zstein.c

Go to the documentation of this file.

/* zstein.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 integer c__2 = 2;
static integer c__1 = 1;
static integer c_n1 = -1;

/* Subroutine */ int zstein_(integer *n, doublereal *d__, doublereal *e, 
        integer *m, doublereal *w, integer *iblock, integer *isplit, 
        doublecomplex *z__, integer *ldz, doublereal *work, integer *iwork, 
        integer *ifail, integer *info)
{
    /* System generated locals */
    integer z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5;
    doublereal d__1, d__2, d__3, d__4, d__5;
    doublecomplex z__1;

    /* Builtin functions */
    double sqrt(doublereal);

    /* Local variables */
    integer i__, j, b1, j1, bn, jr;
    doublereal xj, scl, eps, sep, nrm, tol;
    integer its;
    doublereal xjm, ztr, eps1;
    integer jblk, nblk, jmax;
    extern doublereal dnrm2_(integer *, doublereal *, integer *);
    extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, 
            integer *);
    integer iseed[4], gpind, iinfo;
    extern doublereal dasum_(integer *, doublereal *, integer *);
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
            doublereal *, integer *);
    doublereal ortol;
    integer indrv1, indrv2, indrv3, indrv4, indrv5;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dlagtf_(integer *, doublereal *, doublereal *, 
             doublereal *, doublereal *, doublereal *, doublereal *, integer *
, integer *);
    extern integer idamax_(integer *, doublereal *, integer *);
    extern /* Subroutine */ int xerbla_(char *, integer *), dlagts_(
            integer *, integer *, doublereal *, doublereal *, doublereal *, 
            doublereal *, integer *, doublereal *, doublereal *, integer *);
    integer nrmchk;
    extern /* Subroutine */ int dlarnv_(integer *, integer *, integer *, 
            doublereal *);
    integer blksiz;
    doublereal onenrm, dtpcrt, pertol;


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

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

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

/*  ZSTEIN computes the eigenvectors of a real symmetric tridiagonal */
/*  matrix T corresponding to specified eigenvalues, using inverse */
/*  iteration. */

/*  The maximum number of iterations allowed for each eigenvector is */
/*  specified by an internal parameter MAXITS (currently set to 5). */

/*  Although the eigenvectors are real, they are stored in a complex */
/*  array, which may be passed to ZUNMTR or ZUPMTR for back */
/*  transformation to the eigenvectors of a complex Hermitian matrix */
/*  which was reduced to tridiagonal form. */


/*  Arguments */
/*  ========= */

/*  N       (input) INTEGER */
/*          The order of the matrix.  N >= 0. */

/*  D       (input) DOUBLE PRECISION array, dimension (N) */
/*          The n diagonal elements of the tridiagonal matrix T. */

/*  E       (input) DOUBLE PRECISION array, dimension (N-1) */
/*          The (n-1) subdiagonal elements of the tridiagonal matrix */
/*          T, stored in elements 1 to N-1. */

/*  M       (input) INTEGER */
/*          The number of eigenvectors to be found.  0 <= M <= N. */

/*  W       (input) DOUBLE PRECISION array, dimension (N) */
/*          The first M elements of W contain the eigenvalues for */
/*          which eigenvectors are to be computed.  The eigenvalues */
/*          should be grouped by split-off block and ordered from */
/*          smallest to largest within the block.  ( The output array */
/*          W from DSTEBZ with ORDER = 'B' is expected here. ) */

/*  IBLOCK  (input) INTEGER array, dimension (N) */
/*          The submatrix indices associated with the corresponding */
/*          eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to */
/*          the first submatrix from the top, =2 if W(i) belongs to */
/*          the second submatrix, etc.  ( The output array IBLOCK */
/*          from DSTEBZ is expected here. ) */

/*  ISPLIT  (input) INTEGER array, dimension (N) */
/*          The splitting points, at which T breaks up into submatrices. */
/*          The first submatrix consists of rows/columns 1 to */
/*          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 */
/*          through ISPLIT( 2 ), etc. */
/*          ( The output array ISPLIT from DSTEBZ is expected here. ) */

/*  Z       (output) COMPLEX*16 array, dimension (LDZ, M) */
/*          The computed eigenvectors.  The eigenvector associated */
/*          with the eigenvalue W(i) is stored in the i-th column of */
/*          Z.  Any vector which fails to converge is set to its current */
/*          iterate after MAXITS iterations. */
/*          The imaginary parts of the eigenvectors are set to zero. */

/*  LDZ     (input) INTEGER */
/*          The leading dimension of the array Z.  LDZ >= max(1,N). */

/*  WORK    (workspace) DOUBLE PRECISION array, dimension (5*N) */

/*  IWORK   (workspace) INTEGER array, dimension (N) */

/*  IFAIL   (output) INTEGER array, dimension (M) */
/*          On normal exit, all elements of IFAIL are zero. */
/*          If one or more eigenvectors fail to converge after */
/*          MAXITS iterations, then their indices are stored in */
/*          array IFAIL. */

