cheevr - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian tridiagonal matrix T
SUBROUTINE CHEEVR(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE, UPLO COMPLEX A(LDA,*), Z(LDZ,*), WORK(*) INTEGER N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER ISUPPZ(*), IWORK(*) REAL VL, VU, ABSTOL REAL W(*), RWORK(*) SUBROUTINE CHEEVR_64(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE, UPLO COMPLEX A(LDA,*), Z(LDZ,*), WORK(*) INTEGER*8 N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER*8 ISUPPZ(*), IWORK(*) REAL VL, VU, ABSTOL REAL W(*), RWORK(*) F95 INTERFACE SUBROUTINE HEEVR(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, Z INTEGER :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER, DIMENSION(:) :: ISUPPZ, IWORK REAL :: VL, VU, ABSTOL REAL, DIMENSION(:) :: W, RWORK SUBROUTINE HEEVR_64(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, Z INTEGER(8) :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK REAL :: VL, VU, ABSTOL REAL, DIMENSION(:) :: W, RWORK C INTERFACE #include <sunperf.h> void cheevr(char jobz, char range, char uplo, int n, complex *a, int lda, float vl, float vu, int il, int iu, float abstol, int *m, float *w, complex *z, int ldz, int *isuppz, int *info); void cheevr_64(char jobz, char range, char uplo, long n, complex *a, long lda, float vl, float vu, long il, long iu, float abstol, long *m, float *w, complex *z, long ldz, long *isuppz, long *info);
Oracle Solaris Studio Performance Library cheevr(3P) NAME cheevr - compute selected eigenvalues and, optionally, eigenvectors of a complex Hermitian tridiagonal matrix T SYNOPSIS SUBROUTINE CHEEVR(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE, UPLO COMPLEX A(LDA,*), Z(LDZ,*), WORK(*) INTEGER N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER ISUPPZ(*), IWORK(*) REAL VL, VU, ABSTOL REAL W(*), RWORK(*) SUBROUTINE CHEEVR_64(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER*1 JOBZ, RANGE, UPLO COMPLEX A(LDA,*), Z(LDZ,*), WORK(*) INTEGER*8 N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER*8 ISUPPZ(*), IWORK(*) REAL VL, VU, ABSTOL REAL W(*), RWORK(*) F95 INTERFACE SUBROUTINE HEEVR(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, Z INTEGER :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER, DIMENSION(:) :: ISUPPZ, IWORK REAL :: VL, VU, ABSTOL REAL, DIMENSION(:) :: W, RWORK SUBROUTINE HEEVR_64(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, Z INTEGER(8) :: N, LDA, IL, IU, M, LDZ, LWORK, LRWORK, LIWORK, INFO INTEGER(8), DIMENSION(:) :: ISUPPZ, IWORK REAL :: VL, VU, ABSTOL REAL, DIMENSION(:) :: W, RWORK C INTERFACE #include <sunperf.h> void cheevr(char jobz, char range, char uplo, int n, complex *a, int lda, float vl, float vu, int il, int iu, float abstol, int *m, float *w, complex *z, int ldz, int *isuppz, int *info); void cheevr_64(char jobz, char range, char uplo, long n, complex *a, long lda, float vl, float vu, long il, long iu, float abstol, long *m, float *w, complex *z, long ldz, long *isuppz, long *info); PURPOSE cheevr computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian 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, CHEEVR calls CSTEGR to compute the eigenspectrum using Relatively Robust Representations. CSTEGR 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 : CHEEVR calls CSTEGR 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 CSTEGR 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. UPLO (input) = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) The order of the matrix A. N >= 0. A (input/output) On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangu- lar 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 diag- onal, is destroyed. LDA (input) The leading dimension of the array A. LDA >= max(1,N). 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) 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) 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) 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'. 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 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 guaran- tees 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 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). 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) The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). ISUPPZ (output) 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 LWORK. LWORK (input) 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 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) On exit, if INFO = 0, RWORK(1) returns the optimal (and mini- mal) LRWORK. LRWORK (input) The length of the array RWORK. LRWORK >= max(1,24*N). If LRWORK = -1, then a workspace query is assumed; the rou- tine only calculates the optimal size of the RWORK array, returns this value as the first entry of the RWORK array, and no error message related to LRWORK 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 >= max(1,10*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 cheevr(3P)