.TH "SRC/zheevr_2stage.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/zheevr_2stage.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBzheevr_2stage\fP (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info)"
.br
.RI "\fB ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine zheevr_2stage (character jobz, character range, character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)"

.PP
\fB ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors
!> of a complex Hermitian matrix A using the 2stage technique for
!> the reduction to tridiagonal\&.  Eigenvalues and eigenvectors can
!> be selected by specifying either a range of values or a range of
!> indices for the desired eigenvalues\&.
!>
!> ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call
!> to ZHETRD\&.  Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute
!> eigenspectrum using Relatively Robust Representations\&.  ZSTEMR
!> computes eigenvalues by the dqds algorithm, while orthogonal
!> eigenvectors are computed from various  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 ZSTEMR's documentation and:
!> - Inderjit S\&. Dhillon and Beresford N\&. Parlett: 
!>   Linear Algebra and its Applications, 387(1), pp\&. 1-28, August 2004\&.
!> - Inderjit Dhillon and Beresford Parlett:  SIAM Journal on Matrix Analysis and Applications, Vol\&. 25,
!>   2004\&.  Also LAPACK Working Note 154\&.
!> - Inderjit Dhillon: ,
!>   Computer Science Division Technical Report No\&. UCB/CSD-97-971,
!>   UC Berkeley, May 1997\&.
!>
!>
!> Note 1 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested
!> on machines which conform to the ieee-754 floating point standard\&.
!> ZHEEVR_2STAGE 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\&.
!> 
.fi
.PP
 
.RE
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\fBParameters\fP
.RS 4
\fIJOBZ\fP 
.PP
.nf
!>          JOBZ is CHARACTER*1
!>          = 'N':  Compute eigenvalues only;
!>          = 'V':  Compute eigenvalues and eigenvectors\&.
!>                  Not available in this release\&.
!> 
.fi
.PP
.br
\fIRANGE\fP 
.PP
.nf
!>          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
!> 
.fi
.PP
.br
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          = 'U':  Upper triangle of A is stored;
!>          = 'L':  Lower triangle of A is stored\&.
!> 
.fi
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.br
\fIN\fP 
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.nf
!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIA\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
!>          LDA is INTEGER
!>          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIVL\fP 
.PP
.nf
!>          VL is DOUBLE PRECISION
!>          If RANGE='V', the lower bound of the interval to
!>          be searched for eigenvalues\&. VL < VU\&.
!>          Not referenced if RANGE = 'A' or 'I'\&.
!> 
.fi
.PP
.br
\fIVU\fP 
.PP
.nf
!>          VU is DOUBLE PRECISION
!>          If RANGE='V', the upper bound of the interval to
!>          be searched for eigenvalues\&. VL < VU\&.
!>          Not referenced if RANGE = 'A' or 'I'\&.
!> 
.fi
.PP
.br
\fIIL\fP 
.PP
.nf
!>          IL is INTEGER
!>          If RANGE='I', the index of the
!>          smallest eigenvalue 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'\&.
!> 
.fi
.PP
.br
\fIIU\fP 
.PP
.nf
!>          IU is INTEGER
!>          If RANGE='I', the index of the
!>          largest eigenvalue 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'\&.
!> 
.fi
.PP
.br
\fIABSTOL\fP 
.PP
.nf
!>          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  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
!>          future releases will\&. See J\&. Barlow and J\&. Demmel,
!>          , LAPACK Working Note #7, for a discussion
!>          of which matrices define their eigenvalues to high relative
!>          accuracy\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIW\fP 
.PP
.nf
!>          W is DOUBLE PRECISION array, dimension (N)
!>          The first M elements contain the selected eigenvalues in
!>          ascending order\&.
!> 
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
!>          LDZ is INTEGER
!>          The leading dimension of the array Z\&.  LDZ >= 1, and if
!>          JOBZ = 'V', LDZ >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIISUPPZ\fP 
.PP
.nf
!>          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 )\&. This is an output of ZSTEMR (tridiagonal
!>          matrix)\&. The support of the eigenvectors of A is typically
!>          1:N because of the unitary transformations applied by ZUNMTR\&.
!>          Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          LWORK is INTEGER
!>          The dimension of the array WORK\&.
!>          If N <= 1,               LWORK must be at least 1\&.
!>          If JOBZ = 'N' and N > 1, LWORK must be queried\&.
!>                                   LWORK = MAX(1, 26*N, dimension) where
!>                                   dimension = max(stage1,stage2) + (KD+1)*N + N
!>                                             = N*KD + N*max(KD+1,FACTOPTNB)
!>                                               + max(2*KD*KD, KD*NTHREADS)
!>                                               + (KD+1)*N + N
!>                                   where KD is the blocking size of the reduction,
!>                                   FACTOPTNB is the blocking used by the QR or LQ
!>                                   algorithm, usually FACTOPTNB=128 is a good choice
!>                                   NTHREADS is the number of threads used when
!>                                   openMP compilation is enabled, otherwise =1\&.
!>          If JOBZ = 'V' and N > 1, LWORK must be queried\&. Not yet available
!>
!>          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\&.
!> 
.fi
.PP
.br
\fIRWORK\fP 
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!>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
!>          On exit, if INFO = 0, RWORK(1) returns the optimal
!>          (and minimal) LRWORK\&.
!> 
.fi
.PP
.br
\fILRWORK\fP 
.PP
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!>          LRWORK is INTEGER
!>          The length of the array RWORK\&.
!>          If N <= 1, LRWORK >= 1, else LRWORK >= 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\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal
!>          (and minimal) LIWORK\&.
!> 
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
!>          LIWORK is INTEGER
!>          The dimension of the array IWORK\&.
!>          If N <= 1, LIWORK >= 1, else LIWORK >= 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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  Internal error
!> 
.fi
.PP
 