/*  INFO    (output) INTEGER */
/*          = 0: successful exit */
/*          < 0: if INFO = -i, the i-th argument had an illegal value */
/*          > 0: if INFO = i, then i eigenvectors failed to converge */
/*               in MAXITS iterations.  Their indices are stored in */
/*               array IFAIL. */

/*  Internal Parameters */
/*  =================== */

/*  MAXITS  INTEGER, default = 5 */
/*          The maximum number of iterations performed. */

/*  EXTRA   INTEGER, default = 2 */
/*          The number of iterations performed after norm growth */
/*          criterion is satisfied, should be at least 1. */

/* ===================================================================== */

/*     .. Parameters .. */
/*     .. */
/*     .. Local Scalars .. */
/*     .. */
/*     .. Local Arrays .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. Executable Statements .. */

/*     Test the input parameters. */

    /* Parameter adjustments */
    --d__;
    --e;
    --w;
    --iblock;
    --isplit;
    z_dim1 = *ldz;
    z_offset = 1 + z_dim1;
    z__ -= z_offset;
    --work;
    --iwork;
    --ifail;

    /* Function Body */
    *info = 0;
    i__1 = *m;
    for (i__ = 1; i__ <= i__1; ++i__) {
        ifail[i__] = 0;
/* L10: */
    }

    if (*n < 0) {
        *info = -1;
    } else if (*m < 0 || *m > *n) {
        *info = -4;
    } else if (*ldz < max(1,*n)) {
        *info = -9;
    } else {
        i__1 = *m;
        for (j = 2; j <= i__1; ++j) {
            if (iblock[j] < iblock[j - 1]) {
                *info = -6;
                goto L30;
            }
            if (iblock[j] == iblock[j - 1] && w[j] < w[j - 1]) {
                *info = -5;
                goto L30;
            }
/* L20: */
        }
L30:
        ;
    }

    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("ZSTEIN", &i__1);
        return 0;
    }

/*     Quick return if possible */

    if (*n == 0 || *m == 0) {
        return 0;
    } else if (*n == 1) {
        i__1 = z_dim1 + 1;
        z__[i__1].r = 1., z__[i__1].i = 0.;
        return 0;
    }

/*     Get machine constants. */

    eps = dlamch_("Precision");

/*     Initialize seed for random number generator DLARNV. */

    for (i__ = 1; i__ <= 4; ++i__) {
        iseed[i__ - 1] = 1;
/* L40: */
    }

/*     Initialize pointers. */

    indrv1 = 0;
    indrv2 = indrv1 + *n;
    indrv3 = indrv2 + *n;
    indrv4 = indrv3 + *n;
    indrv5 = indrv4 + *n;

/*     Compute eigenvectors of matrix blocks. */

    j1 = 1;
    i__1 = iblock[*m];
    for (nblk = 1; nblk <= i__1; ++nblk) {

/*        Find starting and ending indices of block nblk. */

        if (nblk == 1) {
            b1 = 1;
        } else {
            b1 = isplit[nblk - 1] + 1;
        }
        bn = isplit[nblk];
        blksiz = bn - b1 + 1;
        if (blksiz == 1) {
            goto L60;
        }
        gpind = b1;

/*        Compute reorthogonalization criterion and stopping criterion. */

        onenrm = (d__1 = d__[b1], abs(d__1)) + (d__2 = e[b1], abs(d__2));
/* Computing MAX */
        d__3 = onenrm, d__4 = (d__1 = d__[bn], abs(d__1)) + (d__2 = e[bn - 1],
                 abs(d__2));
        onenrm = max(d__3,d__4);
        i__2 = bn - 1;
        for (i__ = b1 + 1; i__ <= i__2; ++i__) {
/* Computing MAX */
            d__4 = onenrm, d__5 = (d__1 = d__[i__], abs(d__1)) + (d__2 = e[
                    i__ - 1], abs(d__2)) + (d__3 = e[i__], abs(d__3));
            onenrm = max(d__4,d__5);
/* L50: */
        }
        ortol = onenrm * .001;

        dtpcrt = sqrt(.1 / blksiz);

/*        Loop through eigenvalues of block nblk. */

L60:
        jblk = 0;
        i__2 = *m;
        for (j = j1; j <= i__2; ++j) {
            if (iblock[j] != nblk) {
                j1 = j;
                goto L180;
            }
            ++jblk;
            xj = w[j];

/*           Skip all the work if the block size is one. */

            if (blksiz == 1) {
                work[indrv1 + 1] = 1.;
                goto L140;
            }

