--- /dev/null
+#include "clapack.h"
+
+/* Table of constant values */
+
+static integer c__1 = 1;
+static integer c__2 = 2;
+
+/* Subroutine */ int slarre_(char *range, integer *n, real *vl, real *vu,
+ integer *il, integer *iu, real *d__, real *e, real *e2, real *rtol1,
+ real *rtol2, real *spltol, integer *nsplit, integer *isplit, integer *
+ m, real *w, real *werr, real *wgap, integer *iblock, integer *indexw,
+ real *gers, real *pivmin, real *work, integer *iwork, integer *info)
+{
+ /* System generated locals */
+ integer i__1, i__2;
+ real r__1, r__2, r__3;
+
+ /* Builtin functions */
+ double sqrt(doublereal), log(doublereal);
+
+ /* Local variables */
+ integer i__, j;
+ real s1, s2;
+ integer mb;
+ real gl;
+ integer in, mm;
+ real gu;
+ integer cnt;
+ real eps, tau, tmp, rtl;
+ integer cnt1, cnt2;
+ real tmp1, eabs;
+ integer iend, jblk;
+ real eold;
+ integer indl;
+ real dmax__, emax;
+ integer wend, idum, indu;
+ real rtol;
+ integer iseed[4];
+ real avgap, sigma;
+ extern logical lsame_(char *, char *);
+ integer iinfo;
+ logical norep;
+ extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
+ integer *), slasq2_(integer *, real *, integer *);
+ integer ibegin;
+ logical forceb;
+ integer irange;
+ real sgndef;
+ extern doublereal slamch_(char *);
+ integer wbegin;
+ real safmin, spdiam;
+ extern /* Subroutine */ int slarra_(integer *, real *, real *, real *,
+ real *, real *, integer *, integer *, integer *);
+ logical usedqd;
+ real clwdth, isleft;
+ extern /* Subroutine */ int slarrb_(integer *, real *, real *, integer *,
+ integer *, real *, real *, integer *, real *, real *, real *,
+ real *, integer *, real *, real *, integer *, integer *), slarrc_(
+ char *, integer *, real *, real *, real *, real *, real *,
+ integer *, integer *, integer *, integer *), slarrd_(char
+ *, char *, integer *, real *, real *, integer *, integer *, real *
+, real *, real *, real *, real *, real *, integer *, integer *,
+ integer *, real *, real *, real *, real *, integer *, integer *,
+ real *, integer *, integer *), slarrk_(integer *,
+ integer *, real *, real *, real *, real *, real *, real *, real *,
+ real *, integer *);
+ real isrght, bsrtol, dpivot;
+ extern /* Subroutine */ int slarnv_(integer *, integer *, integer *, real
+ *);
+
+
+/* -- LAPACK auxiliary routine (version 3.1) -- */
+/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
+/* November 2006 */
+
+/* .. Scalar Arguments .. */
+/* .. */
+/* .. Array Arguments .. */
+/* .. */
+
+/* Purpose */
+/* ======= */
+
+/* To find the desired eigenvalues of a given real symmetric */
+/* tridiagonal matrix T, SLARRE 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 */
+/* SSTEMR 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 SLASQ2) to */
+/* conpute all and then discard any unwanted one. */
+/* As an added benefit, SLARRE 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) REAL */
+/* VU (input/output) REAL */
+/* 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', SLARRE 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) REAL 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) REAL 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) REAL 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) REAL */
+/* RTOL2 (input) REAL */
+/* Parameters for bisection. */
+/* An interval [LEFT,RIGHT] has converged if */
+/* RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) */
+
+/* SPLTOL (input) REAL */
