/* 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 */
integer minwsize, i__, j, k, p, q, miniwsize, ii;
doublereal gl;
integer im, in;
doublereal gu, gap, eps, tau, tol, tmp;
integer zto;
doublereal ztz;
integer iend, jblk;
doublereal lgap;
integer done;
doublereal rgap, left;
integer wend, iter;
doublereal bstw;
integer itmp1;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
integer indld;
doublereal fudge;
integer idone;
doublereal sigma;
integer iinfo, iindr;
doublereal resid;
logical eskip;
doublereal right;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
integer nclus, zfrom;
doublereal rqtol;
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 *);
logical stp2ii;
doublereal lambda;
extern doublereal dlamch_(char *);
integer ibegin, indeig;
logical needbs;
integer indlld;
doublereal sgndef, mingma;
extern /* Subroutine */ int dlarrb_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, integer *, integer *);
integer oldien, oldncl, wbegin;
doublereal spdiam;
integer negcnt;
extern /* Subroutine */ int dlarrf_(integer *, doublereal *, doublereal *,
doublereal *, integer *, integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *);
integer oldcls;
doublereal savgap;
integer ndepth;
doublereal ssigma;
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *);
logical usedbs;
integer iindwk, offset;
doublereal gaptol;
integer newcls, oldfst, indwrk, windex, oldlst;
logical usedrq;
integer newfst, newftt, parity, windmn, windpl, isupmn, newlst, zusedl;
doublereal bstres;
integer newsiz, zusedu, zusedw;
doublereal nrminv, rqcorr;
logical tryrqc;
integer isupmx;
/* -- LAPACK auxiliary routine (version 3.1.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* 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 */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* .. */
/* 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_ */
/* Subroutine */ int dlarre_(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)
{
/* -- LAPACK auxiliary routine (version 3.1) --
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
November 2006
Purpose
=======
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.
Arguments
=========
RANGE (input) CHARACTER
= '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.
N (input) INTEGER
The order of the matrix. N > 0.
VL (input/output) DOUBLE PRECISION
VU (input/output) 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.
IL (input) INTEGER
IU (input) INTEGER
If RANGE='I', the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= IL <= IU <= N.
D (input/output) 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.
E (input/output) 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.
E2 (input/output) 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
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|) )
SPLTOL (input) DOUBLE PRECISION
The threshold for splitting.
NSPLIT (output) INTEGER
The number of blocks T splits into. 1 <= NSPLIT <= N.
ISPLIT (output) 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.
M (output) INTEGER
The total number of eigenvalues (of all L_i D_i L_i^T)
found.
W (output) 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).
WERR (output) DOUBLE PRECISION array, dimension (N)
The error bound on the corresponding eigenvalue in W.
WGAP (output) 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
IBLOCK (output) 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 (output) 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
GERS (output) DOUBLE PRECISION array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval
is (GERS(2*i-1), GERS(2*i)).
PIVMIN (output) DOUBLE PRECISION
The minimum pivot in the Sturm sequence for T.
WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
Workspace.
IWORK (workspace) INTEGER array, dimension (5*N)
Workspace.
INFO (output) 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.
Further Details
The base representations are required to suffer very little
element growth and consequently define all their eigenvalues to
high relative accuracy.
