zlarrv.c

Go to the documentation of this file.

/* zlarrv.f -- translated by f2c (version 20061008).
   You must link the resulting object file with libf2c:
        on Microsoft Windows system, link with libf2c.lib;
        on Linux or Unix systems, link with .../path/to/libf2c.a -lm
        or, if you install libf2c.a in a standard place, with -lf2c -lm
        -- in that order, at the end of the command line, as in
                cc *.o -lf2c -lm
        Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,

                http://www.netlib.org/f2c/libf2c.zip
*/

#include "f2c.h"
#include "blaswrap.h"

/* Table of constant values */

static doublecomplex c_b1 = {0.,0.};
static integer c__1 = 1;
static integer c__2 = 2;
static doublereal c_b28 = 0.;

/* Subroutine */ int zlarrv_(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, 
         doublecomplex *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;
    doublecomplex z__1;
    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, 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, indin1, indin2;
    logical stp2ii;
    extern /* Subroutine */ int zlar1v_(integer *, integer *, integer *, 
            doublereal *, doublereal *, doublereal *, doublereal *, 
            doublereal *, doublereal *, doublereal *, doublecomplex *, 
            logical *, integer *, doublereal *, doublereal *, integer *, 
            integer *, doublereal *, doublereal *, doublereal *, doublereal *)
            ;
    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 zdscal_(integer *, doublereal *, 
            doublecomplex *, 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;
    logical tryrqc;
    integer isupmx;
    doublereal rqcorr;
    extern /* Subroutine */ int zlaset_(char *, integer *, integer *, 
            doublecomplex *, doublecomplex *, doublecomplex *, integer *);


/*  -- LAPACK auxiliary routine (version 3.2) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

/*     .. Scalar Arguments .. */
/*     .. */
/*     .. Array Arguments .. */
/*     .. */

/*  Purpose */
/*  ======= */

/*  ZLARRV 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) COMPLEX*16       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 ZLARRV. */
/*          < 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;
    indin1 = *n * 3 + 1;
    indin2 = (*n << 2) + 1;
    indwrk = *n * 5 + 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;
    zlaset_("Full", n, &zusedw, &c_b1, &c_b1, &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;
            i__2 = ibegin + wbegin * z_dim1;
            z__[i__2].r = 1., z__[i__2].i = 0.;
            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;
                        }
                    }
                    i__3 = in - 1;
                    for (k = 1; k <= i__3; ++k) {
                        i__4 = ibegin + k - 1 + j * z_dim1;
                        d__[ibegin + k - 1] = z__[i__4].r;
                        i__4 = ibegin + k - 1 + (j + 1) * z_dim1;
                        l[ibegin + k - 1] = z__[i__4].r;
/* L45: */
                    }
                    i__3 = iend + j * z_dim1;
                    d__[iend] = z__[i__3].r;
                    i__3 = iend + (j + 1) * z_dim1;
                    sigma = z__[i__3].r;
/*                 Set the corresponding entries in Z to zero */
                    zlaset_("Full", &in, &c__2, &c_b1, &c_b1, &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, &work[indin1], &work[
                                indin2], &work[indwrk], &iinfo);
/*                    In the complex case, DLARRF cannot write */
/*                    the new RRR directly into Z and needs an intermediate */
/*                    workspace */
                        i__4 = in - 1;
                        for (k = 1; k <= i__4; ++k) {
                            i__5 = ibegin + k - 1 + newftt * z_dim1;
                            i__6 = indin1 + k - 1;
                            z__1.r = work[i__6], z__1.i = 0.;
                            z__[i__5].r = z__1.r, z__[i__5].i = z__1.i;
                            i__5 = ibegin + k - 1 + (newftt + 1) * z_dim1;
                            i__6 = indin2 + k - 1;
                            z__1.r = work[i__6], z__1.i = 0.;
                            z__[i__5].r = z__1.r, z__[i__5].i = z__1.i;
/* L56: */
                        }
                        i__4 = iend + newftt * z_dim1;
                        i__5 = indin1 + in - 1;
                        z__1.r = work[i__5], z__1.i = 0.;
                        z__[i__4].r = z__1.r, z__[i__4].i = z__1.i;
                        if (iinfo == 0) {
/*                       a new RRR for the cluster was found by DLARRF */
/*                       update shift and store it */
                            ssigma = sigma + tau;
                            i__4 = iend + (newftt + 1) * z_dim1;
                            z__1.r = ssigma, z__1.i = 0.;
                            z__[i__4].r = z__1.r, z__[i__4].i = z__1.i;
/*                       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_b28, &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;
                        zlar1v_(&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;
                                zlar1v_(&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) {
                                i__5 = ii + windex * z_dim1;
                                z__[i__5].r = 0., z__[i__5].i = 0.;
/* L122: */
                            }
                        }
                        if (isupmx > zto) {
                            i__4 = isupmx;
                            for (ii = zto + 1; ii <= i__4; ++ii) {
                                i__5 = ii + windex * z_dim1;
                                z__[i__5].r = 0., z__[i__5].i = 0.;
/* L123: */
                            }
                        }
                        i__4 = zto - zfrom + 1;
                        zdscal_(&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 ZLARRV */

} /* zlarrv_ */