * 3 + 1;
miniwsize = *n * 7;
i__1 = miniwsize;
for (i__ = 1; i__ <= i__1; ++i__) {
iwork[i__] = 0;
/* L10: */
}
zusedl = 1;
if (*dol > 1) {
/* Set lower bound for use of Z */
zusedl = *dol - 1;
}
zusedu = *m;
if (*dou < *m) {
/* Set lower bound for use of Z */
zusedu = *dou + 1;
}
/* The width of the part of Z that is used */
zusedw = zusedu - zusedl + 1;
dlaset_("Full", n, &zusedw, &c_b5, &c_b5, &z__[zusedl * z_dim1 + 1], ldz);
eps = dlamch_("Precision");
rqtol = eps * 2.;
/* Set expert flags for standard code. */
tryrqc = TRUE_;
if (*dol == 1 && *dou == *m) {
} else {
/* Only selected eigenpairs are computed. Since the other evalues
are not refined by RQ iteration, bisection has to compute to full
accuracy. */
*rtol1 = eps * 4.;
*rtol2 = eps * 4.;
}
/* The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
desired eigenvalues. The support of the nonzero eigenvector
entries is contained in the interval IBEGIN:IEND.
Remark that if k eigenpairs are desired, then the eigenvectors
are stored in k contiguous columns of Z.
DONE is the number of eigenvectors already computed */
done = 0;
ibegin = 1;
wbegin = 1;
i__1 = iblock[*m];
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = isplit[jblk];
sigma = l[iend];
/* Find the eigenvectors of the submatrix indexed IBEGIN
through IEND. */
wend = wbegin - 1;
L15:
if (wend < *m) {
if (iblock[wend + 1] == jblk) {
++wend;
goto L15;
}
}
if (wend < wbegin) {
ibegin = iend + 1;
goto L170;
} else if (wend < *dol || wbegin > *dou) {
ibegin = iend + 1;
wbegin = wend + 1;
goto L170;
}
/* Find local spectral diameter of the block */
gl = gers[(ibegin << 1) - 1];
gu = gers[ibegin * 2];
i__2 = iend;
for (i__ = ibegin + 1; 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);
/* L20: */
}
spdiam = gu - gl;
/* OLDIEN is the last index of the previous block */
oldien = ibegin - 1;
/* Calculate the size of the current block */
in = iend - ibegin + 1;
/* The number of eigenvalues in the current block */
im = wend - wbegin + 1;
/* This is for a 1x1 block */
if (ibegin == iend) {
++done;
z__[ibegin + wbegin * z_dim1] = 1.;
isuppz[(wbegin << 1) - 1] = ibegin;
isuppz[wbegin * 2] = ibegin;
w[wbegin] += sigma;
work[wbegin] = w[wbegin];
ibegin = iend + 1;
++wbegin;
goto L170;
}
/* The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
Note that these can be approximations, in this case, the corresp.
entries of WERR give the size of the uncertainty interval.
The eigenvalue approximations will be refined when necessary as
high relative accuracy is required for the computation of the
corresponding eigenvectors. */
dcopy_(&im, &w[wbegin], &c__1, &work[wbegin], &c__1);
/* We store in W the eigenvalue approximations w.r.t. the original
matrix T. */
i__2 = im;
for (i__ = 1; i__ <= i__2; ++i__) {
w[wbegin + i__ - 1] += sigma;
/* L30: */
}
/* NDEPTH is the current depth of the representation tree */
ndepth = 0;
/* PARITY is either 1 or 0 */
parity = 1;
/* NCLUS is the number of clusters for the next level of the
representation tree, we start with NCLUS = 1 for the root */
nclus = 1;
iwork[iindc1 + 1] = 1;
iwork[iindc1 + 2] = im;
/* IDONE is the number of eigenvectors already computed in the current
block */
idone = 0;
/* loop while( IDONE.LT.IM )
generate the representation tree for the current block and
compute the eigenvectors */
L40:
if (idone < im) {
/* This is a crude protection against infinitely deep trees */
if (ndepth > *m) {
*info = -2;
return 0;
}
/* breadth first processing of the current level of the representation
tree: OLDNCL = number of clusters on current level */
oldncl = nclus;
/* reset NCLUS to count the number of child clusters */
nclus = 0;
parity = 1 - parity;
if (parity == 0) {
oldcls = iindc1;
newcls = iindc2;
} else {
oldcls = iindc2;
newcls = iindc1;
}
/* Process the clusters on the current level */
i__2 = oldncl;
for (i__ = 1; i__ <= i__2; ++i__) {
j = oldcls + (i__ << 1);
/* OLDFST, OLDLST = first, last index of current cluster.