===============
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
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c__2 = 2;
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal), log(doublereal);
/* Local variables */
static integer i__, j;
static doublereal s1, s2;
static integer mb;
static doublereal gl;
static integer in, mm;
static doublereal gu;
static integer cnt;
static doublereal eps, tau, tmp, rtl;
static integer cnt1, cnt2;
static doublereal tmp1, eabs;
static integer iend, jblk;
static doublereal eold;
static integer indl;
static doublereal dmax__, emax;
static integer wend, idum, indu;
static doublereal rtol;
static integer iseed[4];
static doublereal avgap, sigma;
extern logical lsame_(char *, char *);
static integer iinfo;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static logical norep;
extern /* Subroutine */ int dlasq2_(integer *, doublereal *, integer *);
extern doublereal dlamch_(char *);
static integer ibegin;
static logical forceb;
static integer irange;
static doublereal sgndef;
extern /* Subroutine */ int dlarra_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *,
integer *), dlarrb_(integer *, doublereal *, doublereal *,
integer *, integer *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *, integer *,
doublereal *, doublereal *, integer *, integer *), dlarrc_(char *
, integer *, doublereal *, doublereal *, doublereal *, doublereal
*, doublereal *, integer *, integer *, integer *, integer *);
static integer wbegin;
extern /* Subroutine */ int dlarrd_(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 *);
static doublereal safmin, spdiam;
extern /* Subroutine */ int dlarrk_(integer *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *);
static logical usedqd;
static doublereal clwdth, isleft;
extern /* Subroutine */ int dlarnv_(integer *, integer *, integer *,
doublereal *);
static doublereal isrght, bsrtol, dpivot;
--iwork;
--work;
--gers;
--indexw;
--iblock;
--wgap;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
/* Function Body */
*info = 0;
/* Decode RANGE */
if (lsame_(range, "A")) {
irange = 1;
} else if (lsame_(range, "V")) {
irange = 3;
} else if (lsame_(range, "I")) {
irange = 2;
}
*m = 0;
/* Get machine constants */
safmin = dlamch_("S");
eps = dlamch_("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 */
dlarra_(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_;
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 */
dlarrd_(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 */
dlarrk_(&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);
dlarrk_(&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) {
dlarrc_("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.;
} 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. */
dcopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
dcopy_(&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;
dlarnv_(&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;
dlarrb_(&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.;
dlasq2_(&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 */
} /* dlarre_ */
int PMR_process_s_task(singleton_t *sng, int tid, counter_t *num_left,
workQ_t *workQ, val_t *Wstruct, vec_t *Zstruct,
tol_t *tolstruct, double *work, int *iwork)
{
/* Inputs */
int begin = sng->begin;
int end = sng->end;
int size = end - begin + 1;
int bl_begin = sng->bl_begin;
int bl_end = sng->bl_end;
int bl_size = bl_end - bl_begin + 1;
double bl_spdiam = sng->bl_spdiam;
rrr_t *RRR = sng->RRR;
double *restrict D = RRR->D;
double *restrict L = RRR->L;
double *restrict DL = RRR->DL;
double *restrict DLL = RRR->DLL;
int n = Wstruct->n;
double *restrict W = Wstruct->W;
double *restrict Werr = Wstruct->Werr;
double *restrict Wgap = Wstruct->Wgap;
int *restrict Windex = Wstruct->Windex;
double *restrict Wshifted = Wstruct->Wshifted;
int ldz = Zstruct->ldz;
double *restrict Z = Zstruct->Z;
int *restrict isuppZ = Zstruct->Zsupp;;
int *restrict Zindex = Zstruct->Zindex;
double pivmin = tolstruct->pivmin;
/* others */
int info, i, k, itmp;
int IONE = 1;
