Ejemplo n.º 1
0
/* Subroutine */ int ssyevr_(char *jobz, char *range, char *uplo, integer *n, 
	real *a, integer *lda, real *vl, real *vu, integer *il, integer *iu, 
	real *abstol, integer *m, real *w, real *z__, integer *ldz, integer *
	isuppz, real *work, integer *lwork, integer *iwork, integer *liwork, 
	integer *info)
{
    /* System generated locals */
    integer a_dim1, a_offset, z_dim1, z_offset, i__1, i__2;
    real r__1, r__2;

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

    /* Local variables */
    integer i__, j, nb, jj;
    real eps, vll, vuu, tmp1;
    integer indd, inde;
    real anrm;
    integer imax;
    real rmin, rmax;
    logical test;
    integer inddd, indee;
    real sigma;
    extern logical lsame_(char *, char *);
    integer iinfo;
    extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
    char order[1];
    integer indwk, lwmin;
    logical lower;
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
	    integer *), sswap_(integer *, real *, integer *, real *, integer *
);
    logical wantz, alleig, indeig;
    integer iscale, ieeeok, indibl, indifl;
    logical valeig;
    extern doublereal slamch_(char *);
    real safmin;
    extern integer ilaenv_(integer *, char *, char *, integer *, integer *, 
	    integer *, integer *);
    extern /* Subroutine */ int xerbla_(char *, integer *);
    real abstll, bignum;
    integer indtau, indisp, indiwo, indwkn, liwmin;
    logical tryrac;
    extern /* Subroutine */ int sstein_(integer *, real *, real *, integer *, 
	    real *, integer *, integer *, real *, integer *, real *, integer *
, integer *, integer *), ssterf_(integer *, real *, real *, 
	    integer *);
    integer llwrkn, llwork, nsplit;
    real smlnum;
    extern doublereal slansy_(char *, char *, integer *, real *, integer *, 
	    real *);
    extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *, 
	    real *, integer *, integer *, real *, real *, real *, integer *, 
	    integer *, real *, integer *, integer *, real *, integer *, 
	    integer *), sstemr_(char *, char *, integer *, 
	    real *, real *, real *, real *, integer *, integer *, integer *, 
	    real *, real *, integer *, integer *, integer *, logical *, real *
, integer *, integer *, integer *, integer *);
    integer lwkopt;
    logical lquery;
    extern /* Subroutine */ int sormtr_(char *, char *, char *, integer *, 
	    integer *, real *, integer *, real *, real *, integer *, real *, 
	    integer *, integer *), ssytrd_(char *, 
	    integer *, real *, integer *, real *, real *, real *, real *, 
	    integer *, integer *);


/*  -- LAPACK driver routine (version 3.1) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

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

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

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

/*  SSYEVR first reduces the matrix A to tridiagonal form T with a call */
/*  to SSYTRD.  Then, whenever possible, SSYEVR calls SSTEMR to compute */
/*  the eigenspectrum using Relatively Robust Representations.  SSTEMR */
/*  computes eigenvalues by the dqds algorithm, while orthogonal */
/*  eigenvectors are computed from various "good" L D L^T representations */
/*  (also known as Relatively Robust Representations). Gram-Schmidt */
/*  orthogonalization is avoided as far as possible. More specifically, */
/*  the various steps of the algorithm are as follows. */

/*  For each unreduced block (submatrix) of T, */
/*     (a) Compute T - sigma I  = L D L^T, so that L and D */
/*         define all the wanted eigenvalues to high relative accuracy. */
/*         This means that small relative changes in the entries of D and L */
/*         cause only small relative changes in the eigenvalues and */
/*         eigenvectors. The standard (unfactored) representation of the */
/*         tridiagonal matrix T does not have this property in general. */
/*     (b) Compute the eigenvalues to suitable accuracy. */
/*         If the eigenvectors are desired, the algorithm attains full */
/*         accuracy of the computed eigenvalues only right before */
/*         the corresponding vectors have to be computed, see steps c) and d). */
/*     (c) For each cluster of close eigenvalues, select a new */
/*         shift close to the cluster, find a new factorization, and refine */
/*         the shifted eigenvalues to suitable accuracy. */
/*     (d) For each eigenvalue with a large enough relative separation compute */
/*         the corresponding eigenvector by forming a rank revealing twisted */
/*         factorization. Go back to (c) for any clusters that remain. */

/*  The desired accuracy of the output can be specified by the input */
/*  parameter ABSTOL. */

/*  For more details, see SSTEMR's documentation and: */
/*  - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations */
/*    to compute orthogonal eigenvectors of symmetric tridiagonal matrices," */
/*    Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. */
/*  - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and */
/*    Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25, */
/*    2004.  Also LAPACK Working Note 154. */
/*  - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric */
/*    tridiagonal eigenvalue/eigenvector problem", */
/*    Computer Science Division Technical Report No. UCB/CSD-97-971, */
/*    UC Berkeley, May 1997. */


/*  Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested */
/*  on machines which conform to the ieee-754 floating point standard. */
/*  SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and */
/*  when partial spectrum requests are made. */

/*  Normal execution of SSTEMR may create NaNs and infinities and */
/*  hence may abort due to a floating point exception in environments */
/*  which do not handle NaNs and infinities in the ieee standard default */
/*  manner. */

/*  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. */
/* ********* For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and */
/* ********* SSTEIN are called */