+/* 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) REAL 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 ( SLARRE may use the */
+/* remaining N-M elements as workspace). */
+
+/* WERR (output) REAL array, dimension (N) */
+/* The error bound on the corresponding eigenvalue in W. */
+
+/* WGAP (output) REAL 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) REAL 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) REAL array, dimension (6*N) */
+/* Workspace. */
+
+/* IWORK (workspace) INTEGER array, dimension (5*N) */
+/* Workspace. */
+
+/* INFO (output) INTEGER */
+/* = 0: successful exit */
+/* > 0: A problem occured in SLARRE. */
+/* < 0: One of the called subroutines signaled an internal problem. */
+/* Needs inspection of the corresponding parameter IINFO */
+/* for further information. */
+
+/* =-1: Problem in SLARRD. */
+/* = 2: No base representation could be found in MAXTRY iterations. */
+/* Increasing MAXTRY and recompilation might be a remedy. */
+/* =-3: Problem in SLARRB when computing the refined root */
+/* representation for SLASQ2. */
+/* =-4: Problem in SLARRB when preforming bisection on the */
+/* desired part of the spectrum. */
+/* =-5: Problem in SLASQ2. */
+/* =-6: Problem in SLASQ2. */
+
+/* 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 */
+
+/* ===================================================================== */
+
+/* .. 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 */
+
+ 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 = slamch_("S");
+ eps = slamch_("P");
+/* Set parameters */
+ rtl = eps * 100.f;
+/* If one were ever to ask for less initial precision in BSRTOL, */
+/* one should keep in mind that for the subset case, the extremal */
+/* eigenvalues must be at least as accurate as the current setting */
+/* (eigenvalues in the middle need not as much accuracy) */
+ bsrtol = sqrt(eps) * 5e-4f;
+/* 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.f;
+ wgap[1] = 0.f;
+ 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.f;
+ 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.f;
+ emax = 0.f;
+ e[*n] = 0.f;
+ i__1 = *n;
+ for (i__ = 1; i__ <= i__1; ++i__) {
+ werr[i__] = 0.f;
+ wgap[i__] = 0.f;
+ eabs = (r__1 = e[i__], dabs(r__1));
+ if (eabs >= emax) {
+ emax = eabs;
+ }
+ tmp1 = eabs + eold;
+ gers[(i__ << 1) - 1] = d__[i__] - tmp1;
+/* Computing MIN */
+ r__1 = gl, r__2 = gers[(i__ << 1) - 1];
+ gl = dmin(r__1,r__2);
+ gers[i__ * 2] = d__[i__] + tmp1;
+/* Computing MAX */
+ r__1 = gu, r__2 = gers[i__ * 2];
+ gu = dmax(r__1,r__2);
+ eold = eabs;
+/* L5: */
+ }
+/* The minimum pivot allowed in the Sturm sequence for T */
+/* Computing MAX */
+/* Computing 2nd power */
+ r__3 = emax;
+ r__1 = 1.f, r__2 = r__3 * r__3;
+ *pivmin = safmin * dmax(r__1,r__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 */
+ slarra_(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 SLARRD 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)) */
+/* SLARRD needs a WORK of size 4*N, IWORK of size 3*N */
+ slarrd_(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.f;
+ werr[i__] = 0.f;
+ 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.f;
+/* The gap for a single block doesn't matter for the later */
+/* algorithm and is assigned an arbitrary large value */
+ wgap[*m] = 0.f;
+ iblock[*m] = jblk;
+ indexw[*m] = 1;
+ ++wbegin;
+ }
+/* E( IEND ) holds the shift for the initial RRR */
+ e[iend] = 0.f;
+ 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.f;
+
+/* 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 */
+ r__1 = gers[(i__ << 1) - 1];
