SSYTRD_2STAGE reduces a real symmetric matrix A to real symmetric


 tridiagonal form T by a orthogonal similarity transformation:


 Q1**T Q2**T* A * Q2 * Q1 = T.

VECT

          VECT is CHARACTER*1


          = 'N':  No need for the Housholder representation, 


                  in particular for the second stage (Band to


                  tridiagonal) and thus LHOUS2 is of size max(1, 4*N);


          = 'V':  the Householder representation is needed to 


                  either generate Q1 Q2 or to apply Q1 Q2, 


                  then LHOUS2 is to be queried and computed.


                  (NOT AVAILABLE IN THIS RELEASE).

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 REAL array, dimension (LDA,N)


          On entry, the symmetric matrix A.  If UPLO = 'U', the leading


          N-by-N upper triangular part of A contains the upper


          triangular part of the matrix A, and the strictly lower


          triangular part of A is not referenced.  If UPLO = 'L', the


          leading N-by-N lower triangular part of A contains the lower


          triangular part of the matrix A, and the strictly upper


          triangular part of A is not referenced.


          On exit, if UPLO = 'U', the band superdiagonal


          of A are overwritten by the corresponding elements of the


          internal band-diagonal matrix AB, and the elements above 


          the KD superdiagonal, with the array TAU, represent the orthogonal


          matrix Q1 as a product of elementary reflectors; if UPLO


          = 'L', the diagonal and band subdiagonal of A are over-


          written by the corresponding elements of the internal band-diagonal


          matrix AB, and the elements below the KD subdiagonal, with


          the array TAU, represent the orthogonal matrix Q1 as a product


          of elementary reflectors. See Further Details.

LDA

          LDA is INTEGER


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

D

          D is REAL array, dimension (N)


          The diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)


          The off-diagonal elements of the tridiagonal matrix T.

TAU

          TAU is REAL array, dimension (N-KD)


          The scalar factors of the elementary reflectors of 


          the first stage (see Further Details).

HOUS2

          HOUS2 is REAL array, dimension LHOUS2, that


          store the Householder representation of the stage2


          band to tridiagonal.

LHOUS2

          LHOUS2 is INTEGER


          The dimension of the array HOUS2. LHOUS2 = MAX(1, dimension)


          If LWORK = -1, or LHOUS2=-1,


          then a query is assumed; the routine


          only calculates the optimal size of the HOUS2 array, returns


          this value as the first entry of the HOUS2 array, and no error


          message related to LHOUS2 is issued by XERBLA.


          LHOUS2 = MAX(1, dimension) where


          dimension = 4*N if VECT='N'


          not available now if VECT='H'

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


          The dimension of the array WORK. LWORK = MAX(1, dimension)


          If LWORK = -1, or LHOUS2=-1,


          then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.


          LWORK = MAX(1, dimension) where


          dimension   = max(stage1,stage2) + (KD+1)*N


                      = N*KD + N*max(KD+1,FACTOPTNB) 


                        + max(2*KD*KD, KD*NTHREADS) 


                        + (KD+1)*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.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017

  Implemented by Azzam Haidar.


  All details are available on technical report, SC11, SC13 papers.


  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