/*  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. */

/*  A       (input/output) REAL array, dimension (LDA, N) */
/*          On entry, the symmetric matrix A.  If UPLO = 'U', the */
/*          leading N-by-N upper triangular part of A contains the */
/*          upper triangular part of the matrix A.  If UPLO = 'L', */
/*          the leading N-by-N lower triangular part of A contains */
/*          the lower triangular part of the matrix A. */
/*          On exit, the lower triangle (if UPLO='L') or the upper */
/*          triangle (if UPLO='U') of A, including the diagonal, is */
/*          destroyed. */

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

/*  VL      (input) REAL */
/*  VU      (input) REAL */
/*          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) REAL */
/*          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 A to tridiagonal form. */

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

/*          If high relative accuracy is important, set ABSTOL to */
/*          SLAMCH( 'Safe minimum' ).  Doing so will guarantee that */
/*          eigenvalues are computed to high relative accuracy when */
/*          possible in future releases.  The current code does not */
/*          make any guarantees about high relative accuracy, but */
/*          future releases will. See J. Barlow and J. Demmel, */
/*          "Computing Accurate Eigensystems of Scaled Diagonally */
/*          Dominant Matrices", LAPACK Working Note #7, for a discussion */
/*          of which matrices define their eigenvalues to high relative */
/*          accuracy. */

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

/*  Z       (output) REAL 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 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. */
/*          Supplying N columns is always safe. */

/*  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 ). */
/* ********* Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 */

/*  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK)) */
/*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */

/*  LWORK   (input) INTEGER */
/*          The dimension of the array WORK.  LWORK >= max(1,26*N). */
/*          For optimal efficiency, LWORK >= (NB+6)*N, */
/*          where NB is the max of the blocksize for SSYTRD and SORMTR */
/*          returned by ILAENV. */

/*          If LWORK = -1, then a workspace query is assumed; the routine */
/*          only calculates the optimal sizes of the WORK and IWORK */
/*          arrays, returns these values as the first entries of the WORK */
/*          and IWORK arrays, and no error message related to LWORK or */
/*          LIWORK is issued by XERBLA. */

/*  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */
/*          On exit, if INFO = 0, IWORK(1) returns the optimal LWORK. */

/*  LIWORK  (input) INTEGER */
/*          The dimension of the array IWORK.  LIWORK >= max(1,10*N). */

/*          If LIWORK = -1, then a workspace query is assumed; the */
/*          routine only calculates the optimal sizes of the WORK and */
/*          IWORK arrays, returns these values as the first entries of */
/*          the WORK and IWORK arrays, and no error message related to */
/*          LWORK or LIWORK is issued by XERBLA. */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value */
/*          > 0:  Internal error */

/*  Further Details */
/*  =============== */

/*  Based on contributions by */
/*     Inderjit Dhillon, IBM Almaden, USA */
/*     Osni Marques, LBNL/NERSC, USA */
/*     Ken Stanley, Computer Science Division, University of */
/*       California at Berkeley, USA */
/*     Jason Riedy, Computer Science Division, University of */
/*       California at Berkeley, USA */

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

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

/*     Test the input parameters. */

    /* Parameter adjustments */
    a_dim1 = *lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    --w;
    z_dim1 = *ldz;
    z_offset = 1 + z_dim1;
    z__ -= z_offset;
    --isuppz;
    --work;
    --iwork;

    /* Function Body */
    ieeeok = ilaenv_(&c__10, "SSYEVR", "N", &c__1, &c__2, &c__3, &c__4);

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

    lquery = *lwork == -1 || *liwork == -1;

/* Computing MAX */
    i__1 = 1, i__2 = *n * 26;
    lwmin = max(i__1,i__2);
/* Computing MAX */
    i__1 = 1, i__2 = *n * 10;
    liwmin = max(i__1,i__2);

    *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 (*lda < max(1,*n)) {
	*info = -6;
    } else {
	if (valeig) {
	    if (*n > 0 && *vu <= *vl) {
		*info = -8;
	    }
	} else if (indeig) {
	    if (*il < 1 || *il > max(1,*n)) {
		*info = -9;
	    } else if (*iu < min(*n,*il) || *iu > *n) {
		*info = -10;
	    }
	}
    }
    if (*info == 0) {
	if (*ldz < 1 || wantz && *ldz < *n) {
	    *info = -15;
	}
    }

    if (*info == 0) {
	nb = ilaenv_(&c__1, "SSYTRD", uplo, n, &c_n1, &c_n1, &c_n1);
/* Computing MAX */
	i__1 = nb, i__2 = ilaenv_(&c__1, "SORMTR", uplo, n, &c_n1, &c_n1, &
		c_n1);
	nb = max(i__1,i__2);
/* Computing MAX */
	i__1 = (nb + 1) * *n;
	lwkopt = max(i__1,lwmin);
	work[1] = (real) lwkopt;
	iwork[1] = liwmin;

	if (*lwork < lwmin && ! lquery) {
	    *info = -18;
	} else if (*liwork < liwmin && ! lquery) {
	    *info = -20;
	}
    }

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

/*     Quick return if possible */

    *m = 0;
    if (*n == 0) {
	work[1] = 1.f;
	return 0;
    }

    if (*n == 1) {
	work[1] = 26.f;
	if (alleig || indeig) {
	    *m = 1;
	    w[1] = a[a_dim1 + 1];
	} else {
	    if (*vl < a[a_dim1 + 1] && *vu >= a[a_dim1 + 1]) {
		*m = 1;
		w[1] = a[a_dim1 + 1];
	    }
	}
	if (wantz) {
	    z__[z_dim1 + 1] = 1.f;
	}
	return 0;
    }

