DSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of


 a real symmetric band matrix A using the 2stage technique for


 the reduction to tridiagonal.

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 triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


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

KD

          KD is INTEGER


          The number of superdiagonals of the matrix A if UPLO = 'U',


          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB, N)


          On entry, the upper or lower triangle of the symmetric band


          matrix A, stored in the first KD+1 rows of the array.  The


          j-th column of A is stored in the j-th column of the array AB


          as follows:


          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;


          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).


          On exit, AB is overwritten by values generated during the


          reduction to tridiagonal form.  If UPLO = 'U', the first


          superdiagonal and the diagonal of the tridiagonal matrix T


          are returned in rows KD and KD+1 of AB, and if UPLO = 'L',


          the diagonal and first subdiagonal of T are returned in the


          first two rows of AB.

LDAB

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= KD + 1.

W

          W is DOUBLE PRECISION array, dimension (N)


          If INFO = 0, the eigenvalues in ascending order.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)


          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal


          eigenvectors of the matrix A, with the i-th column of Z


          holding the eigenvector associated with W(i).


          If JOBZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension 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 = (2KD+1)*N + KD*NTHREADS + N


                                   where KD is the size of the band.


                                   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:  if INFO = i, the algorithm failed to converge; i


                off-diagonal elements of an intermediate tridiagonal


                form did not converge to zero.

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