cluster indices start with 1 and are relative
to WBEGIN when accessing W, WGAP, WERR, Z */
oldfst = iwork[j - 1];
oldlst = iwork[j];
if (ndepth > 0) {
/* Retrieve relatively robust representation (RRR) of cluster
that has been computed at the previous level
The RRR is stored in Z and overwritten once the eigenvectors
have been computed or when the cluster is refined */
if (*dol == 1 && *dou == *m) {
/* Get representation from location of the leftmost evalue
of the cluster */
j = wbegin + oldfst - 1;
} else {
if (wbegin + oldfst - 1 < *dol) {
/* Get representation from the left end of Z array */
j = *dol - 1;
} else if (wbegin + oldfst - 1 > *dou) {
/* Get representation from the right end of Z array */
j = *dou;
} else {
j = wbegin + oldfst - 1;
}
}
dcopy_(&in, &z__[ibegin + j * z_dim1], &c__1, &d__[ibegin]
, &c__1);
i__3 = in - 1;
dcopy_(&i__3, &z__[ibegin + (j + 1) * z_dim1], &c__1, &l[
ibegin], &c__1);
sigma = z__[iend + (j + 1) * z_dim1];
/* Set the corresponding entries in Z to zero */
dlaset_("Full", &in, &c__2, &c_b5, &c_b5, &z__[ibegin + j
* z_dim1], ldz);
}
/* Compute DL and DLL of current RRR */
i__3 = iend - 1;
for (j = ibegin; j <= i__3; ++j) {
tmp = d__[j] * l[j];
work[indld - 1 + j] = tmp;
work[indlld - 1 + j] = tmp * l[j];
/* L50: */
}
if (ndepth > 0) {
/* P and Q are index of the first and last eigenvalue to compute
within the current block */
p = indexw[wbegin - 1 + oldfst];
q = indexw[wbegin - 1 + oldlst];
/* Offset for the arrays WORK, WGAP and WERR, i.e., th P-OFFSET
thru' Q-OFFSET elements of these arrays are to be used.
OFFSET = P-OLDFST */
offset = indexw[wbegin] - 1;
/* perform limited bisection (if necessary) to get approximate
eigenvalues to the precision needed. */
dlarrb_(&in, &d__[ibegin], &work[indlld + ibegin - 1], &p,
&q, rtol1, rtol2, &offset, &work[wbegin], &wgap[
wbegin], &werr[wbegin], &work[indwrk], &iwork[
iindwk], pivmin, &spdiam, &in, &iinfo);
if (iinfo != 0) {
*info = -1;
return 0;
}
/* We also recompute the extremal gaps. W holds all eigenvalues
of the unshifted matrix and must be used for computation
of WGAP, the entries of WORK might stem from RRRs with
different shifts. The gaps from WBEGIN-1+OLDFST to
WBEGIN-1+OLDLST are correctly computed in DLARRB.
However, we only allow the gaps to become greater since
this is what should happen when we decrease WERR */
if (oldfst > 1) {
/* Computing MAX */
d__1 = wgap[wbegin + oldfst - 2], d__2 = w[wbegin +
oldfst - 1] - werr[wbegin + oldfst - 1] - w[
wbegin + oldfst - 2] - werr[wbegin + oldfst -
2];
wgap[wbegin + oldfst - 2] = max(d__1,d__2);
}
if (wbegin + oldlst - 1 < wend) {
/* Computing MAX */
d__1 = wgap[wbegin + oldlst - 1], d__2 = w[wbegin +
oldlst] - werr[wbegin + oldlst] - w[wbegin +
oldlst - 1] - werr[wbegin + oldlst - 1];
wgap[wbegin + oldlst - 1] = max(d__1,d__2);
}
/* Each time the eigenvalues in WORK get refined, we store
the newly found approximation with all shifts applied in W */
i__3 = oldlst;
for (j = oldfst; j <= i__3; ++j) {
w[wbegin + j - 1] = work[wbegin + j - 1] + sigma;
/* L53: */
}
}
/* Process the current node. */
newfst = oldfst;
i__3 = oldlst;
for (j = oldfst; j <= i__3; ++j) {
if (j == oldlst) {
/* we are at the right end of the cluster, this is also the
boundary of the child cluster */
newlst = j;
} else if (wgap[wbegin + j - 1] >= *minrgp * (d__1 = work[
wbegin + j - 1], abs(d__1))) {
/* the right relative gap is big enough, the child cluster
(NEWFST,..,NEWLST) is well separated from the following */
newlst = j;
} else {
/* inside a child cluster, the relative gap is not
big enough. */
goto L140;
}
/* Compute size of child cluster found */
newsiz = newlst - newfst + 1;
/* NEWFTT is the place in Z where the new RRR or the computed
eigenvector is to be stored */
if (*dol == 1 && *dou == *m) {
/* Store representation at location of the leftmost evalue
of the cluster */
newftt = wbegin + newfst - 1;
} else {
if (wbegin + newfst - 1 < *dol) {
/* Store representation at the left end of Z array */
newftt = *dol - 1;
} else if (wbegin + newfst - 1 > *dou) {
/* Store representation at the right end of Z array */
newftt = *dou;
} else {
newftt = wbegin + newfst - 1;
}
}
if (newsiz > 1) {
/* Current child is not a singleton but a cluster.