/*     Get machine constants. */

    safmin = slamch_("Safe minimum");
    eps = slamch_("Precision");
    smlnum = safmin / eps;
    bignum = 1.f / smlnum;
    rmin = sqrt(smlnum);
/* Computing MIN */
    r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
    rmax = dmin(r__1,r__2);

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

    iscale = 0;
    abstll = *abstol;
    if (valeig) {
	vll = *vl;
	vuu = *vu;
    }
    anrm = slansy_("M", uplo, n, &a[a_offset], lda, &work[1]);
    if (anrm > 0.f && anrm < rmin) {
	iscale = 1;
	sigma = rmin / anrm;
    } else if (anrm > rmax) {
	iscale = 1;
	sigma = rmax / anrm;
    }
    if (iscale == 1) {
	if (lower) {
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
		i__2 = *n - j + 1;
		sscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1);
/* L10: */
	    }
	} else {
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
		sscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1);
/* L20: */
	    }
	}
	if (*abstol > 0.f) {
	    abstll = *abstol * sigma;
	}
	if (valeig) {
	    vll = *vl * sigma;
	    vuu = *vu * sigma;
	}
    }
/*     Initialize indices into workspaces.  Note: The IWORK indices are */
/*     used only if SSTERF or SSTEMR fail. */
/*     WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the */
/*     elementary reflectors used in SSYTRD. */
    indtau = 1;
/*     WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries. */
    indd = indtau + *n;
/*     WORK(INDE:INDE+N-1) stores the off-diagonal entries of the */
/*     tridiagonal matrix from SSYTRD. */
    inde = indd + *n;
/*     WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over */
/*     -written by SSTEMR (the SSTERF path copies the diagonal to W). */
    inddd = inde + *n;
/*     WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over */
/*     -written while computing the eigenvalues in SSTERF and SSTEMR. */
    indee = inddd + *n;
/*     INDWK is the starting offset of the left-over workspace, and */
/*     LLWORK is the remaining workspace size. */
    indwk = indee + *n;
    llwork = *lwork - indwk + 1;
/*     IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and */
/*     stores the block indices of each of the M<=N eigenvalues. */
    indibl = 1;
/*     IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and */
/*     stores the starting and finishing indices of each block. */
    indisp = indibl + *n;
/*     IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors */
/*     that corresponding to eigenvectors that fail to converge in */
/*     SSTEIN.  This information is discarded; if any fail, the driver */
/*     returns INFO > 0. */
    indifl = indisp + *n;
/*     INDIWO is the offset of the remaining integer workspace. */
    indiwo = indisp + *n;

/*     Call SSYTRD to reduce symmetric matrix to tridiagonal form. */

    ssytrd_(uplo, n, &a[a_offset], lda, &work[indd], &work[inde], &work[
	    indtau], &work[indwk], &llwork, &iinfo);

/*     If all eigenvalues are desired */
/*     then call SSTERF or SSTEMR and SORMTR. */

    test = FALSE_;
    if (indeig) {
	if (*il == 1 && *iu == *n) {
	    test = TRUE_;
	}
    }
    if ((alleig || test) && ieeeok == 1) {
	if (! wantz) {
	    scopy_(n, &work[indd], &c__1, &w[1], &c__1);
	    i__1 = *n - 1;
	    scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
	    ssterf_(n, &w[1], &work[indee], info);
	} else {
	    i__1 = *n - 1;
	    scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
	    scopy_(n, &work[indd], &c__1, &work[inddd], &c__1);

	    if (*abstol <= *n * 2.f * eps) {
		tryrac = TRUE_;
	    } else {
		tryrac = FALSE_;
	    }
	    sstemr_(jobz, "A", n, &work[inddd], &work[indee], vl, vu, il, iu, 
		    m, &w[1], &z__[z_offset], ldz, n, &isuppz[1], &tryrac, &
		    work[indwk], lwork, &iwork[1], liwork, info);



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

	    if (wantz && *info == 0) {
		indwkn = inde;
		llwrkn = *lwork - indwkn + 1;
		sormtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau]
, &z__[z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
	    }
	}


	if (*info == 0) {
/*           Everything worked.  Skip SSTEBZ/SSTEIN.  IWORK(:) are */
/*           undefined. */
	    *m = *n;
	    goto L30;
	}
	*info = 0;
    }

/*     Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN. */
/*     Also call SSTEBZ and SSTEIN if SSTEMR fails. */

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

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

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

	indwkn = inde;
	llwrkn = *lwork - indwkn + 1;
	sormtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau], &z__[
		z_offset], ldz, &work[indwkn], &llwrkn, &iinfo);
    }

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

/*  Jump here if SSTEMR/SSTEIN succeeded. */
L30:
    if (iscale == 1) {
	if (*info == 0) {
	    imax = *m;
	} else {
	    imax = *info - 1;
	}
	r__1 = 1.f / sigma;
	sscal_(&imax, &r__1, &w[1], &c__1);
    }

/*     If eigenvalues are not in order, then sort them, along with */
/*     eigenvectors.  Note: We do not sort the IFAIL portion of IWORK. */
/*     It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do */
/*     not return this detailed information to the user. */

    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) {
		w[i__] = w[j];
		w[j] = tmp1;
		sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[j * z_dim1 + 1], 
			 &c__1);
	    }
/* L50: */
	}
    }

