zheevr (l) - Linux Manuals
zheevr: computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A
NAME
ZHEEVR - computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix ASYNOPSIS
- SUBROUTINE ZHEEVR(
- JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO )
- CHARACTER JOBZ, RANGE, UPLO
- INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, M, N
- DOUBLE PRECISION ABSTOL, VL, VU
- INTEGER ISUPPZ( * ), IWORK( * )
- DOUBLE PRECISION RWORK( * ), W( * )
- COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
PURPOSE
ZHEEVR 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.ZHEEVR first reduces the matrix A to tridiagonal form T with a call to ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute eigenspectrum using Relatively Robust Representations. ZSTEMR 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)
(b)
(c)
(d)
For more details, see DSTEMRaqs documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and
when partial spectrum requests are made.
Normal execution of ZSTEMR 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
-
= aqNaq: Compute eigenvalues only;
= aqVaq: Compute eigenvalues and eigenvectors. - RANGE (input) CHARACTER*1
-
= aqAaq: all eigenvalues will be found.
= aqVaq: all eigenvalues in the half-open interval (VL,VU] will be found. = aqIaq: the IL-th through IU-th eigenvalues will be found. - UPLO (input) CHARACTER*1
-
= aqUaq: Upper triangle of A is stored;
= aqLaq: 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 = aqUaq, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = aqLaq, 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=aqLaq) or the upper triangle (if UPLO=aqUaq) 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=aqVaq, the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = aqAaq or aqIaq.
- IL (input) INTEGER
- IU (input) INTEGER If RANGE=aqIaq, 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 = aqAaq or aqVaq.
- 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( aqSafe minimumaq ). 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 = aqAaq, M = N, and if RANGE = aqIaq, M = IU-IL+1.
- W (output) DOUBLE PRECISION array, dimension (N)
- The first M elements contain the selected eigenvalues in ascending order.
- Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
- If JOBZ = aqVaq, 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 = aqNaq, 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 = aqVaq, 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 = aqVaq, 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 ).
- 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 >= max(1,2*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 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) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
- On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK. 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. 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 byInderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of
Jason Riedy, Computer Science Division, University of