double DZERO = 0.0;
double tol, lambda, left, right;
int i_local;
size_t zind;
double gap, lgap, rgap, gaptol, savedgap, tmp;
bool usedBS, usedRQ, needBS, wantNC, step2II;
int r, offset;
double twoeps = 2*DBL_EPSILON, RQtol = 2*DBL_EPSILON;
double residual, bstres, bstw;
int i_supmn, i_supmx;
double RQcorr;
int negcount;
int sgndef, suppsize;
double sigma;
int i_Zfrom, i_Zto;
double ztz, norminv, mingma;
int count=1;
/* set tolerance parameter */
tol = 4.0 * log( (double) bl_size ) * DBL_EPSILON;
/* decrement counter */
PMR_decrement_counter(num_left, size);
/* loop over all singletons in the task */
for (i=begin; i<=end; i++) {
if (bl_size == 1) {
/* set eigenvector to column of identity matrix */
zind = Zindex[i];
memset(&Z[zind*ldz], 0.0, n*sizeof(double) );
Z[zind*ldz + bl_begin] = 1.0;
isuppZ[2*zind ] = bl_begin + 1;
isuppZ[2*zind + 1] = bl_begin + 1;
continue;
}
lambda = Wshifted[i];
left = Wshifted[i] - Werr[i];
right = Wshifted[i] + Werr[i];
i_local = Windex[i];
r = 0;
/* compute left and right gap */
if (i == bl_begin)
lgap = DBL_EPSILON * fmax(fabs(left), fabs(right));
else if (i == begin)
lgap = sng->lgap;
else
lgap = Wgap[i-1];
if (i == bl_end) {
rgap = DBL_EPSILON * fmax(fabs(left), fabs(right));
} else {
rgap = Wgap[i];
}
gap = fmin(lgap, rgap);
if ( i == bl_begin || i == bl_end ) {
gaptol = 0.0;
} else {
gaptol = gap * DBL_EPSILON;
}
/* initialize lower and upper value of support */
i_supmn = bl_size;
i_supmx = 1;
/* update Wgap so that it holds minimum gap and save the
* old value */
savedgap = Wgap[i];
Wgap[i] = gap;
/* initialize flags indicating if bisection or Rayleigh-Quotient
* correction was used */
usedBS = false;
usedRQ = false;
/* the need for bisection is initially turned off */
needBS = !TRY_RQC;
/* IEEE floating point is assumed, so that all 0 bits are 0.0 */
zind = Zindex[i];
memset(&Z[zind*ldz], 0.0, n*sizeof(double));
/* inverse iteration with twisted factorization */
for (k=1; k<=MAXITER; k++) {
if (needBS == true) {
usedBS = true;
itmp = r;
offset = Windex[i] - 1;
tmp = Wgap[i];
Wgap[i] = 0.0;
dlarrb_(&bl_size, D, DLL, &i_local, &i_local, &DZERO,
&twoeps, &offset, &Wshifted[i], &Wgap[i],
&Werr[i], work, iwork, &pivmin, &bl_spdiam,
&itmp, &info);
assert(info == 0);
Wgap[i] = tmp;
lambda = Wshifted[i];
r = 0;
}
wantNC = (usedBS == true) ? false : true;
/* compute the eigenvector corresponding to lambda */
dlar1v_(&bl_size, &IONE, &bl_size, &lambda, D, L, DL, DLL,
&pivmin, &gaptol, &Z[zind*ldz+bl_begin], &wantNC,
&negcount, &ztz, &mingma, &r, &isuppZ[2*zind],
&norminv, &residual, &RQcorr, work);
if (k == 1) {
bstres = residual;
bstw = lambda;
} else if (residual < bstres) {
bstres = residual;
bstw = lambda;
}
/* update support held */
i_supmn = imin(i_supmn, isuppZ[2*zind ]);
i_supmx = imax(i_supmx, isuppZ[2*zind + 1]);
/* Convergence test for Rayleigh Quotient Iteration
* not done if bisection was used */
if ( !usedBS && residual > tol*gap
&& fabs(RQcorr) > RQtol*fabs(lambda) ) {
if (i_local <= negcount) {
sgndef = -1; /* wanted eigenvalue lies to the left */
} else {
sgndef = 1; /* wanted eigenvalue lies to the right */
}
if ( RQcorr*sgndef >= 0.0
&& lambda+RQcorr <= right
&& lambda+RQcorr >= left ) {
usedRQ = true;
if ( sgndef == 1 )
left = lambda;
else
right = lambda;
Wshifted[i] = 0.5*(left + right);
lambda += RQcorr;
} else { /* bisection is needed */
needBS = true;
}
if ( right-left < RQtol*fabs(lambda) ) {
/* eigenvalue computed to bisection accuracy
* => compute eigenvector */
usedBS = true;
} else if ( k == MAXITER-1 ) {
/* for last iteration use bisection */
needBS = true;
}
} else {
/* go to next iteration */
break;
}
} /* end k */
/* if necessary call dlar1v to improve error angle by 2nd step */
step2II = false;
if ( usedRQ && usedBS && (bstres <= residual) ) {
lambda = bstw;
step2II = true;
}
if ( step2II == true ) {
dlar1v_(&bl_size, &IONE, &bl_size, &lambda, D, L, DL, DLL,
&pivmin, &gaptol, &Z[zind*ldz+bl_begin], &wantNC,
&negcount, &ztz, &mingma, &r, &isuppZ[2*zind],
&norminv, &residual, &RQcorr, work);
}
Wshifted[i] = lambda;