/*           If eigenvalues j and j-1 are too close, add a relatively */
/*           small perturbation. */

            if (jblk > 1) {
                eps1 = (d__1 = eps * xj, abs(d__1));
                pertol = eps1 * 10.;
                sep = xj - xjm;
                if (sep < pertol) {
                    xj = xjm + pertol;
                }
            }

            its = 0;
            nrmchk = 0;

/*           Get random starting vector. */

            dlarnv_(&c__2, iseed, &blksiz, &work[indrv1 + 1]);

/*           Copy the matrix T so it won't be destroyed in factorization. */

            dcopy_(&blksiz, &d__[b1], &c__1, &work[indrv4 + 1], &c__1);
            i__3 = blksiz - 1;
            dcopy_(&i__3, &e[b1], &c__1, &work[indrv2 + 2], &c__1);
            i__3 = blksiz - 1;
            dcopy_(&i__3, &e[b1], &c__1, &work[indrv3 + 1], &c__1);

/*           Compute LU factors with partial pivoting  ( PT = LU ) */

            tol = 0.;
            dlagtf_(&blksiz, &work[indrv4 + 1], &xj, &work[indrv2 + 2], &work[
                    indrv3 + 1], &tol, &work[indrv5 + 1], &iwork[1], &iinfo);

/*           Update iteration count. */

L70:
            ++its;
            if (its > 5) {
                goto L120;
            }

/*           Normalize and scale the righthand side vector Pb. */

/* Computing MAX */
            d__2 = eps, d__3 = (d__1 = work[indrv4 + blksiz], abs(d__1));
            scl = blksiz * onenrm * max(d__2,d__3) / dasum_(&blksiz, &work[
                    indrv1 + 1], &c__1);
            dscal_(&blksiz, &scl, &work[indrv1 + 1], &c__1);

/*           Solve the system LU = Pb. */

            dlagts_(&c_n1, &blksiz, &work[indrv4 + 1], &work[indrv2 + 2], &
                    work[indrv3 + 1], &work[indrv5 + 1], &iwork[1], &work[
                    indrv1 + 1], &tol, &iinfo);

/*           Reorthogonalize by modified Gram-Schmidt if eigenvalues are */
/*           close enough. */

            if (jblk == 1) {
                goto L110;
            }
            if ((d__1 = xj - xjm, abs(d__1)) > ortol) {
                gpind = j;
            }
            if (gpind != j) {
                i__3 = j - 1;
                for (i__ = gpind; i__ <= i__3; ++i__) {
                    ztr = 0.;
                    i__4 = blksiz;
                    for (jr = 1; jr <= i__4; ++jr) {
                        i__5 = b1 - 1 + jr + i__ * z_dim1;
                        ztr += work[indrv1 + jr] * z__[i__5].r;
/* L80: */
                    }
                    i__4 = blksiz;
                    for (jr = 1; jr <= i__4; ++jr) {
                        i__5 = b1 - 1 + jr + i__ * z_dim1;
                        work[indrv1 + jr] -= ztr * z__[i__5].r;
/* L90: */
                    }
/* L100: */
                }
            }

/*           Check the infinity norm of the iterate. */

L110:
            jmax = idamax_(&blksiz, &work[indrv1 + 1], &c__1);
            nrm = (d__1 = work[indrv1 + jmax], abs(d__1));

/*           Continue for additional iterations after norm reaches */
/*           stopping criterion. */

            if (nrm < dtpcrt) {
                goto L70;
            }
            ++nrmchk;
            if (nrmchk < 3) {
                goto L70;
            }

            goto L130;

/*           If stopping criterion was not satisfied, update info and */
/*           store eigenvector number in array ifail. */

L120:
            ++(*info);
            ifail[*info] = j;

/*           Accept iterate as jth eigenvector. */

L130:
            scl = 1. / dnrm2_(&blksiz, &work[indrv1 + 1], &c__1);
            jmax = idamax_(&blksiz, &work[indrv1 + 1], &c__1);
            if (work[indrv1 + jmax] < 0.) {
                scl = -scl;
            }
            dscal_(&blksiz, &scl, &work[indrv1 + 1], &c__1);
L140:
            i__3 = *n;
            for (i__ = 1; i__ <= i__3; ++i__) {
                i__4 = i__ + j * z_dim1;
                z__[i__4].r = 0., z__[i__4].i = 0.;
/* L150: */
            }
            i__3 = blksiz;
            for (i__ = 1; i__ <= i__3; ++i__) {
                i__4 = b1 + i__ - 1 + j * z_dim1;
                i__5 = indrv1 + i__;
                z__1.r = work[i__5], z__1.i = 0.;
                z__[i__4].r = z__1.r, z__[i__4].i = z__1.i;
/* L160: */
            }

/*           Save the shift to check eigenvalue spacing at next */
/*           iteration. */

            xjm = xj;

/* L170: */
        }
L180:
        ;
    }

    return 0;

/*     End of ZSTEIN */

} /* zstein_ */