dsbevx.c

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/* dsbevx.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 doublereal c_b14 = 1.;
static integer c__1 = 1;
static doublereal c_b34 = 0.;

/* Subroutine */ int dsbevx_(char *jobz, char *range, char *uplo, integer *n, 
        integer *kd, doublereal *ab, integer *ldab, doublereal *q, integer *
        ldq, doublereal *vl, doublereal *vu, integer *il, integer *iu, 
        doublereal *abstol, integer *m, doublereal *w, doublereal *z__, 
        integer *ldz, doublereal *work, integer *iwork, integer *ifail, 
        integer *info)
{
    /* System generated locals */
    integer ab_dim1, ab_offset, q_dim1, q_offset, z_dim1, z_offset, i__1, 
            i__2;
    doublereal d__1, d__2;

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

    /* Local variables */
    integer i__, j, jj;
    doublereal eps, vll, vuu, tmp1;
    integer indd, inde;
    doublereal anrm;
    integer imax;
    doublereal rmin, rmax;
    logical test;
    integer itmp1, indee;
    extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, 
            integer *);
    doublereal sigma;
    extern logical lsame_(char *, char *);
    extern /* Subroutine */ int dgemv_(char *, integer *, integer *, 
            doublereal *, doublereal *, integer *, doublereal *, integer *, 
            doublereal *, doublereal *, integer *);
    integer iinfo;
    char order[1];
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
            doublereal *, integer *), dswap_(integer *, doublereal *, integer 
            *, doublereal *, integer *);
    logical lower, wantz;
    extern doublereal dlamch_(char *);
    logical alleig, indeig;
    integer iscale, indibl;
    extern /* Subroutine */ int dlascl_(char *, integer *, integer *, 
            doublereal *, doublereal *, integer *, integer *, doublereal *, 
            integer *, integer *);
    extern doublereal dlansb_(char *, char *, integer *, integer *, 
            doublereal *, integer *, doublereal *);
    logical valeig;
    extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, 
            doublereal *, integer *, doublereal *, integer *);
    doublereal safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *);
    doublereal abstll, bignum;
    extern /* Subroutine */ int dsbtrd_(char *, char *, integer *, integer *, 
            doublereal *, integer *, doublereal *, doublereal *, doublereal *, 
             integer *, doublereal *, integer *);
    integer indisp;
    extern /* Subroutine */ int dstein_(integer *, doublereal *, doublereal *, 
             integer *, doublereal *, integer *, integer *, doublereal *, 
            integer *, doublereal *, integer *, integer *, integer *), 
            dsterf_(integer *, doublereal *, doublereal *, integer *);
    integer indiwo;
    extern /* Subroutine */ int dstebz_(char *, char *, integer *, doublereal 
            *, doublereal *, integer *, integer *, doublereal *, doublereal *, 
             doublereal *, integer *, integer *, doublereal *, integer *, 
            integer *, doublereal *, integer *, integer *);
    integer indwrk;
    extern /* Subroutine */ int dsteqr_(char *, integer *, doublereal *, 
            doublereal *, doublereal *, integer *, doublereal *, integer *);
    integer nsplit;
    doublereal smlnum;


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

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

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

/*  DSBEVX computes selected eigenvalues and, optionally, eigenvectors */
/*  of a real symmetric band matrix A.  Eigenvalues and eigenvectors can */
/*  be selected by specifying either a range of values or a range of */
/*  indices for the desired eigenvalues. */

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

/*  JOBZ    (input) CHARACTER*1 */
/*          = 'N':  Compute eigenvalues only; */
/*          = 'V':  Compute eigenvalues and eigenvectors. */

/*  RANGE   (input) CHARACTER*1 */
/*          = 'A': all eigenvalues will be found; */
/*          = 'V': all eigenvalues in the half-open interval (VL,VU] */
/*                 will be found; */
/*          = 'I': the IL-th through IU-th eigenvalues will be found. */

/*  UPLO    (input) CHARACTER*1 */
/*          = 'U':  Upper triangle of A is stored; */
/*          = 'L':  Lower triangle of A is stored. */