.RE
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\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

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Univ\&. of Colorado Denver 

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NAG Ltd\&. 
.RE
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\fBContributors:\fP
.RS 4
Inderjit Dhillon, IBM Almaden, USA 
.br
 Osni Marques, LBNL/NERSC, USA 
.br
 Ken Stanley, Computer Science Division, University of California at Berkeley, USA 
.br
 Jason Riedy, Computer Science Division, University of California at Berkeley, USA 
.br
 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  All details about the 2stage techniques are available in:
!>
!>  Azzam Haidar, Hatem Ltaief, and Jack Dongarra\&.
!>  Parallel reduction to condensed forms for symmetric eigenvalue problems
!>  using aggregated fine-grained and memory-aware kernels\&. In Proceedings
!>  of 2011 International Conference for High Performance Computing,
!>  Networking, Storage and Analysis (SC '11), New York, NY, USA,
!>  Article 8 , 11 pages\&.
!>  http://doi\&.acm\&.org/10\&.1145/2063384\&.2063394
!>
!>  A\&. Haidar, J\&. Kurzak, P\&. Luszczek, 2013\&.
!>  An improved parallel singular value algorithm and its implementation
!>  for multicore hardware, In Proceedings of 2013 International Conference
!>  for High Performance Computing, Networking, Storage and Analysis (SC '13)\&.
!>  Denver, Colorado, USA, 2013\&.
!>  Article 90, 12 pages\&.
!>  http://doi\&.acm\&.org/10\&.1145/2503210\&.2503292
!>
!>  A\&. Haidar, R\&. Solca, S\&. Tomov, T\&. Schulthess and J\&. Dongarra\&.
!>  A novel hybrid CPU-GPU generalized eigensolver for electronic structure
!>  calculations based on fine-grained memory aware tasks\&.
!>  International Journal of High Performance Computing Applications\&.
!>  Volume 28 Issue 2, Pages 196-209, May 2014\&.
!>  http://hpc\&.sagepub\&.com/content/28/2/196
!>
!> 
.fi
.PP
 
.RE
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Definition at line \fB405\fP of file \fBzheevr_2stage\&.f\fP\&.
.SH "Author"
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Generated automatically by Doxygen for LAPACK from the source code\&.