cheevr-f(3)

cheevr.f –

Synopsis

Functions/Subroutines

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)
CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Function/Subroutine Documentation

subroutine cheevr (characterJOBZ, characterRANGE, characterUPLO, integerN, complex, dimension( lda, * )A, integerLDA, realVL, realVU, integerIL, integerIU, realABSTOL, integerM, real, dimension( * )W, complex, dimension( ldz, * )Z, integerLDZ, integer, dimension( * )ISUPPZ, complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK, integerLRWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)

CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

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.

Parameters:

JOBZ

 JOBZ is CHARACTER*1
 = 'N': Compute eigenvalues only;
 = 'V': Compute eigenvalues and eigenvectors.

RANGE

 RANGE is 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

 UPLO is CHARACTER*1
 = 'U': Upper triangle of A is stored;
 = 'L': Lower triangle of A is stored.

 N is INTEGER
 The order of the matrix A. N >= 0.

 A is 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

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max(1,N).

 VL is REAL

 VU is 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 is INTEGER

 IU is 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

 ABSTOL is 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 is INTEGER
 The total number of eigenvalues found. 0 <= M <= N.
 If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

 W is REAL array, dimension (N)
 The first M elements contain the selected eigenvalues in
 ascending order.

 Z is 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

 LDZ is INTEGER
 The leading dimension of the array Z. LDZ >= 1, and if
 JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

 ISUPPZ is 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

 WORK is COMPLEX array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

 LWORK is 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

 RWORK is REAL array, dimension (MAX(1,LRWORK))
 On exit, if INFO = 0, RWORK(1) returns the optimal
 (and minimal) LRWORK.

LRWORK

 LRWORK is 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

 IWORK is INTEGER array, dimension (MAX(1,LIWORK))
 On exit, if INFO = 0, IWORK(1) returns the optimal
 (and minimal) LIWORK.

LIWORK

 LIWORK is 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

 INFO is INTEGER
 = 0: successful exit
 < 0: if INFO = -i, the i-th argument had an illegal value
 > 0: Internal error

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Contributors:

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

Definition at line 347 of file cheevr.f.

Author

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