Compute and store new representation of child.
Compute left and right cluster gap.
LGAP and RGAP are not computed from WORK because
the eigenvalue approximations may stem from RRRs
different shifts. However, W hold all eigenvalues
of the unshifted matrix. Still, the entries in WGAP
have to be computed from WORK since the entries
in W might be of the same order so that gaps are not
exhibited correctly for very close eigenvalues. */
if (newfst == 1) {
/* Computing MAX */
d__1 = 0., d__2 = w[wbegin] - werr[wbegin] - *vl;
lgap = max(d__1,d__2);
} else {
lgap = wgap[wbegin + newfst - 2];
}
rgap = wgap[wbegin + newlst - 1];
/* Compute left- and rightmost eigenvalue of child
to high precision in order to shift as close
as possible and obtain as large relative gaps
as possible */
for (k = 1; k <= 2; ++k) {
if (k == 1) {
p = indexw[wbegin - 1 + newfst];
} else {
p = indexw[wbegin - 1 + newlst];
}
offset = indexw[wbegin] - 1;
dlarrb_(&in, &d__[ibegin], &work[indlld + ibegin
- 1], &p, &p, &rqtol, &rqtol, &offset, &
work[wbegin], &wgap[wbegin], &werr[wbegin]
, &work[indwrk], &iwork[iindwk], pivmin, &
spdiam, &in, &iinfo);
/* L55: */
}
if (wbegin + newlst - 1 < *dol || wbegin + newfst - 1
> *dou) {
/* if the cluster contains no desired eigenvalues
skip the computation of that branch of the rep. tree
We could skip before the refinement of the extremal
eigenvalues of the child, but then the representation
tree could be different from the one when nothing is
skipped. For this reason we skip at this place. */
idone = idone + newlst - newfst + 1;
goto L139;
}
/* Compute RRR of child cluster.
Note that the new RRR is stored in Z
DLARRF needs LWORK = 2*N */
dlarrf_(&in, &d__[ibegin], &l[ibegin], &work[indld +
ibegin - 1], &newfst, &newlst, &work[wbegin],
&wgap[wbegin], &werr[wbegin], &spdiam, &lgap,
&rgap, pivmin, &tau, &z__[ibegin + newftt *
z_dim1], &z__[ibegin + (newftt + 1) * z_dim1],
&work[indwrk], &iinfo);
if (iinfo == 0) {
/* a new RRR for the cluster was found by DLARRF
update shift and store it */
ssigma = sigma + tau;
z__[iend + (newftt + 1) * z_dim1] = ssigma;
/* WORK() are the midpoints and WERR() the semi-width
Note that the entries in W are unchanged. */
i__4 = newlst;
for (k = newfst; k <= i__4; ++k) {
fudge = eps * 3. * (d__1 = work[wbegin + k -
1], abs(d__1));
work[wbegin + k - 1] -= tau;
fudge += eps * 4. * (d__1 = work[wbegin + k -
1], abs(d__1));
/* Fudge errors */
werr[wbegin + k - 1] += fudge;
/* Gaps are not fudged. Provided that WERR is small
when eigenvalues are close, a zero gap indicates
that a new representation is needed for resolving
the cluster. A fudge could lead to a wrong decision
of judging eigenvalues 'separated' which in
reality are not. This could have a negative impact
on the orthogonality of the computed eigenvectors.