/*     Set WORK(1) to optimal workspace size. */

    work[1] = (real) lwkopt;
    iwork[1] = liwmin;

    return 0;

/*     End of SSYEVR */

} /* ssyevr_ */
Ejemplo n.º 2
0
/* Subroutine */ int ssbevx_(char *jobz, char *range, char *uplo, integer *n, 
	integer *kd, real *ab, integer *ldab, real *q, integer *ldq, real *vl,
	 real *vu, integer *il, integer *iu, real *abstol, integer *m, real *
	w, real *z, integer *ldz, real *work, integer *iwork, integer *ifail, 
	integer *info)
{
/*  -- LAPACK driver routine (version 2.0) --   
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,   
       Courant Institute, Argonne National Lab, and Rice University   
       September 30, 1994   


    Purpose   
    =======   

    SSBEVX 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) REAL 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) REAL 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) REAL   
    VU      (input) REAL   
            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) REAL   
            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*SLAMCH('S'), not zero. 
  
            If this routine returns with INFO>0, indicating that some   
            eigenvectors did not converge, try setting ABSTOL to   
            2*SLAMCH('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) REAL array, dimension (N)   
            The first M elements contain the selected eigenvalues in   
            ascending order.   

    Z       (output) REAL 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) REAL 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.   

    ===================================================================== 
  


       Test the input parameters.   

    
   Parameter adjustments   
       Function Body */
    /* Table of constant values */
    static real c_b14 = 1.f;
    static integer c__1 = 1;
    static real c_b34 = 0.f;
    
    /* System generated locals */
    integer ab_dim1, ab_offset, q_dim1, q_offset, z_dim1, z_offset, i__1, 
	    i__2;
    real r__1, r__2;
    /* Builtin functions */
    double sqrt(doublereal);
    /* Local variables */
    static integer indd, inde;
    static real anrm;
    static integer imax;
    static real rmin, rmax;
    static integer itmp1, i, j, indee;
    static real sigma;
    extern logical lsame_(char *, char *);
    static integer iinfo;
    extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
    static char order[1];
    extern /* Subroutine */ int sgemv_(char *, integer *, integer *, real *, 
	    real *, integer *, real *, integer *, real *, real *, integer *);
    static logical lower;
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
	    integer *), sswap_(integer *, real *, integer *, real *, integer *
	    );
    static logical wantz;
    static integer jj;
    static logical alleig, indeig;
    static integer iscale, indibl;
    static logical valeig;
    extern doublereal slamch_(char *);
    static real safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *);
    static real abstll, bignum;
    extern doublereal slansb_(char *, char *, integer *, integer *, real *, 
	    integer *, real *);
    extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, 
	    real *, integer *, integer *, real *, integer *, integer *);
    static integer indisp, indiwo;
    extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, 
	    integer *, real *, integer *);
    static integer indwrk;
    extern /* Subroutine */ int ssbtrd_(char *, char *, integer *, integer *, 
	    real *, integer *, real *, real *, real *, integer *, real *, 
	    integer *), sstein_(integer *, real *, real *, 
	    integer *, real *, integer *, integer *, real *, integer *, real *
	    , integer *, integer *, integer *), ssterf_(integer *, real *, 
	    real *, integer *);
    static integer nsplit;
    static real smlnum;
    extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *, 
	    real *, integer *, integer *, real *, real *, real *, integer *, 
	    integer *, real *, integer *, integer *, real *, integer *, 
	    integer *), ssteqr_(char *, integer *, real *, 
	    real *, real *, integer *, real *, integer *);
    static real eps, vll, vuu, tmp1;



#define W(I) w[(I)-1]
#define WORK(I) work[(I)-1]
#define IWORK(I) iwork[(I)-1]
#define IFAIL(I) ifail[(I)-1]

#define AB(I,J) ab[(I)-1 + ((J)-1)* ( *ldab)]
#define Q(I,J) q[(I)-1 + ((J)-1)* ( *ldq)]
#define Z(I,J) z[(I)-1 + ((J)-1)* ( *ldz)]

    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 (*ldq < *n) {
	*info = -9;
    } else if (valeig && *n > 0 && *vu <= *vl) {
	*info = -11;
    } else if (indeig && *il < 1) {
	*info = -12;
    } else if (indeig && (*iu < min(*n,*il) || *iu > *n)) {
	*info = -13;
    } else if (*ldz < 1 || wantz && *ldz < *n) {
	*info = -18;
    }

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

/*     Quick return if possible */

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

    if (*n == 1) {
	if (alleig || indeig) {
	    *m = 1;
	    W(1) = AB(1,1);
	} else {
	    if (*vl < AB(1,1) && *vu >= AB(1,1)) {
		*m = 1;
		W(1) = AB(1,1);
	    }
	}
	if (wantz) {
	    Z(1,1) = 1.f;
	}
	return 0;
    }

/*     Get machine constants. */

    safmin = slamch_("Safe minimum");
    eps = slamch_("Precision");
    smlnum = safmin / eps;
    bignum = 1.f / smlnum;
    rmin = sqrt(smlnum);
/* Computing MIN */
    r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
    rmax = dmin(r__1,r__2);

