#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int zheevx_(char *jobz, char *range, char *uplo, integer *n, doublecomplex *a, integer *lda, doublereal *vl, doublereal *vu, integer *il, integer *iu, doublereal *abstol, integer *m, doublereal * w, doublecomplex *z__, integer *ldz, doublecomplex *work, integer * lwork, doublereal *rwork, integer *iwork, integer *ifail, integer * info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= ZHEEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian 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. A (input/output) COMPLEX*16 array, dimension (LDA, N) On entry, the Hermitian 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) 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 A 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) On normal exit, the first M elements contain the selected eigenvalues in ascending order. Z (output) COMPLEX*16 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/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for ZHETRD and for ZUNMTR as returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (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. ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c_n1 = -1; /* System generated locals */ integer a_dim1, a_offset, z_dim1, z_offset, i__1, i__2; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, j, nb, jj; static doublereal eps, vll, vuu, tmp1; static integer indd, inde; static doublereal anrm; static integer imax; static doublereal rmin, rmax; static logical test; static integer itmp1, indee; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); static doublereal sigma; extern logical lsame_(char *, char *); static integer iinfo; static char order[1]; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *); static logical lower, wantz; extern /* Subroutine */ int zswap_(integer *, doublecomplex *, integer *, doublecomplex *, integer *); extern doublereal dlamch_(char *); static logical alleig, indeig; static integer iscale, indibl; static logical valeig; static doublereal safmin; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); extern /* Subroutine */ int xerbla_(char *, integer *), zdscal_( integer *, doublereal *, doublecomplex *, integer *); static doublereal abstll, bignum; extern doublereal zlanhe_(char *, char *, integer *, doublecomplex *, integer *, doublereal *); static integer indiwk, indisp, indtau; extern /* Subroutine */ int dsterf_(integer *, doublereal *, doublereal *, integer *), dstebz_(char *, char *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); static integer indrwk, indwrk; extern /* Subroutine */ int zhetrd_(char *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublecomplex *, doublecomplex *, integer *, integer *); static integer lwkmin; extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *); static integer llwork, nsplit; static doublereal smlnum; extern /* Subroutine */ int zstein_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *, doublecomplex *, integer *, doublereal *, integer *, integer *, integer *); static integer lwkopt; static logical lquery; extern /* Subroutine */ int zsteqr_(char *, integer *, doublereal *, doublereal *, doublecomplex *, integer *, doublereal *, integer *), zungtr_(char *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, integer *), zunmtr_(char *, char *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, doublecomplex *, integer *, doublecomplex *, integer *, integer *); a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --w; z_dim1 = *ldz; z_offset = 1 + z_dim1; z__ -= z_offset; --work; --rwork; --iwork; --ifail; /* Function Body */ lower = lsame_(uplo, "L"); wantz = lsame_(jobz, "V"); alleig = lsame_(range, "A"); valeig = lsame_(range, "V"); indeig = lsame_(range, "I"); lquery = *lwork == -1; *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) { if (*n <= 1) { lwkmin = 1; work[1].r = (doublereal) lwkmin, work[1].i = 0.; } else { lwkmin = *n << 1; nb = ilaenv_(&c__1, "ZHETRD", uplo, n, &c_n1, &c_n1, &c_n1, ( ftnlen)6, (ftnlen)1); /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "ZUNMTR", uplo, n, &c_n1, &c_n1, &c_n1, (ftnlen)6, (ftnlen)1); nb = max(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = (nb + 1) * *n; lwkopt = max(i__1,i__2); work[1].r = (doublereal) lwkopt, work[1].i = 0.; } /* Computing MAX */ i__1 = 1, i__2 = *n << 1; if (*lwork < max(i__1,i__2) && ! lquery) { *info = -17; } } if (*info != 0) { i__1 = -(*info); xerbla_("ZHEEVX", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ *m = 0; if (*n == 0) { return 0; } if (*n == 1) { if (alleig || indeig) { *m = 1; i__1 = a_dim1 + 1; w[1] = a[i__1].r; } else if (valeig) { i__1 = a_dim1 + 1; i__2 = a_dim1 + 1; if (*vl < a[i__1].r && *vu >= a[i__2].r) { *m = 1; i__1 = a_dim1 + 1; w[1] = a[i__1].r; } } if (wantz) { i__1 = z_dim1 + 1; z__[i__1].r = 1., z__[i__1].i = 0.; } 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; } anrm = zlanhe_("M", uplo, n, &a[a_offset], lda, &rwork[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) { i__1 = *n; for (j = 1; j <= i__1; ++j) { i__2 = *n - j + 1; zdscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1); /* L10: */ } } else { i__1 = *n; for (j = 1; j <= i__1; ++j) { zdscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1); /* L20: */ } } if (*abstol > 0.) { abstll = *abstol * sigma; } if (valeig) { vll = *vl * sigma; vuu = *vu * sigma; } } /* Call ZHETRD to reduce Hermitian matrix to tridiagonal form. */ indd = 1; inde = indd + *n; indrwk = inde + *n; indtau = 1; indwrk = indtau + *n; llwork = *lwork - indwrk + 1; zhetrd_(uplo, n, &a[a_offset], lda, &rwork[indd], &rwork[inde], &work[ indtau], &work[indwrk], &llwork, &iinfo); /* If all eigenvalues are desired and ABSTOL is less than or equal to zero, then call DSTERF or ZUNGTR and ZSTEQR. 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, &rwork[indd], &c__1, &w[1], &c__1); indee = indrwk + (*n << 1); if (! wantz) { i__1 = *n - 1; dcopy_(&i__1, &rwork[inde], &c__1, &rwork[indee], &c__1); dsterf_(n, &w[1], &rwork[indee], info); } else { zlacpy_("A", n, n, &a[a_offset], lda, &z__[z_offset], ldz); zungtr_(uplo, n, &z__[z_offset], ldz, &work[indtau], &work[indwrk] , &llwork, &iinfo); i__1 = *n - 1; dcopy_(&i__1, &rwork[inde], &c__1, &rwork[indee], &c__1); zsteqr_(jobz, n, &w[1], &rwork[indee], &z__[z_offset], ldz, & rwork[indrwk], info); if (*info == 0) { i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { ifail[i__] = 0; /* L30: */ } } } if (*info == 0) { *m = *n; goto L40; } *info = 0; } /* Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN. */ if (wantz) { *(unsigned char *)order = 'B'; } else { *(unsigned char *)order = 'E'; } indibl = 1; indisp = indibl + *n; indiwk = indisp + *n; dstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &rwork[indd], & rwork[inde], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], & rwork[indrwk], &iwork[indiwk], info); if (wantz) { zstein_(n, &rwork[indd], &rwork[inde], m, &w[1], &iwork[indibl], & iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[ indiwk], &ifail[1], info); /* Apply unitary matrix used in reduction to tridiagonal form to eigenvectors returned by ZSTEIN. */ zunmtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau], &z__[ z_offset], ldz, &work[indwrk], &llwork, &iinfo); } /* If matrix was scaled, then rescale eigenvalues appropriately. */ L40: 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]; } /* L50: */ } 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; zswap_(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; } } /* L60: */ } } /* Set WORK(1) to optimal complex workspace size. */ work[1].r = (doublereal) lwkopt, work[1].i = 0.; return 0; /* End of ZHEEVX */ } /* zheevx_ */