L116: */
}
++nclus;
k = newcls + (nclus << 1);
iwork[k - 1] = newfst;
iwork[k] = newlst;
} else {
*info = -2;
return 0;
}
} else {
/* Compute eigenvector of singleton */
iter = 0;
tol = log((doublereal) in) * 4. * eps;
k = newfst;
windex = wbegin + k - 1;
/* Computing MAX */
i__4 = windex - 1;
windmn = max(i__4,1);
/* Computing MIN */
i__4 = windex + 1;
windpl = min(i__4,*m);
lambda = work[windex];
++done;
/* Check if eigenvector computation is to be skipped */
if (windex < *dol || windex > *dou) {
eskip = TRUE_;
goto L125;
} else {
eskip = FALSE_;
}
left = work[windex] - werr[windex];
right = work[windex] + werr[windex];
indeig = indexw[windex];
/* Note that since we compute the eigenpairs for a child,
all eigenvalue approximations are w.r.t the same shift.
In this case, the entries in WORK should be used for
computing the gaps since they exhibit even very small
differences in the eigenvalues, as opposed to the
entries in W which might "look" the same. */
if (k == 1) {
/* In the case RANGE='I' and with not much initial
accuracy in LAMBDA and VL, the formula
LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
can lead to an overestimation of the left gap and
thus to inadequately early RQI 'convergence'.
Prevent this by forcing a small left gap.
Computing MAX */
d__1 = abs(left), d__2 = abs(right);
lgap = eps * max(d__1,d__2);
} else {
lgap = wgap[windmn];
}
if (k == im) {
/* In the case RANGE='I' and with not much initial
accuracy in LAMBDA and VU, the formula
can lead to an overestimation of the right gap and
thus to inadequately early RQI 'convergence'.
Prevent this by forcing a small right gap.
Computing MAX */
d__1 = abs(left), d__2 = abs(right);
rgap = eps * max(d__1,d__2);
} else {
rgap = wgap[windex];
}
gap = min(lgap,rgap);
if (k == 1 || k == im) {
/* The eigenvector support can become wrong
because significant entries could be cut off due to a
large GAPTOL parameter in LAR1V. Prevent this. */
gaptol = 0.;
} else {
gaptol = gap * eps;
}
isupmn = in;
isupmx = 1;
/* Update WGAP so that it holds the minimum gap
to the left or the right. This is crucial in the
case where bisection is used to ensure that the
eigenvalue is refined up to the required precision.
The correct value is restored afterwards. */
savgap = wgap[windex];
wgap[windex] = gap;
/* We want to use the Rayleigh Quotient Correction
as often as possible since it converges quadratically
when we are close enough to the desired eigenvalue.
However, the Rayleigh Quotient can have the wrong sign
and lead us away from the desired eigenvalue. In this
case, the best we can do is to use bisection. */
usedbs = FALSE_;
usedrq = FALSE_;
/* Bisection is initially turned off unless it is forced */
needbs = ! tryrqc;
L120:
/* Check if bisection should be used to refine eigenvalue */
if (needbs) {
/* Take the bisection as new iterate */
usedbs = TRUE_;
itmp1 = iwork[iindr + windex];
offset = indexw[wbegin] - 1;
d__1 = eps * 2.;
dlarrb_(&in, &d__[ibegin], &work[indlld + ibegin
- 1], &indeig, &indeig, &c_b5, &d__1, &
offset, &work[wbegin], &wgap[wbegin], &
werr[wbegin], &work[indwrk], &iwork[
iindwk], pivmin, &spdiam, &itmp1, &iinfo);
if (iinfo != 0) {
*info = -3;
return 0;
}
lambda = work[windex];
/* Reset twist index from inaccurate LAMBDA to
force computation of true MINGMA */
iwork[iindr + windex] = 0;
}
/* Given LAMBDA, compute the eigenvector. */
L__1 = ! usedbs;
dlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin], &l[
ibegin], &work[indld + ibegin - 1], &work[
indlld + ibegin - 1], pivmin, &gaptol, &z__[
ibegin + windex * z_dim1], &L__1, &negcnt, &
ztz, &mingma, &iwork[iindr + windex], &isuppz[
(windex << 1) - 1], &nrminv, &resid, &rqcorr,
&work[indwrk]);
if (iter == 0) {
bstres = resid;
bstw = lambda;
} else if (resid < bstres) {
bstres = resid;
bstw = lambda;
}
/* Computing MIN */
i__4 = isupmn, i__5 = isuppz[(windex << 1) - 1];
isupmn = min(i__4,i__5);
/* Computing MAX */
i__4 = isupmx, i__5 = isuppz[windex * 2];
isupmx = max(i__4,i__5);
++iter;
/* sin alpha <= |resid|/gap
Note that both the residual and the gap are
proportional to the matrix, so ||T|| doesn't play
a role in the quotient
Convergence test for Rayleigh-Quotient iteration
(omitted when Bisection has been used) */
if (resid > tol * gap && abs(rqcorr) > rqtol * abs(
lambda) && ! usedbs) {
/* We need to check that the RQCORR update doesn't
move the eigenvalue away from the desired one and
towards a neighbor. -> protection with bisection */
if (indeig <= negcnt) {
/* The wanted eigenvalue lies to the left */
sgndef = -1.;
} else {
/* The wanted eigenvalue lies to the right */
sgndef = 1.;
}
/* We only use the RQCORR if it improves the
the iterate reasonably. */
if (rqcorr * sgndef >= 0. && lambda + rqcorr <=
right && lambda + rqcorr >= left) {
usedrq = TRUE_;
/* Store new midpoint of bisection interval in WORK */
if (sgndef == 1.) {
/* The current LAMBDA is on the left of the true
eigenvalue */
left = lambda;
/* We prefer to assume that the error estimate
is correct. We could make the interval not
as a bracket but to be modified if the RQCORR
chooses to. In this case, the RIGHT side should
be modified as follows:
RIGHT = MAX(RIGHT, LAMBDA + RQCORR) */
} else {
/* The current LAMBDA is on the right of the true
eigenvalue */
right = lambda;
/* See comment about assuming the error estimate is
correct above.