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

    iscale = 0;
    abstll = *abstol;
    if (valeig) {
	vll = *vl;
	vuu = *vu;
    }
    anrm = slansb_("M", uplo, n, kd, &AB(1,1), ldab, &WORK(1));
    if (anrm > 0.f && anrm < rmin) {
	iscale = 1;
	sigma = rmin / anrm;
    } else if (anrm > rmax) {
	iscale = 1;
	sigma = rmax / anrm;
    }
    if (iscale == 1) {
	if (lower) {
	    slascl_("B", kd, kd, &c_b14, &sigma, n, n, &AB(1,1), ldab, 
		    info);
	} else {
	    slascl_("Q", kd, kd, &c_b14, &sigma, n, n, &AB(1,1), ldab, 
		    info);
	}
	if (*abstol > 0.f) {
	    abstll = *abstol * sigma;
	}
	if (valeig) {
	    vll = *vl * sigma;
	    vuu = *vu * sigma;
	}
    }

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

    indd = 1;
    inde = indd + *n;
    indwrk = inde + *n;
    ssbtrd_(jobz, uplo, n, kd, &AB(1,1), ldab, &WORK(indd), &WORK(inde),
	     &Q(1,1), ldq, &WORK(indwrk), &iinfo);

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

    if ((alleig || indeig && *il == 1 && *iu == *n) && *abstol <= 0.f) {
	scopy_(n, &WORK(indd), &c__1, &W(1), &c__1);
	indee = indwrk + (*n << 1);
	if (! wantz) {
	    i__1 = *n - 1;
	    scopy_(&i__1, &WORK(inde), &c__1, &WORK(indee), &c__1);
	    ssterf_(n, &W(1), &WORK(indee), info);
	} else {
	    slacpy_("A", n, n, &Q(1,1), ldq, &Z(1,1), ldz);
	    i__1 = *n - 1;
	    scopy_(&i__1, &WORK(inde), &c__1, &WORK(indee), &c__1);
	    ssteqr_(jobz, n, &W(1), &WORK(indee), &Z(1,1), ldz, &WORK(
		    indwrk), info);
	    if (*info == 0) {
		i__1 = *n;
		for (i = 1; i <= *n; ++i) {
		    IFAIL(i) = 0;
/* L10: */
		}
	    }
	}
	if (*info == 0) {
	    *m = *n;
	    goto L30;
	}
	*info = 0;
    }

/*     Otherwise, call SSTEBZ 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;
    sstebz_(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) {
	sstein_(n, &WORK(indd), &WORK(inde), m, &W(1), &IWORK(indibl), &IWORK(
		indisp), &Z(1,1), ldz, &WORK(indwrk), &IWORK(indiwo), &
		IFAIL(1), info);

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

	i__1 = *m;
	for (j = 1; j <= *m; ++j) {
	    scopy_(n, &Z(1,j), &c__1, &WORK(1), &c__1);
	    sgemv_("N", n, n, &c_b14, &Q(1,1), ldq, &WORK(1), &c__1, &
		    c_b34, &Z(1,j), &c__1);
/* L20: */
	}
    }

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

L30:
    if (iscale == 1) {
	if (*info == 0) {
	    imax = *m;
	} else {
	    imax = *info - 1;
	}
	r__1 = 1.f / sigma;
	sscal_(&imax, &r__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 <= *m-1; ++j) {
	    i = 0;
	    tmp1 = W(j);
	    i__2 = *m;
	    for (jj = j + 1; jj <= *m; ++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;
		sswap_(n, &Z(1,i), &c__1, &Z(1,j), &
			c__1);
		if (*info != 0) {
		    itmp1 = IFAIL(i);
		    IFAIL(i) = IFAIL(j);
		    IFAIL(j) = itmp1;
		}
	    }
/* L50: */
	}
    }

    return 0;

/*     End of SSBEVX */

} /* ssbevx_ */
Ejemplo n.º 3
0
/* Subroutine */ int sspevx_(char *jobz, char *range, char *uplo, integer *n, 
	real *ap, real *vl, real *vu, integer *il, integer *iu, real *abstol, 
	integer *m, real *w, real *z__, integer *ldz, real *work, integer *
	iwork, integer *ifail, integer *info, ftnlen jobz_len, ftnlen 
	range_len, ftnlen uplo_len)
{
    /* System generated locals */
    integer z_dim1, z_offset, i__1, i__2;
    real r__1, r__2;

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

    /* Local variables */
    static integer i__, j, jj;
    static real eps, vll, vuu, tmp1;
    static integer indd, inde;
    static real anrm;
    static integer imax;
    static real rmin, rmax;
    static integer itmp1, indee;
    static real sigma;
    extern logical lsame_(char *, char *, ftnlen, ftnlen);
    static integer iinfo;
    extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
    static char order[1];
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
	    integer *), sswap_(integer *, real *, integer *, real *, integer *
	    );
    static logical wantz, alleig, indeig;
    static integer iscale, indibl;
    static logical valeig;
    extern doublereal slamch_(char *, ftnlen);
    static real safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen);
    static real abstll, bignum;
    static integer indtau, indisp, indiwo, indwrk;
    extern doublereal slansp_(char *, char *, integer *, real *, real *, 
	    ftnlen, ftnlen);
    extern /* Subroutine */ int sstein_(integer *, real *, real *, integer *, 
	    real *, integer *, integer *, real *, integer *, real *, integer *
	    , integer *, integer *), ssterf_(integer *, real *, real *, 
	    integer *);
    static integer nsplit;
    extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *, 
	    real *, integer *, integer *, real *, real *, real *, integer *, 
	    integer *, real *, integer *, integer *, real *, integer *, 
	    integer *, ftnlen, ftnlen);
    static real smlnum;
    extern /* Subroutine */ int sopgtr_(char *, integer *, real *, real *, 
	    real *, integer *, real *, integer *, ftnlen), ssptrd_(char *, 
	    integer *, real *, real *, real *, real *, integer *, ftnlen), 
	    ssteqr_(char *, integer *, real *, real *, real *, integer *, 
	    real *, integer *, ftnlen), sopmtr_(char *, char *, char *, 
	    integer *, integer *, real *, real *, real *, integer *, real *, 
	    integer *, ftnlen, ftnlen, ftnlen);