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

/*  KD      (input) INTEGER */
/*          The number of superdiagonals of the matrix A if UPLO = 'U', */
/*          or the number of subdiagonals if UPLO = 'L'.  KD >= 0. */

/*  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB, N) */
/*          On entry, the upper or lower triangle of the symmetric band */
/*          matrix A, stored in the first KD+1 rows of the array.  The */
/*          j-th column of A is stored in the j-th column of the array AB */
/*          as follows: */
/*          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/*          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd). */

/*          On exit, AB is overwritten by values generated during the */
/*          reduction to tridiagonal form.  If UPLO = 'U', the first */
/*          superdiagonal and the diagonal of the tridiagonal matrix T */
/*          are returned in rows KD and KD+1 of AB, and if UPLO = 'L', */
/*          the diagonal and first subdiagonal of T are returned in the */
/*          first two rows of AB. */

/*  LDAB    (input) INTEGER */
/*          The leading dimension of the array AB.  LDAB >= KD + 1. */

/*  Q       (output) DOUBLE PRECISION array, dimension (LDQ, N) */
/*          If JOBZ = 'V', the N-by-N orthogonal matrix used in the */
/*                         reduction to tridiagonal form. */
/*          If JOBZ = 'N', the array Q is not referenced. */

/*  LDQ     (input) INTEGER */
/*          The leading dimension of the array Q.  If JOBZ = 'V', then */
/*          LDQ >= max(1,N). */

/*  VL      (input) DOUBLE PRECISION */
/*  VU      (input) DOUBLE PRECISION */
/*          If RANGE='V', the lower and upper bounds of the interval to */
/*          be searched for eigenvalues. VL < VU. */
/*          Not referenced if RANGE = 'A' or 'I'. */

/*  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, if N > 0; IL = 1 and IU = 0 if N = 0. */
/*          Not referenced if RANGE = 'A' or 'V'. */

/*  ABSTOL  (input) DOUBLE PRECISION */
/*          The absolute error tolerance for the eigenvalues. */
/*          An approximate eigenvalue is accepted as converged */
/*          when it is determined to lie in an interval [a,b] */
/*          of width less than or equal to */

/*                  ABSTOL + EPS *   max( |a|,|b| ) , */

/*          where EPS is the machine precision.  If ABSTOL is less than */
/*          or equal to zero, then  EPS*|T|  will be used in its place, */
/*          where |T| is the 1-norm of the tridiagonal matrix obtained */
/*          by reducing AB to tridiagonal form. */

/*          Eigenvalues will be computed most accurately when ABSTOL is */
/*          set to twice the underflow threshold 2*DLAMCH('S'), not zero. */
/*          If this routine returns with INFO>0, indicating that some */
/*          eigenvectors did not converge, try setting ABSTOL to */
/*          2*DLAMCH('S'). */

/*          See "Computing Small Singular Values of Bidiagonal Matrices */
/*          with Guaranteed High Relative Accuracy," by Demmel and */
/*          Kahan, LAPACK Working Note #3. */

/*  M       (output) INTEGER */
/*          The total number of eigenvalues found.  0 <= M <= N. */
/*          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. */

/*  W       (output) DOUBLE PRECISION array, dimension (N) */
/*          The first M elements contain the selected eigenvalues in */
/*          ascending order. */

/*  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) */
/*          If JOBZ = 'V', then if INFO = 0, the first M columns of Z */
/*          contain the orthonormal eigenvectors of the matrix A */
/*          corresponding to the selected eigenvalues, with the i-th */
/*          column of Z holding the eigenvector associated with W(i). */
/*          If an eigenvector fails to converge, then that column of Z */
/*          contains the latest approximation to the eigenvector, and the */
/*          index of the eigenvector is returned in IFAIL. */
/*          If JOBZ = 'N', then Z is not referenced. */
/*          Note: the user must ensure that at least max(1,M) columns are */
/*          supplied in the array Z; if RANGE = 'V', the exact value of M */
/*          is not known in advance and an upper bound must be used. */

