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zheevr.f(3)				      LAPACK				      zheevr.f(3)

NAME
       zheevr.f -

SYNOPSIS
   Functions/Subroutines
       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)
	    ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors
	   for HE matrices

Function/Subroutine Documentation
   subroutine zheevr (characterJOBZ, characterRANGE, characterUPLO, integerN, complex*16,
       dimension( lda, * )A, integerLDA, double precisionVL, double precisionVU, integerIL,
       integerIU, double precisionABSTOL, integerM, double precision, dimension( * )W,
       complex*16, dimension( ldz, * )Z, integerLDZ, integer, dimension( * )ISUPPZ, complex*16,
       dimension( * )WORK, integerLWORK, double precision, dimension( * )RWORK, integerLRWORK,
       integer, dimension( * )IWORK, integerLIWORK, integerINFO)
	ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       HE matrices

       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) 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 : 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.

       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, DSTEBZ and
		     ZSTEIN are called

	   UPLO

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

	   N

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

	   A

		     A is COMPLEX*16 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

		     VL is DOUBLE PRECISION

	   VU

		     VU is 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

		     IL is INTEGER

	   IU

		     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 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
		     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

		     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

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

	   Z

		     Z is COMPLEX*16 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*16 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 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

		     RWORK is DOUBLE PRECISION 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 zheevr.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      zheevr.f(3)
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