/*  -- LAPACK driver routine (version 3.0) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/*     Courant Institute, Argonne National Lab, and Rice University */
/*     June 30, 1999 */

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

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

/*  SSPEVX computes selected eigenvalues and, optionally, eigenvectors */
/*  of a real symmetric matrix A in packed storage.  Eigenvalues/vectors */
/*  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. */

/*  AP      (input/output) REAL array, dimension (N*(N+1)/2) */
/*          On entry, the upper or lower triangle of the symmetric matrix */
/*          A, packed columnwise in a linear array.  The j-th column of A */
/*          is stored in the array AP as follows: */
/*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */
/*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. */

/*          On exit, AP is overwritten by values generated during the */
/*          reduction to tridiagonal form.  If UPLO = 'U', the diagonal */
/*          and first superdiagonal of the tridiagonal matrix T overwrite */
/*          the corresponding elements of A, and if UPLO = 'L', the */
/*          diagonal and first subdiagonal of T overwrite the */
/*          corresponding elements of A. */

/*  VL      (input) REAL */
/*  VU      (input) REAL */
/*          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) REAL */
/*          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 AP to tridiagonal form. */

/*          Eigenvalues will be computed most accurately when ABSTOL is */
/*          set to twice the underflow threshold 2*SLAMCH('S'), not zero. */
/*          If this routine returns with INFO>0, indicating that some */
/*          eigenvectors did not converge, try setting ABSTOL to */
/*          2*SLAMCH('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) REAL array, dimension (N) */
/*          If INFO = 0, the selected eigenvalues in ascending order. */

/*  Z       (output) REAL 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) REAL array, dimension (8*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 */
    --ap;
    --w;
    z_dim1 = *ldz;
    z_offset = 1 + z_dim1;
    z__ -= z_offset;
    --work;
    --iwork;
    --ifail;

    /* Function Body */
    wantz = lsame_(jobz, "V", (ftnlen)1, (ftnlen)1);
    alleig = lsame_(range, "A", (ftnlen)1, (ftnlen)1);
    valeig = lsame_(range, "V", (ftnlen)1, (ftnlen)1);
    indeig = lsame_(range, "I", (ftnlen)1, (ftnlen)1);

    *info = 0;
    if (! (wantz || lsame_(jobz, "N", (ftnlen)1, (ftnlen)1))) {
	*info = -1;
    } else if (! (alleig || valeig || indeig)) {
	*info = -2;
    } else if (! (lsame_(uplo, "L", (ftnlen)1, (ftnlen)1) || lsame_(uplo, 
	    "U", (ftnlen)1, (ftnlen)1))) {
	*info = -3;
    } else if (*n < 0) {
	*info = -4;
    } else {
	if (valeig) {
	    if (*n > 0 && *vu <= *vl) {
		*info = -7;
	    }
	} else if (indeig) {
	    if (*il < 1 || *il > max(1,*n)) {
		*info = -8;
	    } else if (*iu < min(*n,*il) || *iu > *n) {
		*info = -9;
	    }
	}
    }
    if (*info == 0) {
	if (*ldz < 1 || wantz && *ldz < *n) {
	    *info = -14;
	}
    }

    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("SSPEVX", &i__1, (ftnlen)6);
	return 0;
    }

/*     Quick return if possible */

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

    if (*n == 1) {
	if (alleig || indeig) {
	    *m = 1;
	    w[1] = ap[1];
	} else {
	    if (*vl < ap[1] && *vu >= ap[1]) {
		*m = 1;
		w[1] = ap[1];
	    }
	}
	if (wantz) {
	    z__[z_dim1 + 1] = 1.f;
	}
	return 0;
    }

/*     Get machine constants. */

    safmin = slamch_("Safe minimum", (ftnlen)12);
    eps = slamch_("Precision", (ftnlen)9);
    smlnum = safmin / eps;
    bignum = 1.f / smlnum;
    rmin = sqrt(smlnum);
/* Computing MIN */
    r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
    rmax = dmin(r__1,r__2);

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

    iscale = 0;
    abstll = *abstol;
    if (valeig) {
	vll = *vl;
	vuu = *vu;
    } else {
	vll = 0.f;
	vuu = 0.f;
    }
    anrm = slansp_("M", uplo, n, &ap[1], &work[1], (ftnlen)1, (ftnlen)1);
    if (anrm > 0.f && anrm < rmin) {
	iscale = 1;
	sigma = rmin / anrm;
    } else if (anrm > rmax) {
	iscale = 1;
	sigma = rmax / anrm;
    }
    if (iscale == 1) {
	i__1 = *n * (*n + 1) / 2;
	sscal_(&i__1, &sigma, &ap[1], &c__1);
	if (*abstol > 0.f) {
	    abstll = *abstol * sigma;
	}
	if (valeig) {
	    vll = *vl * sigma;
	    vuu = *vu * sigma;
	}
    }