/* compute support w.r.t. whole matrix
* block beginning is offset for each support */
isuppZ[2*zind ] += bl_begin;
isuppZ[2*zind + 1] += bl_begin;
/* ensure vector is okay if support changed in RQI
* minus ones because of indices starting from zero */
i_Zfrom = isuppZ[2*zind ] - 1;
i_Zto = isuppZ[2*zind + 1] - 1;
i_supmn += bl_begin - 1;
i_supmx += bl_begin - 1;
if ( i_supmn < i_Zfrom ) {
for ( k=i_supmn; k < i_Zfrom; k++ ) {
Z[k + zind*ldz] = 0.0;
}
}
if ( i_supmx > i_Zto ) {
for ( k=i_Zto+1; k <= i_supmx; k++ ) {
Z[k + zind*ldz] = 0.0;
}
}
/* normalize eigenvector */
suppsize = i_Zto - i_Zfrom + 1;
mrrr_dscal(&suppsize, &norminv, &Z[i_Zfrom + zind*ldz], &IONE);
sigma = L[bl_size-1];
W[i] = lambda + sigma;
if (i < end)
Wgap[i] = fmax(savedgap, W[i+1]-Werr[i+1] - W[i]-Werr[i]);
/* Every x iterations process all pendinf r-tasks */
count++;
if (count % EMPTY_RQ_ITER == 0)
PMR_process_r_queue(tid, workQ, Wstruct, tolstruct, work,
iwork);
} /* end i */
/* clean up */
PMR_try_destroy_rrr(RRR);
free(sng);
return(0);
}
/* Subroutine */ int dlarrv_(integer *n, doublereal *d__, doublereal *l,
integer *isplit, integer *m, doublereal *w, integer *iblock,
doublereal *gersch, doublereal *tol, 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, i__6;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer iend, jblk;
extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *,
integer *);
static integer iter, temp[1], ktot;
extern doublereal dnrm2_(integer *, doublereal *, integer *);
static integer itmp1, itmp2, i__, j, k, p, q;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
static integer indld;
static doublereal sigma;
static integer ndone, iinfo, iindr;
static doublereal resid;
extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
doublereal *, integer *);
static integer nclus;
extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *);
static integer iindc1, iindc2;
extern /* Subroutine */ int dlar1v_(integer *, integer *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *, doublereal *);
static doublereal lambda;
static integer im, in;
extern doublereal dlamch_(char *);
static integer ibegin, indgap, indlld;
extern /* Subroutine */ int dlarrb_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, integer *, integer *);
static doublereal mingma;
static integer oldien, oldncl;
static doublereal relgap;
extern /* Subroutine */ int dlarrf_(integer *, doublereal *, doublereal *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
doublereal *, doublereal *, doublereal *, integer *, integer *),
dlaset_(char *, integer *, integer *, doublereal *, doublereal *,
doublereal *, integer *);
static integer oldcls, ndepth, inderr, iindwk;
extern /* Subroutine */ int dstein_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *, doublereal *,
integer *, doublereal *, integer *, integer *, integer *);
static logical mgscls;
static integer lsbdpt, newcls, oldfst;
static doublereal minrgp;
static integer indwrk, oldlst;
static doublereal reltol;
static integer maxitr, newfrs, newftt;
static doublereal mgstol;
static integer nsplit;
static doublereal nrminv, rqcorr;
static integer newlst, newsiz;
static doublereal gap, eps, ztz, tmp1;
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
/* -- LAPACK auxiliary routine (instru to count ops, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Common block to return operation count
Purpose
=======
DLARRV computes the eigenvectors of the tridiagonal matrix
T = L D L^T given L, D and the eigenvalues of L D L^T.
The input eigenvalues should have high relative accuracy with
respect to the entries of L and D. The desired accuracy of the
output can be specified by the input parameter TOL.
Arguments
=========
N (input) INTEGER
The order of the matrix. N >= 0.
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-1)
On entry, the (n-1) subdiagonal elements of the unit
bidiagonal matrix L in elements 1 to N-1 of L. L(N) need
not be set. On exit, L is overwritten.
ISPLIT (input) INTEGER array, dimension (N)
The splitting points, at which T breaks up into submatrices.