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

/*  WORK    (workspace) DOUBLE PRECISION array, dimension (7*N) */

/*  IWORK   (workspace) INTEGER array, dimension (5*N) */

/*  IFAIL   (output) INTEGER array, dimension (N) */
/*          If JOBZ = 'V', then if INFO = 0, the first M elements of */
/*          IFAIL are zero.  If INFO > 0, then IFAIL contains the */
/*          indices of the eigenvectors that failed to converge. */
/*          If JOBZ = 'N', then IFAIL is not referenced. */

/*  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. */
/*                Their indices are stored in array IFAIL. */

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

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

/*     Test the input parameters. */

    /* Parameter adjustments */
    ab_dim1 = *ldab;
    ab_offset = 1 + ab_dim1;
    ab -= ab_offset;
    q_dim1 = *ldq;
    q_offset = 1 + q_dim1;
    q -= q_offset;
    --w;
    z_dim1 = *ldz;
    z_offset = 1 + z_dim1;
    z__ -= z_offset;
    --work;
    --iwork;
    --ifail;

    /* Function Body */
    wantz = lsame_(jobz, "V");
    alleig = lsame_(range, "A");
    valeig = lsame_(range, "V");
    indeig = lsame_(range, "I");
    lower = lsame_(uplo, "L");

    *info = 0;
    if (! (wantz || lsame_(jobz, "N"))) {
        *info = -1;
    } else if (! (alleig || valeig || indeig)) {
        *info = -2;
    } else if (! (lower || lsame_(uplo, "U"))) {
        *info = -3;
    } else if (*n < 0) {
        *info = -4;
    } else if (*kd < 0) {
        *info = -5;
    } else if (*ldab < *kd + 1) {
        *info = -7;
    } else if (wantz && *ldq < max(1,*n)) {
        *info = -9;
    } else {
        if (valeig) {
            if (*n > 0 && *vu <= *vl) {
                *info = -11;
            }
        } else if (indeig) {
            if (*il < 1 || *il > max(1,*n)) {
                *info = -12;
            } else if (*iu < min(*n,*il) || *iu > *n) {
                *info = -13;
            }
        }
    }
    if (*info == 0) {
        if (*ldz < 1 || wantz && *ldz < *n) {
            *info = -18;
        }
    }

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

/*     Quick return if possible */

    *m = 0;
    if (*n == 0) {
        return 0;
    }

    if (*n == 1) {
        *m = 1;
        if (lower) {
            tmp1 = ab[ab_dim1 + 1];
        } else {
            tmp1 = ab[*kd + 1 + ab_dim1];
        }
        if (valeig) {
            if (! (*vl < tmp1 && *vu >= tmp1)) {
                *m = 0;
            }
        }
        if (*m == 1) {
            w[1] = tmp1;
            if (wantz) {
                z__[z_dim1 + 1] = 1.;
            }
        }
        return 0;
    }

/*     Get machine constants. */

    safmin = dlamch_("Safe minimum");
    eps = dlamch_("Precision");
    smlnum = safmin / eps;
    bignum = 1. / smlnum;
    rmin = sqrt(smlnum);
/* Computing MIN */
    d__1 = sqrt(bignum), d__2 = 1. / sqrt(sqrt(safmin));
    rmax = min(d__1,d__2);

/*     Scale matrix to allowable range, if necessary. */

    iscale = 0;
    abstll = *abstol;
    if (valeig) {
        vll = *vl;
        vuu = *vu;
    } else {
        vll = 0.;
        vuu = 0.;
    }
    anrm = dlansb_("M", uplo, n, kd, &ab[ab_offset], ldab, &work[1]);
    if (anrm > 0. && anrm < rmin) {
        iscale = 1;
        sigma = rmin / anrm;
    } else if (anrm > rmax) {
        iscale = 1;
        sigma = rmax / anrm;
    }
    if (iscale == 1) {
        if (lower) {
            dlascl_("B", kd, kd, &c_b14, &sigma, n, n, &ab[ab_offset], ldab, 
                    info);
        } else {
            dlascl_("Q", kd, kd, &c_b14, &sigma, n, n, &ab[ab_offset], ldab, 
                    info);
        }
        if (*abstol > 0.) {
            abstll = *abstol * sigma;
        }
        if (valeig) {
            vll = *vl * sigma;
            vuu = *vu * sigma;
        }
    }