/*     Call SSPTRD to reduce symmetric packed matrix to tridiagonal form. */

    indtau = 1;
    inde = indtau + *n;
    indd = inde + *n;
    indwrk = indd + *n;
    ssptrd_(uplo, n, &ap[1], &work[indd], &work[inde], &work[indtau], &iinfo, 
	    (ftnlen)1);

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

    if ((alleig || indeig && *il == 1 && *iu == *n) && *abstol <= 0.f) {
	scopy_(n, &work[indd], &c__1, &w[1], &c__1);
	indee = indwrk + (*n << 1);
	if (! wantz) {
	    i__1 = *n - 1;
	    scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
	    ssterf_(n, &w[1], &work[indee], info);
	} else {
	    sopgtr_(uplo, n, &ap[1], &work[indtau], &z__[z_offset], ldz, &
		    work[indwrk], &iinfo, (ftnlen)1);
	    i__1 = *n - 1;
	    scopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1);
	    ssteqr_(jobz, n, &w[1], &work[indee], &z__[z_offset], ldz, &work[
		    indwrk], info, (ftnlen)1);
	    if (*info == 0) {
		i__1 = *n;
		for (i__ = 1; i__ <= i__1; ++i__) {
		    ifail[i__] = 0;
/* L10: */
		}
	    }
	}
	if (*info == 0) {
	    *m = *n;
	    goto L20;
	}
	*info = 0;
    }

/*     Otherwise, call SSTEBZ 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;
    sstebz_(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, (ftnlen)1, (ftnlen)1);

    if (wantz) {
	sstein_(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 SSTEIN. */

	sopmtr_("L", uplo, "N", n, m, &ap[1], &work[indtau], &z__[z_offset], 
		ldz, &work[indwrk], info, (ftnlen)1, (ftnlen)1, (ftnlen)1);
    }

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

L20:
    if (iscale == 1) {
	if (*info == 0) {
	    imax = *m;
	} else {
	    imax = *info - 1;
	}
	r__1 = 1.f / sigma;
	sscal_(&imax, &r__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];
		}
/* L30: */
	    }

	    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;
		sswap_(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;
		}
	    }
/* L40: */
	}
    }

    return 0;

/*     End of SSPEVX */

} /* sspevx_ */
Ejemplo n.º 4
0
/* Subroutine */ int sstevx_(char *jobz, char *range, integer *n, real *d, 
	real *e, real *vl, real *vu, integer *il, integer *iu, real *abstol, 
	integer *m, real *w, real *z, integer *ldz, real *work, integer *
	iwork, integer *ifail, integer *info)
{
/*  -- LAPACK driver routine (version 2.0) --   
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,   
       Courant Institute, Argonne National Lab, and Rice University   
       September 30, 1994   


    Purpose   
    =======   

    SSTEVX computes selected eigenvalues and, optionally, eigenvectors   
    of a real symmetric tridiagonal 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.   

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

    D       (input/output) REAL array, dimension (N)   
            On entry, the n diagonal elements of the tridiagonal matrix   
            A.   
            On exit, D may be multiplied by a constant factor chosen   
            to avoid over/underflow in computing the eigenvalues.   

    E       (input/output) REAL array, dimension (N)   
            On entry, the (n-1) subdiagonal elements of the tridiagonal   
            matrix A in elements 1 to N-1 of E; E(N) need not be set.   
            On exit, E may be multiplied by a constant factor chosen   
            to avoid over/underflow in computing the eigenvalues.   

    VL      (input) REAL   
    VU      (input) REAL   
            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) REAL   
            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.   

            Eigenvalues will be computed most accurately when ABSTOL is   
            set to twice the underflow threshold 2*SLAMCH('S'), not zero. 
  
            If this routine returns with INFO>0, indicating that some   
            eigenvectors did not converge, try setting ABSTOL to   
            2*SLAMCH('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) REAL array, dimension (N)   
            The first M elements contain the selected eigenvalues in   
            ascending order.   

    Z       (output) REAL 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 (INFO > 0), 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) REAL array, dimension (5*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.   

    ===================================================================== 
  


       Test the input parameters.   

    
   Parameter adjustments   
       Function Body */
    /* Table of constant values */
    static integer c__1 = 1;
    
    /* System generated locals */
    integer z_dim1, z_offset, i__1, i__2;
    real r__1, r__2;
    /* Builtin functions */
    double sqrt(doublereal);
    /* Local variables */
    static integer imax;
    static real rmin, rmax, tnrm;
    static integer itmp1, i, j;
    static real sigma;
    extern logical lsame_(char *, char *);
    extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
    static char order[1];
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
	    integer *), sswap_(integer *, real *, integer *, real *, integer *
	    );
    static logical wantz;
    static integer jj;
    static logical alleig, indeig;
    static integer iscale, indibl;
    static logical valeig;
    extern doublereal slamch_(char *);
    static real safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *);
    static real bignum;
    static integer indisp, indiwo, indwrk;
    extern doublereal slanst_(char *, integer *, real *, real *);
    extern /* Subroutine */ int sstein_(integer *, real *, real *, integer *, 
	    real *, integer *, integer *, real *, integer *, real *, integer *
	    , integer *, integer *), ssterf_(integer *, real *, real *, 
	    integer *);
    static integer nsplit;
    extern /* Subroutine */ int sstebz_(char *, char *, integer *, real *, 
	    real *, integer *, integer *, real *, real *, real *, integer *, 
	    integer *, real *, integer *, integer *, real *, integer *, 
	    integer *);
    static real smlnum;
    extern /* Subroutine */ int ssteqr_(char *, integer *, real *, real *, 
	    real *, integer *, real *, integer *);
    static real eps, vll, vuu, tmp1;