LEFT = MIN(LEFT, LAMBDA + RQCORR) */
}
work[windex] = (right + left) * .5;
/* Take RQCORR since it has the correct sign and
improves the iterate reasonably */
lambda += rqcorr;
/* Update width of error interval */
werr[windex] = (right - left) * .5;
} else {
needbs = TRUE_;
}
if (right - left < rqtol * abs(lambda)) {
/* The eigenvalue is computed to bisection accuracy
compute eigenvector and stop */
usedbs = TRUE_;
goto L120;
} else if (iter < 10) {
goto L120;
} else if (iter == 10) {
needbs = TRUE_;
goto L120;
} else {
*info = 5;
return 0;
}
} else {
stp2ii = FALSE_;
if (usedrq && usedbs && bstres <= resid) {
lambda = bstw;
stp2ii = TRUE_;
}
if (stp2ii) {
/* improve error angle by second step */
L__1 = ! usedbs;
dlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin]
, &l[ibegin], &work[indld + ibegin -
1], &work[indlld + ibegin - 1],
pivmin, &gaptol, &z__[ibegin + windex
* z_dim1], &L__1, &negcnt, &ztz, &
mingma, &iwork[iindr + windex], &
isuppz[(windex << 1) - 1], &nrminv, &
resid, &rqcorr, &work[indwrk]);
}
work[windex] = lambda;
}
/* Compute FP-vector support w.r.t. whole matrix */
isuppz[(windex << 1) - 1] += oldien;
isuppz[windex * 2] += oldien;
zfrom = isuppz[(windex << 1) - 1];
zto = isuppz[windex * 2];
isupmn += oldien;
isupmx += oldien;
/* Ensure vector is ok if support in the RQI has changed */
if (isupmn < zfrom) {
i__4 = zfrom - 1;
for (ii = isupmn; ii <= i__4; ++ii) {
z__[ii + windex * z_dim1] = 0.;
/* L122: */
}
}
if (isupmx > zto) {
i__4 = isupmx;
for (ii = zto + 1; ii <= i__4; ++ii) {
z__[ii + windex * z_dim1] = 0.;
/* L123: */
}
}
i__4 = zto - zfrom + 1;
dscal_(&i__4, &nrminv, &z__[zfrom + windex * z_dim1],
&c__1);
L125:
/* Update W */
w[windex] = lambda + sigma;
/* Recompute the gaps on the left and right
But only allow them to become larger and not
smaller (which can only happen through "bad"
cancellation and doesn't reflect the theory
where the initial gaps are underestimated due
to WERR being too crude.) */
if (! eskip) {
if (k > 1) {
/* Computing MAX */
d__1 = wgap[windmn], d__2 = w[windex] - werr[
windex] - w[windmn] - werr[windmn];
wgap[windmn] = max(d__1,d__2);
}
if (windex < wend) {
/* Computing MAX */
d__1 = savgap, d__2 = w[windpl] - werr[windpl]
- w[windex] - werr[windex];
wgap[windex] = max(d__1,d__2);
}
}
++idone;
}
/* here ends the code for the current child */
L139:
/* Proceed to any remaining child nodes */
newfst = j + 1;
L140:
;
}
/* L150: */
}
++ndepth;
goto L40;
}
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return 0;
/* End of DLARRV */
} /* dlarrv_ */