DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of


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


 the reduction to tridiagonal. If eigenvectors are desired, it uses


 a divide and conquer algorithm.


 The divide and conquer algorithm makes very mild assumptions about


 floating point arithmetic. It will work on machines with a guard


 digit in add/subtract, or on those binary machines without guard


 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or


 Cray-2. It could conceivably fail on hexadecimal or decimal machines


 without guard digits, but we know of none.

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 sizes of the WORK and IWORK


          arrays, returns these values as the first entries of the WORK


          and IWORK arrays, and no error message related to LWORK 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 LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.


          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.


          If JOBZ  = 'V' and N > 2, LIWORK must be at least 3 + 5*N.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates the optimal sizes of the WORK and


          IWORK arrays, returns these values as the first entries of


          the WORK and IWORK arrays, and no error message related to


          LWORK 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:  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