The first submatrix consists of rows/columns 1 to
ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
through ISPLIT( 2 ), etc.
TOL (input) DOUBLE PRECISION
The absolute error tolerance for the
eigenvalues/eigenvectors.
Errors in the input eigenvalues must be bounded by TOL.
The eigenvectors output have residual norms
bounded by TOL, and the dot products between different
eigenvectors are bounded by TOL. TOL must be at least
N*EPS*|T|, where EPS is the machine precision and |T| is
the 1-norm of the tridiagonal matrix.
M (input) INTEGER
The total number of eigenvalues found. 0 <= M <= N.
If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (input) DOUBLE PRECISION array, dimension (N)
The first M elements of W contain the 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 ).
Errors in W must be bounded by TOL (see above).
IBLOCK (input) INTEGER array, dimension (N)
The submatrix indices associated with the corresponding
eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
the first submatrix from the top, =2 if W(i) belongs to
the second submatrix, etc.
Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
If JOBZ = 'V', then if INFO = 0, the first M columns of Z
contain the orthonormal eigenvectors of the matrix T
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
If JOBZ = 'N', then Z is not referenced.
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z; if RANGE = 'V', the exact value of M
is not known in advance and an upper bound must be used.
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 (13*N)
IWORK (workspace) INTEGER array, dimension (6*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = 1, internal error in DLARRB
if INFO = 2, internal error in DSTEIN
Further Details
===============
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--l;
--isplit;
--w;
--iblock;
--gersch;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--isuppz;
--work;
--iwork;
/* Function Body */
inderr = *n + 1;
indld = *n << 1;
indlld = *n * 3;
indgap = *n << 2;
indwrk = *n * 5 + 1;
iindr = *n;
iindc1 = *n << 1;
iindc2 = *n * 3;
iindwk = (*n << 2) + 1;
eps = dlamch_("Precision");
i__1 = *n << 1;
for (i__ = 1; i__ <= i__1; ++i__) {
iwork[i__] = 0;
/* L10: */
}
latime_1.ops += (doublereal) (*m + 1);
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
work[inderr + i__ - 1] = eps * (d__1 = w[i__], abs(d__1));
/* L20: */
}
dlaset_("Full", n, n, &c_b6, &c_b6, &z__[z_offset], ldz);
mgstol = eps * 5.;
nsplit = iblock[*m];
ibegin = 1;
i__1 = nsplit;
for (jblk = 1; jblk <= i__1; ++jblk) {
iend = isplit[jblk];
/* Find the eigenvectors of the submatrix indexed IBEGIN
through IEND. */
if (ibegin == iend) {
z___ref(ibegin, ibegin) = 1.;
isuppz[(ibegin << 1) - 1] = ibegin;
isuppz[ibegin * 2] = ibegin;
ibegin = iend + 1;
goto L170;
}
oldien = ibegin - 1;
in = iend - oldien;
latime_1.ops += 1.;
/* Computing MIN */
d__1 = .01, d__2 = 1. / (doublereal) in;
reltol = min(d__1,d__2);
im = in;
dcopy_(&im, &w[ibegin], &c__1, &work[1], &c__1);
latime_1.ops += (doublereal) (in - 1);
i__2 = in - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
work[indgap + i__] = work[i__ + 1] - work[i__];
/* L30: */
}
/* Computing MAX */
d__2 = (d__1 = work[in], abs(d__1));
work[indgap + in] = max(d__2,eps);
ndone = 0;
ndepth = 0;
lsbdpt = 1;
nclus = 1;
iwork[iindc1 + 1] = 1;
iwork[iindc1 + 2] = in;
/* While( NDONE.LT.IM ) do */
L40:
if (ndone < im) {
oldncl = nclus;