+ gl = dmin(r__1,gl);
+/* Computing MAX */
+ r__1 = gers[i__ * 2];
+ gu = dmax(r__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.f;
+ ibegin = iend + 1;
+ goto L170;
+ } else {
+/* Decide whether dqds or bisection is more efficient */
+ usedqd = (real) mb > in * .5f && ! 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.f;
+ i__2 = wend - 1;
+ for (i__ = wbegin; i__ <= i__2; ++i__) {
+/* Computing MAX */
+ r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
+ werr[i__]);
+ wgap[i__] = dmax(r__1,r__2);
+/* L30: */
+ }
+/* Computing MAX */
+ r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
+ wgap[wend] = dmax(r__1,r__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 */
+ slarrk_(&in, &c__1, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
+ rtl, &tmp, &tmp1, &iinfo);
+ if (iinfo != 0) {
+ *info = -1;
+ return 0;
+ }
+/* Computing MAX */
+ r__2 = gl, r__3 = tmp - tmp1 - eps * 100.f * (r__1 = tmp - tmp1,
+ dabs(r__1));
+ isleft = dmax(r__2,r__3);
+ slarrk_(&in, &in, &gl, &gu, &d__[ibegin], &e2[ibegin], pivmin, &
+ rtl, &tmp, &tmp1, &iinfo);
+ if (iinfo != 0) {
+ *info = -1;
+ return 0;
+ }
+/* Computing MIN */
+ r__2 = gu, r__3 = tmp + tmp1 + eps * 100.f * (r__1 = tmp + tmp1,
+ dabs(r__1));
+ isrght = dmin(r__2,r__3);
+/* Improve the estimate of the spectral diameter */
+ spdiam = isrght - isleft;
+ } else {
+/* Case of bisection */
+/* Find approximations to the wanted extremal eigenvalues */
+/* Computing MAX */
+ r__2 = gl, r__3 = w[wbegin] - werr[wbegin] - eps * 100.f * (r__1 =
+ w[wbegin] - werr[wbegin], dabs(r__1));
+ isleft = dmax(r__2,r__3);
+/* Computing MIN */
+ r__2 = gu, r__3 = w[wend] + werr[wend] + eps * 100.f * (r__1 = w[
+ wend] + werr[wend], dabs(r__1));
+ isrght = dmin(r__2,r__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 SLARRE 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 * .25f;
+ s2 = isrght - spdiam * .25f;
+ } else {
+/* SLARRD has computed IBLOCK and INDEXW for each eigenvalue */
+/* approximation. */
+/* choose sigma */
+ if (usedqd) {
+ s1 = isleft + spdiam * .25f;
+ s2 = isrght - spdiam * .25f;
+ } else {
+ tmp = dmin(isrght,*vu) - dmax(isleft,*vl);
+ s1 = dmax(isleft,*vl) + tmp * .25f;
+ s2 = dmin(isrght,*vu) - tmp * .25f;
+ }
+ }
+/* Compute the negcount at the 1/4 and 3/4 points */
+ if (mb > 1) {
+ slarrc_("T", &in, &s1, &s2, &d__[ibegin], &e[ibegin], pivmin, &
+ cnt, &cnt1, &cnt2, &iinfo);
+ }
+ if (mb == 1) {
+ sigma = gl;
+ sgndef = 1.f;
+ } else if (cnt1 - indl >= indu - cnt2) {
+ if (irange == 1 && ! forceb) {
+ sigma = dmax(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 = dmax(isleft,*vl);
+ }
+ sgndef = 1.f;
+ } else {
+ if (irange == 1 && ! forceb) {
+ sigma = dmin(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 = dmin(isrght,*vu);
+ }
+ sgndef = -1.f;
+ }
+/* 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.f;
+ } else {
+ if (mb > 1) {
+ clwdth = w[wend] + werr[wend] - w[wbegin] - werr[wbegin];
+ avgap = (r__1 = clwdth / (real) (wend - wbegin), dabs(r__1));
+ if (sgndef == 1.f) {
+/* Computing MAX */
+ r__1 = wgap[wbegin];
+ tau = dmax(r__1,avgap) * .5f;
+/* Computing MAX */
+ r__1 = tau, r__2 = werr[wbegin];
+ tau = dmax(r__1,r__2);
+ } else {
+/* Computing MAX */
+ r__1 = wgap[wend - 1];
+ tau = dmax(r__1,avgap) * .5f;
+/* Computing MAX */
+ r__1 = tau, r__2 = werr[wend];
+ tau = dmax(r__1,r__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__ = dabs(work[1]);
+ j = ibegin;
+ i__2 = in - 1;
+ for (i__ = 1; i__ <= i__2; ++i__) {