/*     Call DSBTRD to reduce symmetric band matrix to tridiagonal form. */

    indd = 1;
    inde = indd + *n;
    indwrk = inde + *n;
    dsbtrd_(jobz, uplo, n, kd, &ab[ab_offset], ldab, &work[indd], &work[inde], 
             &q[q_offset], ldq, &work[indwrk], &iinfo);

/*     If all eigenvalues are desired and ABSTOL is less than or equal */
/*     to zero, then call DSTERF or SSTEQR.  If this fails for some */
/*     eigenvalue, then try DSTEBZ. */

    test = FALSE_;
    if (indeig) {
        if (*il == 1 && *iu == *n) {
            test = TRUE_;
        }
    }
    if ((alleig || test) && *abstol <= 0.) {
        dcopy_(n, &work[indd], &c__1, &w[1], &c__1);
        indee = indwrk + (*n << 1);
        if (! wantz) {
            i__1 = *n - 1;
            dcopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
            dsterf_(n, &w[1], &work[indee], info);
        } else {
            dlacpy_("A", n, n, &q[q_offset], ldq, &z__[z_offset], ldz);
            i__1 = *n - 1;
            dcopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
            dsteqr_(jobz, n, &w[1], &work[indee], &z__[z_offset], ldz, &work[
                    indwrk], info);
            if (*info == 0) {
                i__1 = *n;
                for (i__ = 1; i__ <= i__1; ++i__) {
                    ifail[i__] = 0;
/* L10: */
                }
            }
        }
        if (*info == 0) {
            *m = *n;
            goto L30;
        }
        *info = 0;
    }

/*     Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN. */

    if (wantz) {
        *(unsigned char *)order = 'B';
    } else {
        *(unsigned char *)order = 'E';
    }
    indibl = 1;
    indisp = indibl + *n;
    indiwo = indisp + *n;
    dstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &work[indd], &work[
            inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], &work[
            indwrk], &iwork[indiwo], info);

    if (wantz) {
        dstein_(n, &work[indd], &work[inde], m, &w[1], &iwork[indibl], &iwork[
                indisp], &z__[z_offset], ldz, &work[indwrk], &iwork[indiwo], &
                ifail[1], info);

/*        Apply orthogonal matrix used in reduction to tridiagonal */
/*        form to eigenvectors returned by DSTEIN. */

        i__1 = *m;
        for (j = 1; j <= i__1; ++j) {
            dcopy_(n, &z__[j * z_dim1 + 1], &c__1, &work[1], &c__1);
            dgemv_("N", n, n, &c_b14, &q[q_offset], ldq, &work[1], &c__1, &
                    c_b34, &z__[j * z_dim1 + 1], &c__1);
/* L20: */
        }
    }

/*     If matrix was scaled, then rescale eigenvalues appropriately. */

L30:
    if (iscale == 1) {
        if (*info == 0) {
            imax = *m;
        } else {
            imax = *info - 1;
        }
        d__1 = 1. / sigma;
        dscal_(&imax, &d__1, &w[1], &c__1);
    }

/*     If eigenvalues are not in order, then sort them, along with */
/*     eigenvectors. */

    if (wantz) {
        i__1 = *m - 1;
        for (j = 1; j <= i__1; ++j) {
            i__ = 0;
            tmp1 = w[j];
            i__2 = *m;
            for (jj = j + 1; jj <= i__2; ++jj) {
                if (w[jj] < tmp1) {
                    i__ = jj;
                    tmp1 = w[jj];
                }
/* L40: */
            }

            if (i__ != 0) {
                itmp1 = iwork[indibl + i__ - 1];
                w[i__] = w[j];
                iwork[indibl + i__ - 1] = iwork[indibl + j - 1];
                w[j] = tmp1;
                iwork[indibl + j - 1] = itmp1;
                dswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1], 
                         &c__1);
                if (*info != 0) {
                    itmp1 = ifail[i__];
                    ifail[i__] = ifail[j];
                    ifail[j] = itmp1;
                }
            }
/* L50: */
        }
    }

    return 0;

/*     End of DSBEVX */

} /* dsbevx_ */