DSYGV_2STAGE computes all the eigenvalues, and optionally, the eigenvectors


 of a real generalized symmetric-definite eigenproblem, of the form


 A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.


 Here A and B are assumed to be symmetric and B is also


 positive definite.


 This routine use the 2stage technique for the reduction to tridiagonal


 which showed higher performance on recent architecture and for large


 sizes N>2000.

ITYPE

          ITYPE is INTEGER


          Specifies the problem type to be solved:


          = 1:  A*x = (lambda)*B*x


          = 2:  A*B*x = (lambda)*x


          = 3:  B*A*x = (lambda)*x

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.


                  Not available in this release.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangles of A and B are stored;


          = 'L':  Lower triangles of A and B are stored.

N

          N is INTEGER


          The order of the matrices A and B.  N >= 0.

A

          A is DOUBLE PRECISION 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.  If UPLO = 'L',


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


          the lower triangular part of the matrix A.


          On exit, if JOBZ = 'V', then if INFO = 0, A contains the


          matrix Z of eigenvectors.  The eigenvectors are normalized


          as follows:


          if ITYPE = 1 or 2, Z**T*B*Z = I;


          if ITYPE = 3, Z**T*inv(B)*Z = I.


          If JOBZ = 'N', then on exit the upper triangle (if UPLO='U')


          or the lower triangle (if UPLO='L') of A, including the


          diagonal, is destroyed.

LDA

          LDA is INTEGER


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

B

          B is DOUBLE PRECISION array, dimension (LDB, N)


          On entry, the symmetric positive definite matrix B.


          If UPLO = 'U', the leading N-by-N upper triangular part of B


          contains the upper triangular part of the matrix B.


          If UPLO = 'L', the leading N-by-N lower triangular part of B


          contains the lower triangular part of the matrix B.


          On exit, if INFO <= N, the part of B containing the matrix is


          overwritten by the triangular factor U or L from the Cholesky


          factorization B = U**T*U or B = L*L**T.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

W

          W is DOUBLE PRECISION array, dimension (N)


          If INFO = 0, the eigenvalues in ascending order.

WORK

          WORK is DOUBLE PRECISION 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 >= 1, when N <= 1;


          otherwise  


          If JOBZ = 'N' and N > 1, LWORK must be queried.


                                   LWORK = MAX(1, dimension) where


                                   dimension = max(stage1,stage2) + (KD+1)*N + 2*N


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


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


                                               + (KD+1)*N + 2*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 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.

INFO

          INFO is INTEGER


          = 0:  successful exit


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


          > 0:  DPOTRF or DSYEV returned an error code:


             <= N:  if INFO = i, DSYEV failed to converge;


                    i off-diagonal elements of an intermediate


                    tridiagonal form did not converge to zero;


             > N:   if INFO = N + i, for 1 <= i <= N, then the leading


                    minor of order i of B is not positive definite.


                    The factorization of B could not be completed and


                    no eigenvalues or eigenvectors were computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2017

  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