+ work[(in << 1) + i__] = 1.f / 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 */
+ r__1 = dmax__, r__2 = dabs(dpivot);
+ dmax__ = dmax(r__1,r__2);
+ ++j;
+/* L70: */
+ }
+/* check for element growth */
+ if (dmax__ > spdiam * 64.f) {
+ 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.f) {
+ 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.f) {
+/* The fudged Gerschgorin shift should succeed */
+ sigma = gl - spdiam * 2.f * eps * *n - *pivmin * 4.f;
+ } else {
+ sigma = gu + spdiam * 2.f * eps * *n + *pivmin * 4.f;
+ }
+ } else {
+ sigma -= sgndef * tau;
+ tau *= 2.f;
+ }
+ } 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. */
+ scopy_(&in, &work[1], &c__1, &d__[ibegin], &c__1);
+ i__2 = in - 1;
+ scopy_(&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;
+ slarnv_(&c__2, iseed, &i__2, &work[1]);
+ i__2 = in - 1;
+ for (i__ = 1; i__ <= i__2; ++i__) {
+ d__[ibegin + i__ - 1] *= eps * 4.f * work[i__] + 1.f;
+ e[ibegin + i__ - 1] *= eps * 4.f * work[in + i__] + 1.f;
+/* L125: */
+ }
+ d__[iend] *= eps * 4.f * work[in] + 1.f;
+
+ }
+
+/* Don't update the Gerschgorin intervals because keeping track */
+/* of the updates would be too much work in SLARRV. */
+/* 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 SLARRD has been used, shift the eigenvalue approximations */
+/* according to their representation. This is necessary for */
+/* a uniform SLARRV since dqds computes eigenvalues of the */
+/* shifted representation. In SLARRV, W will always hold the */
+/* UNshifted eigenvalue approximation. */
+ i__2 = wend;
+ for (j = wbegin; j <= i__2; ++j) {
+ w[j] -= sigma;
+ werr[j] += (r__1 = w[j], dabs(r__1)) * eps;
+/* L134: */
+ }
+/* call SLARRB to reduce eigenvalue error of the approximations */
+/* from SLARRD */
+ i__2 = iend - 1;
+ for (i__ = ibegin; i__ <= i__2; ++i__) {
+/* Computing 2nd power */
+ r__1 = e[i__];
+ work[i__] = d__[i__] * (r__1 * r__1);
+/* L135: */
+ }
+/* use bisection to find EV from INDL to INDU */
+ i__2 = indl - 1;
+ slarrb_(&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;
+ }
+/* SLARRB computes all gaps correctly except for the last one */
+/* Record distance to VU/GU */
+/* Computing MAX */
+ r__1 = 0.f, r__2 = *vu - sigma - (w[wend] + werr[wend]);
+ wgap[wend] = dmax(r__1,r__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 SLASQ2 */
+/* 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((real) in) * 4.f * eps;
+ j = ibegin;
+ i__2 = in - 1;
+ for (i__ = 1; i__ <= i__2; ++i__) {
+ work[(i__ << 1) - 1] = (r__1 = d__[j], dabs(r__1));
+ work[i__ * 2] = e[j] * e[j] * work[(i__ << 1) - 1];
+ ++j;
+/* L140: */
+ }
+ work[(in << 1) - 1] = (r__1 = d__[iend], dabs(r__1));
+ work[in * 2] = 0.f;
+ slasq2_(&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.f) {
+ *info = -6;
+ return 0;
+ }
+/* L149: */
+ }
+ }
+ if (sgndef > 0.f) {
+ 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 SLASQ2 */
+ werr[i__] = rtol * (r__1 = w[i__], dabs(r__1));
+/* L165: */
+ }
+ i__2 = *m - 1;
+ for (i__ = *m - mb + 1; i__ <= i__2; ++i__) {
+/* compute the right gap between the intervals */
+/* Computing MAX */
+ r__1 = 0.f, r__2 = w[i__ + 1] - werr[i__ + 1] - (w[i__] +
+ werr[i__]);
+ wgap[i__] = dmax(r__1,r__2);
+/* L166: */
+ }
+/* Computing MAX */
+ r__1 = 0.f, r__2 = *vu - sigma - (w[*m] + werr[*m]);
+ wgap[*m] = dmax(r__1,r__2);
+ }
+/* proceed with next block */
+ ibegin = iend + 1;
+ wbegin = wend + 1;
+L170:
+ ;
+ }
+
+ return 0;
+
+/* end of SLARRE */
+
+} /* slarre_ */
projects / opencv / blobdiff