#define D(I) d[(I)-1]
#define E(I) e[(I)-1]
#define W(I) w[(I)-1]
#define WORK(I) work[(I)-1]
#define IWORK(I) iwork[(I)-1]
#define IFAIL(I) ifail[(I)-1]

#define Z(I,J) z[(I)-1 + ((J)-1)* ( *ldz)]

    wantz = lsame_(jobz, "V");
    alleig = lsame_(range, "A");
    valeig = lsame_(range, "V");
    indeig = lsame_(range, "I");

    *info = 0;
    if (! (wantz || lsame_(jobz, "N"))) {
	*info = -1;
    } else if (! (alleig || valeig || indeig)) {
	*info = -2;
    } else if (*n < 0) {
	*info = -3;
    } else if (valeig && *n > 0 && *vu <= *vl) {
	*info = -7;
    } else if (indeig && *il < 1) {
	*info = -8;
    } else if (indeig && (*iu < min(*n,*il) || *iu > *n)) {
	*info = -9;
    } else if (*ldz < 1 || wantz && *ldz < *n) {
	*info = -14;
    }

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

/*     Quick return if possible */

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

    if (*n == 1) {
	if (alleig || indeig) {
	    *m = 1;
	    W(1) = D(1);
	} else {
	    if (*vl < D(1) && *vu >= D(1)) {
		*m = 1;
		W(1) = D(1);
	    }
	}
	if (wantz) {
	    Z(1,1) = 1.f;
	}
	return 0;
    }

/*     Get machine constants. */

    safmin = slamch_("Safe minimum");
    eps = slamch_("Precision");
    smlnum = safmin / eps;
    bignum = 1.f / smlnum;
    rmin = sqrt(smlnum);
/* Computing MIN */
    r__1 = sqrt(bignum), r__2 = 1.f / sqrt(sqrt(safmin));
    rmax = dmin(r__1,r__2);

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

    iscale = 0;
    if (valeig) {
	vll = *vl;
	vuu = *vu;
    }
    tnrm = slanst_("M", n, &D(1), &E(1));
    if (tnrm > 0.f && tnrm < rmin) {
	iscale = 1;
	sigma = rmin / tnrm;
    } else if (tnrm > rmax) {
	iscale = 1;
	sigma = rmax / tnrm;
    }
    if (iscale == 1) {
	sscal_(n, &sigma, &D(1), &c__1);
	i__1 = *n - 1;
	sscal_(&i__1, &sigma, &E(1), &c__1);
	if (valeig) {
	    vll = *vl * sigma;
	    vuu = *vu * sigma;
	}
    }

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

    if ((alleig || indeig && *il == 1 && *iu == *n) && *abstol <= 0.f) {
	scopy_(n, &D(1), &c__1, &W(1), &c__1);
	i__1 = *n - 1;
	scopy_(&i__1, &E(1), &c__1, &WORK(1), &c__1);
	indwrk = *n + 1;
	if (! wantz) {
	    ssterf_(n, &W(1), &WORK(1), info);
	} else {
	    ssteqr_("I", n, &W(1), &WORK(1), &Z(1,1), ldz, &WORK(indwrk),
		     info);
	    if (*info == 0) {
		i__1 = *n;
		for (i = 1; i <= *n; ++i) {
		    IFAIL(i) = 0;
/* L10: */
		}
	    }
	}
	if (*info == 0) {
	    *m = *n;
	    goto L20;
	}
	*info = 0;
    }

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

    if (wantz) {
	*(unsigned char *)order = 'B';
    } else {
	*(unsigned char *)order = 'E';
    }
    indwrk = 1;
    indibl = 1;
    indisp = indibl + *n;
    indiwo = indisp + *n;
    sstebz_(range, order, n, &vll, &vuu, il, iu, abstol, &D(1), &E(1), m, &
	    nsplit, &W(1), &IWORK(indibl), &IWORK(indisp), &WORK(indwrk), &
	    IWORK(indiwo), info);

    if (wantz) {
	sstein_(n, &D(1), &E(1), m, &W(1), &IWORK(indibl), &IWORK(indisp), &Z(1,1), ldz, &WORK(indwrk), &IWORK(indiwo), &IFAIL(1), 
		info);
    }

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

L20:
    if (iscale == 1) {
	if (*info == 0) {
	    imax = *m;
	} else {
	    imax = *info - 1;
	}
	r__1 = 1.f / sigma;
	sscal_(&imax, &r__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 <= *m-1; ++j) {
	    i = 0;
	    tmp1 = W(j);
	    i__2 = *m;
	    for (jj = j + 1; jj <= *m; ++jj) {
		if (W(jj) < tmp1) {
		    i = jj;
		    tmp1 = W(jj);
		}
/* L30: */
	    }

	    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;
		sswap_(n, &Z(1,i), &c__1, &Z(1,j), &
			c__1);
		if (*info != 0) {
		    itmp1 = IFAIL(i);
		    IFAIL(i) = IFAIL(j);
		    IFAIL(j) = itmp1;
		}
	    }
/* L40: */
	}
    }

    return 0;

/*     End of SSTEVX */

} /* sstevx_ */