#include "blaswrap.h" #include "f2c.h" /* 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) { /* -- 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, 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 ===================================================================== Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c__2 = 2; /* System generated locals */ integer i__1, i__2; real r__1, r__2, r__3; /* Builtin functions */ double sqrt(doublereal), log(doublereal); /* Local variables */ static integer i__, j; static real s1, s2; static integer mb; static real gl; static integer in, mm; static real gu; static integer cnt; static real eps, tau, tmp, rtl; static integer cnt1, cnt2; static real tmp1, eabs; static integer iend, jblk; static real eold; static integer indl; static real dmax__, emax; static integer wend, idum, indu; static real rtol; static integer iseed[4]; static real avgap, sigma; extern logical lsame_(char *, char *); static integer iinfo; static logical norep; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *), slasq2_(integer *, real *, integer *); static integer ibegin; static logical forceb; static integer irange; static real sgndef; extern doublereal slamch_(char *); static integer wbegin; static real safmin, spdiam; extern /* Subroutine */ int slarra_(integer *, real *, real *, real *, real *, real *, integer *, integer *, integer *); static logical usedqd; static 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 *); static real isrght, bsrtol, dpivot; extern /* Subroutine */ int slarnv_(integer *, integer *, integer *, real *); --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_ */