nclus = 0;
lsbdpt = 1 - lsbdpt;
i__2 = oldncl;
for (i__ = 1; i__ <= i__2; ++i__) {
if (lsbdpt == 0) {
oldcls = iindc1;
newcls = iindc2;
} else {
oldcls = iindc2;
newcls = iindc1;
}
/* If NDEPTH > 1, retrieve the relatively robust
representation (RRR) and perform limited bisection
(if necessary) to get approximate eigenvalues. */
j = oldcls + (i__ << 1);
oldfst = iwork[j - 1];
oldlst = iwork[j];
if (ndepth > 0) {
j = oldien + oldfst;
dcopy_(&in, &z___ref(ibegin, j), &c__1, &d__[ibegin], &
c__1);
dcopy_(&in, &z___ref(ibegin, j + 1), &c__1, &l[ibegin], &
c__1);
sigma = l[iend];
}
k = ibegin;
latime_1.ops += (doublereal) (in - 1 << 1);
i__3 = in - 1;
for (j = 1; j <= i__3; ++j) {
work[indld + j] = d__[k] * l[k];
work[indlld + j] = work[indld + j] * l[k];
++k;
/* L50: */
}
if (ndepth > 0) {
dlarrb_(&in, &d__[ibegin], &l[ibegin], &work[indld + 1], &
work[indlld + 1], &oldfst, &oldlst, &sigma, &
reltol, &work[1], &work[indgap + 1], &work[inderr]
, &work[indwrk], &iwork[iindwk], &iinfo);
if (iinfo != 0) {
*info = 1;
return 0;
}
}
/* Classify eigenvalues of the current representation (RRR)
as (i) isolated, (ii) loosely clustered or (iii) tightly
clustered */
newfrs = oldfst;
i__3 = oldlst;
for (j = oldfst; j <= i__3; ++j) {
latime_1.ops += 1.;
if (j == oldlst || work[indgap + j] >= reltol * (d__1 =
work[j], abs(d__1))) {
newlst = j;
} else {
/* continue (to the next loop) */
latime_1.ops += 1.;
relgap = work[indgap + j] / (d__1 = work[j], abs(d__1)
);
if (j == newfrs) {
minrgp = relgap;
} else {
minrgp = min(minrgp,relgap);
}
goto L140;
}
newsiz = newlst - newfrs + 1;
maxitr = 10;
newftt = oldien + newfrs;
if (newsiz > 1) {
mgscls = newsiz <= 20 && minrgp >= mgstol;
if (! mgscls) {
dlarrf_(&in, &d__[ibegin], &l[ibegin], &work[
indld + 1], &work[indlld + 1], &newfrs, &
newlst, &work[1], &z___ref(ibegin, newftt)
, &z___ref(ibegin, newftt + 1), &work[
indwrk], &iwork[iindwk], info);
if (*info == 0) {
++nclus;
k = newcls + (nclus << 1);
iwork[k - 1] = newfrs;
iwork[k] = newlst;
} else {
*info = 0;
if (minrgp >= mgstol) {
mgscls = TRUE_;
} else {
/* Call DSTEIN to process this tight cluster.
This happens only if MINRGP <= MGSTOL
and DLARRF returns INFO = 1. The latter
means that a new RRR to "break" the
cluster could not be found. */
work[indwrk] = d__[ibegin];
latime_1.ops += (doublereal) (in - 1);
i__4 = in - 1;
for (k = 1; k <= i__4; ++k) {
work[indwrk + k] = d__[ibegin + k] +
work[indlld + k];
/* L60: */
}
i__4 = newsiz;
for (k = 1; k <= i__4; ++k) {
iwork[iindwk + k - 1] = 1;
/* L70: */
}
i__4 = newlst;
for (k = newfrs; k <= i__4; ++k) {
isuppz[(ibegin + k << 1) - 3] = 1;
isuppz[(ibegin + k << 1) - 2] = in;
/* L80: */
}
temp[0] = in;
dstein_(&in, &work[indwrk], &work[indld +
1], &newsiz, &work[newfrs], &
iwork[iindwk], temp, &z___ref(
ibegin, newftt), ldz, &work[
indwrk + in], &iwork[iindwk + in],
&iwork[iindwk + (in << 1)], &
iinfo);
if (iinfo != 0) {
*info = 2;
return 0;
}
ndone += newsiz;
}
}
}
} else {
mgscls = FALSE_;
}
if (newsiz == 1 || mgscls) {
ktot = newftt;
i__4 = newlst;
for (k = newfrs; k <= i__4; ++k) {
iter = 0;
L90:
lambda = work[k];
dlar1v_(&in, &c__1, &in, &lambda, &d__[ibegin], &
l[ibegin], &work[indld + 1], &work[indlld
+ 1], &gersch[(oldien << 1) + 1], &
z___ref(ibegin, ktot), &ztz, &mingma, &
iwork[iindr + ktot], &isuppz[(ktot << 1)
- 1], &work[indwrk]);
latime_1.ops += 4.;
tmp1 = 1. / ztz;
nrminv = sqrt(tmp1);
