#include "clapack.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 dsyevr_(char *jobz, char *range, char *uplo, integer *n, doublereal *a, integer *lda, doublereal *vl, doublereal *vu, integer * il, integer *iu, doublereal *abstol, integer *m, doublereal *w, doublereal *z__, integer *ldz, integer *isuppz, doublereal *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; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ integer i__, j, nb, jj; doublereal eps, vll, vuu, tmp1; integer indd, inde; doublereal anrm; integer imax; doublereal rmin, rmax; integer inddd, indee; extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); doublereal sigma; extern logical lsame_(char *, char *); integer iinfo; char order[1]; integer indwk; extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), dswap_(integer *, doublereal *, integer *, doublereal *, integer *); integer lwmin; logical lower, wantz; extern doublereal dlamch_(char *); logical alleig, indeig; integer iscale, ieeeok, indibl, indifl; logical valeig; doublereal safmin; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *); extern /* Subroutine */ int xerbla_(char *, integer *); doublereal abstll, bignum; integer indtau, indisp; extern /* Subroutine */ int dstein_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *), dsterf_(integer *, doublereal *, doublereal *, integer *); integer indiwo, indwkn; extern doublereal dlansy_(char *, char *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dstebz_(char *, char *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dstemr_(char *, char *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, integer *, logical *, doublereal *, integer *, integer *, integer *, integer *); integer liwmin; logical tryrac; extern /* Subroutine */ int dormtr_(char *, char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); integer llwrkn, llwork, nsplit; doublereal smlnum; extern /* Subroutine */ int dsytrd_(char *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *); integer lwkopt; logical lquery; /* -- LAPACK driver routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* DSYEVR 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. */ /* DSYEVR first reduces the matrix A to tridiagonal form T with a call */ /* to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute */ /* the eigenspectrum using Relatively Robust Representations. DSTEMR */ /* 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 : DSYEVR calls DSTEMR when the full spectrum is requested */ /* on machines which conform to the ieee-754 floating point standard. */ /* DSYEVR calls DSTEBZ and SSTEIN on non-ieee machines and */ /* when partial spectrum requests are made. */ /* Normal execution of DSTEMR 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, DSTEBZ and */ /* ********* DSTEIN 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) DOUBLE PRECISION 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) 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. */ /* 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 */ /* DLAMCH( '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) DOUBLE PRECISION array, dimension (N) */ /* The first M elements contain the selected eigenvalues in */ /* ascending order. */ /* Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M)) */ /* If JOBZ = 'V', then if INFO = 0, the first M columns of Z */ /* contain the orthonormal eigenvectors of the matrix A */ /* corresponding to the selected eigenvalues, with the i-th */ /* column of Z holding the eigenvector associated with W(i). */ /* If 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) DOUBLE PRECISION 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 DSYTRD and DORMTR */ /* 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. */ /* 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 size of the IWORK array, */ /* returns this value as the first entry of the IWORK array, and */ /* no error message related to 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, "DSYEVR", "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; } else if (*lwork < lwmin && ! lquery) { *info = -18; } else if (*liwork < liwmin && ! lquery) { *info = -20; } } if (*info == 0) { nb = ilaenv_(&c__1, "DSYTRD", uplo, n, &c_n1, &c_n1, &c_n1); /* Computing MAX */ i__1 = nb, i__2 = ilaenv_(&c__1, "DORMTR", 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] = (doublereal) lwkopt; iwork[1] = liwmin; } if (*info != 0) { i__1 = -(*info); xerbla_("DSYEVR", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ *m = 0; if (*n == 0) { work[1] = 1.; return 0; } if (*n == 1) { work[1] = 7.; 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.; } 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; vll = *vl; vuu = *vu; anrm = dlansy_("M", uplo, n, &a[a_offset], lda, &work[1]); if (anrm > 0. && anrm < rmin) { iscale = 1; sigma = rmin / anrm; } else if (anrm > rmax) { iscale = 1; sigma = rmax / anrm; } if (iscale == 1) { if (lower) { i__1 = *n; for (j = 1; j <= i__1; ++j) { i__2 = *n - j + 1; dscal_(&i__2, &sigma, &a[j + j * a_dim1], &c__1); /* L10: */ } } else { i__1 = *n; for (j = 1; j <= i__1; ++j) { dscal_(&j, &sigma, &a[j * a_dim1 + 1], &c__1); /* L20: */ } } if (*abstol > 0.) { abstll = *abstol * sigma; } if (valeig) { vll = *vl * sigma; vuu = *vu * sigma; } } /* Initialize indices into workspaces. Note: The IWORK indices are */ /* used only if DSTERF or DSTEMR fail. */ /* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the */ /* elementary reflectors used in DSYTRD. */ 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 DSYTRD. */ inde = indd + *n; /* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over */ /* -written by DSTEMR (the DSTERF 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 DSTERF and DSTEMR. */ 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 DSTEBZ and */ /* stores the block indices of each of the M<=N eigenvalues. */ indibl = 1; /* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ 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 */ /* DSTEIN. 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 DSYTRD to reduce symmetric matrix to tridiagonal form. */ dsytrd_(uplo, n, &a[a_offset], lda, &work[indd], &work[inde], &work[ indtau], &work[indwk], &llwork, &iinfo); /* If all eigenvalues are desired */ /* then call DSTERF or DSTEMR and DORMTR. */ if ((alleig || indeig && *il == 1 && *iu == *n) && ieeeok == 1) { if (! wantz) { dcopy_(n, &work[indd], &c__1, &w[1], &c__1); i__1 = *n - 1; dcopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1); dsterf_(n, &w[1], &work[indee], info); } else { i__1 = *n - 1; dcopy_(&i__1, &work[inde], &c__1, &work[indee], &c__1); dcopy_(n, &work[indd], &c__1, &work[inddd], &c__1); if (*abstol <= *n * 0. * eps) { tryrac = TRUE_; } else { tryrac = FALSE_; } dstemr_(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 DSTEIN. */ if (wantz && *info == 0) { indwkn = inde; llwrkn = *lwork - indwkn + 1; dormtr_("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 DSTEBZ/DSTEIN. IWORK(:) are */ /* undefined. */ *m = *n; goto L30; } *info = 0; } /* Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN. */ /* Also call DSTEBZ and DSTEIN if DSTEMR fails. */ if (wantz) { *(unsigned char *)order = 'B'; } else { *(unsigned char *)order = 'E'; } dstebz_(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) { dstein_(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 DSTEIN. */ indwkn = inde; llwrkn = *lwork - indwkn + 1; dormtr_("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 DSTEMR/DSTEIN succeeded. */ L30: if (iscale == 1) { if (*info == 0) { imax = *m; } else { imax = *info - 1; } d__1 = 1. / sigma; dscal_(&imax, &d__1, &w[1], &c__1); } /* If eigenvalues are not in order, then sort them, along with */ /* eigenvectors. Note: We do not sort the IFAIL portion of IWORK. */ /* It may not be initialized (if DSTEMR/DSTEIN 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; dswap_(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] = (doublereal) lwkopt; iwork[1] = liwmin; return 0; /* End of DSYEVR */ } /* dsyevr_ */