dstevr - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T
SUBROUTINE DSTEVR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER ISUPPZ(*), IWORK(*) DOUBLE PRECISION VL, VU, ABSTOL DOUBLE PRECISION D(*), E(*), W(*), Z(LDZ,*), WORK(*) SUBROUTINE DSTEVR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER*8 ISUPPZ(*), IWORK(*) DOUBLE PRECISION VL, VU, ABSTOL DOUBLE PRECISION D(*), E(*), W(*), Z(LDZ,*), WORK(*) F95 INTERFACE SUBROUTINE STEVR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER, DIMENSION(:) :: ISUPPZ, IWORK REAL(8) :: VL, VU, ABSTOL REAL(8), DIMENSION(:) :: D, E, W, WORK REAL(8), DIMENSION(:,:) :: Z SUBROUTINE STEVR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK REAL(8) :: VL, VU, ABSTOL REAL(8), DIMENSION(:) :: D, E, W, WORK REAL(8), DIMENSION(:,:) :: Z C INTERFACE #include <sunperf.h> void dstevr(char jobz, char range, int n, double *d, double *e, double vl, double vu, int il, int iu, double abstol, int *m, double *w, double *z, int ldz, int *isuppz, int *info); void dstevr_64(char jobz, char range, long n, double *d, double *e, double vl, double vu, long il, long iu, double abstol, long *m, double *w, double *z, long ldz, long *isuppz, long *info);
Oracle Solaris Studio Performance Library dstevr(3P) NAME dstevr - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T SYNOPSIS SUBROUTINE DSTEVR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE INTEGER N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER ISUPPZ(*), IWORK(*) DOUBLE PRECISION VL, VU, ABSTOL DOUBLE PRECISION D(*), E(*), W(*), Z(LDZ,*), WORK(*) SUBROUTINE DSTEVR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE INTEGER*8 N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER*8 ISUPPZ(*), IWORK(*) DOUBLE PRECISION VL, VU, ABSTOL DOUBLE PRECISION D(*), E(*), W(*), Z(LDZ,*), WORK(*) F95 INTERFACE SUBROUTINE STEVR(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE INTEGER :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER, DIMENSION(:) :: ISUPPZ, IWORK REAL(8) :: VL, VU, ABSTOL REAL(8), DIMENSION(:) :: D, E, W, WORK REAL(8), DIMENSION(:,:) :: Z SUBROUTINE STEVR_64(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE INTEGER(8) :: N, IL, IU, M, LDZ, LWORK, LIWORK, INFO INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK REAL(8) :: VL, VU, ABSTOL REAL(8), DIMENSION(:) :: D, E, W, WORK REAL(8), DIMENSION(:,:) :: Z C INTERFACE #include <sunperf.h> void dstevr(char jobz, char range, int n, double *d, double *e, double vl, double vu, int il, int iu, double abstol, int *m, double *w, double *z, int ldz, int *isuppz, int *info); void dstevr_64(char jobz, char range, long n, double *d, double *e, double vl, double vu, long il, long iu, double abstol, long *m, double *w, double *z, long ldz, long *isuppz, long *info); PURPOSE dstevr computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Whenever possible, DSTEVR calls DSTEGR to compute the eigenspectrum using Relatively Robust Representations. DSTEGR 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 the i-th unreduced block of T, (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation, (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm, (c) If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a), (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input param- eter ABSTOL. For more details, see "A new O(n^2) algorithm for the symmetric tridi- agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997. Note 1 : DSTEVR calls DSTEGR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and when partial spectrum requests are made. Normal execution of DSTEGR 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) = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE (input) = '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) The order of the matrix. N >= 0. D (input/output) 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) 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) 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'. VU (input) See the description of VL. IL (input) If RANGE='I', the indices (in ascending order) of the small- est 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'. IU (input) See the description of IL. ABSTOL (input) The absolute error tolerance for the eigenvalues. An approx- imate eigenvalue is accepted as converged when it is deter- mined 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 guaran- tees 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 eigenval- ues to high relative accuracy. M (output) The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W (output) The first M elements contain the selected eigenvalues in ascending order. Z (output) If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corre- sponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). 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) 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 indi- cating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). WORK (workspace) On exit, if INFO = 0, WORK(1) returns the optimal (and mini- mal) LWORK. LWORK (input) The dimension of the array WORK. LWORK >= 20*N. 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) On exit, if INFO = 0, IWORK(1) returns the optimal (and mini- mal) LIWORK. LIWORK (input) The dimension of the array IWORK. LIWORK >= 10*N. If LIWORK = -1, then a workspace query is assumed; the rou- tine 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) = 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 7 Nov 2015 dstevr(3P)