CHEEVR_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.


 CHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call


 to CHETRD.  Then, whenever possible, CHEEVR_2STAGE calls CSTEMR to compute


 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_2STAGE calls CSTEMR when the full spectrum is requested


 on machines which conform to the ieee-754 floating point standard.


 CHEEVR_2STAGE 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.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.


                  Not available in this release.

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

          N is INTEGER


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

A

          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

          VL is REAL


          If RANGE='V', the lower bound of the interval to


          be searched for eigenvalues. VL < VU.


          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is REAL


          If RANGE='V', the upper bound of the interval to


          be searched for eigenvalues. VL < VU.


          Not referenced if RANGE = 'A' or 'I'.

IL

          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'.

IU

          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'.

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

          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 REAL array, dimension (N)


          The first M elements contain the selected eigenvalues in


          ascending order.

Z

          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 ). This is an output of CSTEMR (tridiagonal


          matrix). The support of the eigenvectors of A is typically 


          1:N because of the unitary transformations applied by CUNMTR.


          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 dimension of the array WORK.  


          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.

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

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

  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