resid = abs(mingma) * nrminv;
rqcorr = mingma * tmp1;
if (k == in) {
gap = work[indgap + k - 1];
} else if (k == 1) {
gap = work[indgap + k];
} else {
/* Computing MIN */
d__1 = work[indgap + k - 1], d__2 = work[
indgap + k];
gap = min(d__1,d__2);
}
++iter;
latime_1.ops += 3.;
if (resid > *tol * gap && abs(rqcorr) > eps * 4. *
abs(lambda)) {
latime_1.ops += 1.;
work[k] = lambda + rqcorr;
if (iter < maxitr) {
goto L90;
}
}
iwork[ktot] = 1;
if (newsiz == 1) {
++ndone;
}
latime_1.ops += (doublereal) in;
dscal_(&in, &nrminv, &z___ref(ibegin, ktot), &
c__1);
++ktot;
/* L100: */
}
if (newsiz > 1) {
itmp1 = isuppz[(newftt << 1) - 1];
itmp2 = isuppz[newftt * 2];
ktot = oldien + newlst;
i__4 = ktot;
for (p = newftt + 1; p <= i__4; ++p) {
i__5 = p - 1;
for (q = newftt; q <= i__5; ++q) {
latime_1.ops += (doublereal) (in << 2);
tmp1 = -ddot_(&in, &z___ref(ibegin, p), &
c__1, &z___ref(ibegin, q), &c__1);
daxpy_(&in, &tmp1, &z___ref(ibegin, q), &
c__1, &z___ref(ibegin, p), &c__1);
/* L110: */
}
latime_1.ops += (doublereal) (in * 3 + 1);
tmp1 = 1. / dnrm2_(&in, &z___ref(ibegin, p), &
c__1);
dscal_(&in, &tmp1, &z___ref(ibegin, p), &c__1)
;
/* Computing MIN */
i__5 = itmp1, i__6 = isuppz[(p << 1) - 1];
itmp1 = min(i__5,i__6);
/* Computing MAX */
i__5 = itmp2, i__6 = isuppz[p * 2];
itmp2 = max(i__5,i__6);
/* L120: */
}
i__4 = ktot;
for (p = newftt; p <= i__4; ++p) {
isuppz[(p << 1) - 1] = itmp1;
isuppz[p * 2] = itmp2;
/* L130: */
}
ndone += newsiz;
}
}
newfrs = j + 1;
L140:
;
}
/* L150: */
}
++ndepth;
goto L40;
}
j = ibegin << 1;
i__2 = iend;
for (i__ = ibegin; i__ <= i__2; ++i__) {
isuppz[j - 1] += oldien;
isuppz[j] += oldien;
j += 2;
/* L160: */
}
ibegin = iend + 1;
L170:
;
}
return 0;
/* End of DLARRV */
} /* dlarrv_ */
int PMR_process_r_task(refine_t *rf, int tid, val_t *Wstruct,
tol_t *tolstruct, double *work, int *iwork)
{
int rf_begin = rf->begin;
double *D = rf->D;
double *DLL = rf->DLL;
int p = rf->p;
int q = rf->q;
int bl_size = rf->bl_size;
double bl_spdiam = rf->bl_spdiam;
subtasks_t *sts = rf->sts;
double *Wshifted = Wstruct->Wshifted;
double *Werr = Wstruct->Werr;
double *Wgap = Wstruct->Wgap;
int *Windex = Wstruct->Windex;
double rtol1 = tolstruct->rtol1;
double rtol2 = tolstruct->rtol2;
double pivmin = tolstruct->pivmin;
/* Others */
int info, offset, taskcount, rf_end, i;
double sigma;
double *restrict L;
double *restrict W;
offset = Windex[rf_begin] - 1;
/* Bisection to refine the eigenvalues */
dlarrb_(&bl_size, D, DLL, &p, &q, &rtol1, &rtol2,
&offset, &Wshifted[rf_begin], &Wgap[rf_begin], &Werr[rf_begin],
work, iwork, &pivmin, &bl_spdiam, &bl_size, &info);
assert(info == 0);
taskcount = PMR_decrement_counter(sts->counter, 1);
if (taskcount == 0) {
L = sts->RRR->L;
W = Wstruct->W;
rf_begin = sts->cl->begin;
for (i=0; i<sts->num_tasks; i++) {
rf_end = rf_begin + sts->chunk - 1;
Wgap[rf_end] = Wshifted[rf_end + 1] - Werr[rf_end + 1]
- Wshifted[rf_end] - Werr[rf_end];
rf_begin = rf_end + 1;
}
sigma = L[bl_size-1];
/* refined eigenvalues with all shifts applied in W */
for ( i=sts->cl->begin; i<=sts->cl->end; i++ ) {
W[i] = Wshifted[i] + sigma;
}
/* create subtasks */
info = PMR_create_subtasks(sts->cl, tid, sts->nthreads, sts->num_left,
sts->workQ, sts->RRR, Wstruct,
sts->Zstruct, tolstruct, work, iwork);
assert(info == 0);
PMR_destroy_counter(sts->counter);
free(sts);
}
free(rf);
return(0);
}
/* Subroutine */
