#include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static integer c__10 = 10; static integer c__1 = 1; static integer c__2 = 2; static integer c__3 = 3; static integer c__4 = 4; static integer c_n1 = -1; /* Subroutine */ int cheevr_(char *jobz, char *range, char *uplo, integer *n, complex *a, integer *lda, real *vl, real *vu, integer *il, integer * iu, real *abstol, integer *m, real *w, complex *z__, integer *ldz, integer *isuppz, complex *work, integer *lwork, real *rwork, integer * lrwork, 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, anrm; integer imax; real rmin, rmax; logical test; integer itmp1, indrd, indre; real sigma; extern logical lsame_(char *, char *); integer iinfo; extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *); char order[1]; integer indwk; extern /* Subroutine */ int cswap_(integer *, complex *, integer *, complex *, integer *); integer lwmin; logical lower; extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, integer *); logical wantz, alleig, indeig; integer iscale, ieeeok, indibl, indrdd, indifl, indree; logical valeig; extern doublereal slamch_(char *); extern /* Subroutine */ int chetrd_(char *, integer *, complex *, integer *, real *, real *, complex *, complex *, integer *, integer *), csscal_(integer *, real *, complex *, integer *); real safmin; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); extern /* Subroutine */ int xerbla_(char *, integer *); real abstll, bignum; integer indtau, indisp; extern /* Subroutine */ int cstein_(integer *, real *, real *, integer *, real *, integer *, integer *, complex *, integer *, real *, integer *, integer *, integer *); integer indiwo, indwkn; extern doublereal clansy_(char *, char *, integer *, complex *, integer *, real *); extern /* Subroutine */ int cstemr_(char *, char *, integer *, real *, real *, real *, real *, integer *, integer *, integer *, real *, complex *, integer *, integer *, integer *, logical *, real *, integer *, integer *, integer *, integer *); integer indrwk, liwmin; logical tryrac; extern /* Subroutine */ int ssterf_(integer *, real *, real *, integer *); integer lrwmin, llwrkn, llwork, nsplit; real smlnum; extern /* Subroutine */ int cunmtr_(char *, char *, char *, integer *, integer *, complex *, integer *, complex *, complex *, integer *, complex *, integer *, integer *), sstebz_( char *, char *, integer *, real *, real *, integer *, integer *, real *, real *, real *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *); logical lquery; integer lwkopt, llrwork; /* -- LAPACK driver routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* CHEEVR 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. */ /* CHEEVR first reduces the matrix A to tridiagonal form T with a call */ /* to CHETRD. Then, whenever possible, CHEEVR calls CSTEMR to compute */ /* the eigenspectrum using Relatively Robust Representations. CSTEMR */ /* 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 DSTEMR'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 : CHEEVR calls CSTEMR when the full spectrum is requested */ /* on machines which conform to the ieee-754 floating point standard. */ /* CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and */ /* when partial spectrum requests are made. */ /* Normal execution of CSTEMR 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 */ /* ********* CSTEIN 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) COMPLEX 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) 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 */ /* furutre 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) COMPLEX 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. */ /* 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) COMPLEX 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 >= max(1,2*N). */ /* For optimal efficiency, LWORK >= (NB+1)*N, */ /* where NB is the max of the blocksize for CHETRD and for */ /* CUNMTR as returned by ILAENV. */ /* If LWORK = -1, then a workspace query is assumed; the routine */ /* only calculates the optimal sizes of the WORK, RWORK and */ /* IWORK arrays, returns these values as the first entries of */ /* the WORK, RWORK and IWORK arrays, and no error message */ /* related to LWORK or LRWORK or LIWORK is issued by XERBLA. */ /* RWORK (workspace/output) REAL array, dimension (MAX(1,LRWORK)) */ /* On exit, if INFO = 0, RWORK(1) returns the optimal */ /* (and minimal) LRWORK. */ /* LRWORK (input) INTEGER */ /* The length of the array RWORK. LRWORK >= max(1,24*N). */ /* If LRWORK = -1, then a workspace query is assumed; the */ /* routine only calculates the optimal sizes of the WORK, RWORK */ /* and IWORK arrays, returns these values as the first entries */ /* of the WORK, RWORK and IWORK arrays, and no error message */ /* related to LWORK or LRWORK 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 */ /* (and minimal) LIWORK. */ /* 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, RWORK */ /* and IWORK arrays, returns these values as the first entries */ /* of the WORK, RWORK and IWORK arrays, and no error message */ /* related to LWORK or LRWORK 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; --rwork; --iwork; /* Function Body */ ieeeok = ilaenv_(&c__10, "CHEEVR", "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 || *lrwork == -1 || *liwork == -1; /* Computing MAX */ i__1 = 1, i__2 = *n * 24; lrwmin = max(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = *n * 10; liwmin = max(i__1,i__2); /* Computing MAX */ i__1 = 1, i__2 = *n << 1; lwmin = 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, "CHETRD", uplo, n, &c_n1, &c_n1, &c_n1); /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "CUNMTR", 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].r = (real) lwkopt, work[1].i = 0.f; rwork[1] = (real) lrwmin; iwork[1] = liwmin; if (*lwork < lwmin && ! lquery) { *info = -18; } else if (*lrwork < lrwmin && ! lquery) { *info = -20; } else if (*liwork < liwmin && ! lquery) { *info = -22; } } if (*info != 0) { i__1 = -(*info); xerbla_("CHEEVR", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ *m = 0; if (*n == 0) { work[1].r = 1.f, work[1].i = 0.f; return 0; } if (*n == 1) { work[1].r = 2.f, work[1].i = 0.f; if (alleig || indeig) { *m = 1; i__1 = a_dim1 + 1; w[1] = a[i__1].r; } else { 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.f, z__[i__1].i = 0.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 = clansy_("M", uplo, n, &a[a_offset], lda, &rwork[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; csscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1); /* L10: */ } } else { i__1 = *n; for (j = 1; j <= i__1; ++j) { csscal_(&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 CSTEMR fail. */ /* WORK(INDTAU:INDTAU+N-1) stores the complex scalar factors of the */ /* elementary reflectors used in CHETRD. */ indtau = 1; /* INDWK is the starting offset of the remaining complex workspace, */ /* and LLWORK is the remaining complex workspace size. */ indwk = indtau + *n; llwork = *lwork - indwk + 1; /* RWORK(INDRD:INDRD+N-1) stores the real tridiagonal's diagonal */ /* entries. */ indrd = 1; /* RWORK(INDRE:INDRE+N-1) stores the off-diagonal entries of the */ /* tridiagonal matrix from CHETRD. */ indre = indrd + *n; /* RWORK(INDRDD:INDRDD+N-1) is a copy of the diagonal entries over */ /* -written by CSTEMR (the SSTERF path copies the diagonal to W). */ indrdd = indre + *n; /* RWORK(INDREE:INDREE+N-1) is a copy of the off-diagonal entries over */ /* -written while computing the eigenvalues in SSTERF and CSTEMR. */ indree = indrdd + *n; /* INDRWK is the starting offset of the left-over real workspace, and */ /* LLRWORK is the remaining workspace size. */ indrwk = indree + *n; llrwork = *lrwork - indrwk + 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 CHETRD to reduce Hermitian matrix to tridiagonal form. */ chetrd_(uplo, n, &a[a_offset], lda, &rwork[indrd], &rwork[indre], &work[ indtau], &work[indwk], &llwork, &iinfo); /* If all eigenvalues are desired */ /* then call SSTERF or CSTEMR and CUNMTR. */ test = FALSE_; if (indeig) { if (*il == 1 && *iu == *n) { test = TRUE_; } } if ((alleig || test) && ieeeok == 1) { if (! wantz) { scopy_(n, &rwork[indrd], &c__1, &w[1], &c__1); i__1 = *n - 1; scopy_(&i__1, &rwork[indre], &c__1, &rwork[indree], &c__1); ssterf_(n, &w[1], &rwork[indree], info); } else { i__1 = *n - 1; scopy_(&i__1, &rwork[indre], &c__1, &rwork[indree], &c__1); scopy_(n, &rwork[indrd], &c__1, &rwork[indrdd], &c__1); if (*abstol <= *n * 1.f * eps) { tryrac = TRUE_; } else { tryrac = FALSE_; } cstemr_(jobz, "A", n, &rwork[indrdd], &rwork[indree], vl, vu, il, iu, m, &w[1], &z__[z_offset], ldz, n, &isuppz[1], &tryrac, &rwork[indrwk], &llrwork, &iwork[1], liwork, info); /* Apply unitary matrix used in reduction to tridiagonal */ /* form to eigenvectors returned by CSTEIN. */ if (wantz && *info == 0) { indwkn = indwk; llwrkn = *lwork - indwkn + 1; cunmtr_("L", uplo, "N", n, m, &a[a_offset], lda, &work[indtau] , &z__[z_offset], ldz, &work[indwkn], &llwrkn, &iinfo); } } if (*info == 0) { *m = *n; goto L30; } *info = 0; } /* Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN. */ /* Also call SSTEBZ and CSTEIN if CSTEMR fails. */ if (wantz) { *(unsigned char *)order = 'B'; } else { *(unsigned char *)order = 'E'; } sstebz_(range, order, n, &vll, &vuu, il, iu, &abstll, &rwork[indrd], & rwork[indre], m, &nsplit, &w[1], &iwork[indibl], &iwork[indisp], & rwork[indrwk], &iwork[indiwo], info); if (wantz) { cstein_(n, &rwork[indrd], &rwork[indre], m, &w[1], &iwork[indibl], & iwork[indisp], &z__[z_offset], ldz, &rwork[indrwk], &iwork[ indiwo], &iwork[indifl], info); /* Apply unitary matrix used in reduction to tridiagonal */ /* form to eigenvectors returned by CSTEIN. */ indwkn = indwk; llwrkn = *lwork - indwkn + 1; cunmtr_("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. */ 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 <= 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; cswap_(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].r = (real) lwkopt, work[1].i = 0.f; rwork[1] = (real) lrwmin; iwork[1] = liwmin; return 0; /* End of CHEEVR */ } /* cheevr_ */