int dlarre_(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 lsame_(char *, char *);
integer iinfo;
extern /* Subroutine */
int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *);
logical norep;
extern /* Subroutine */
int dlasq2_(integer *, doublereal *, integer *);
extern doublereal dlamch_(char *);
integer ibegin;
logical forceb;
integer irange;
doublereal sgndef;
extern /* Subroutine */
int dlarra_(integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *), dlarrb_(integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlarrc_(char * , integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *);
integer wbegin;
extern /* Subroutine */
int dlarrd_(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 dlarrk_(integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *);
logical usedqd;
doublereal clwdth, isleft;
extern /* Subroutine */
int dlarnv_(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 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
--iwork;
--work;
--gers;
--indexw;
--iblock;
--wgap;
--werr;
--w;
--isplit;
--e2;
--e;
--d__;
/* Function Body */
*info = 0;
/* Decode RANGE */
irange = 0;
if (lsame_(range, "A"))
{
irange = 1;
}
else if (lsame_(range, "V"))
{
irange = 3;
}
else if (lsame_(range, "I"))
{
irange = 2;
}
*m = 0;
/* Get machine constants */
safmin = dlamch_("S");
eps = dlamch_("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]; // , expr subst
gl = min(d__1,d__2);
gers[i__ * 2] = d__[i__] + tmp1;
/* Computing MAX */
d__1 = gu;
d__2 = gers[i__ * 2]; // , expr subst
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; // , expr subst
*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 */
dlarra_(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 */
dlarrd_(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__]); // , expr subst
wgap[i__] = max(d__1,d__2);
/* L30: */
}
/* Computing MAX */
d__1 = 0.;
d__2 = *vu - sigma - (w[wend] + werr[wend]); // , expr subst
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 */
dlarrk_(&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)); // , expr subst
isleft = max(d__2,d__3);
dlarrk_(&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)); // , expr subst
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)); // , expr subst
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)); // , expr subst
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)
{
dlarrc_("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); // , expr subst
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]; // , expr subst
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]; // , expr subst
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); // , expr subst
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. */
dcopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
i__2 = in - 1;
dcopy_(&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;
dlarnv_(&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;
dlarrb_(&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]); // , expr subst
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.;
dlasq2_(&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__]); // , expr subst
wgap[i__] = max(d__1,d__2);
/* L166: */
}
/* Computing MAX */
d__1 = 0.;
d__2 = *vu - sigma - (w[*m] + werr[*m]); // , expr subst
wgap[*m] = max(d__1,d__2);
}
/* proceed with next block */
ibegin = iend + 1;
wbegin = wend + 1;
L170:
;
}
return 0;
/* end of DLARRE */
}
