DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors


 of a real symmetric band 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.

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.

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.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)


          If JOBZ = 'V', the N-by-N orthogonal matrix used in the


                         reduction to tridiagonal form.


          If JOBZ = 'N', the array Q is not referenced.

LDQ

          LDQ is INTEGER


          The leading dimension of the array Q.  If JOBZ = 'V', then


          LDQ >= max(1,N).

VL

          VL is DOUBLE PRECISION


          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 DOUBLE PRECISION


          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 DOUBLE PRECISION


          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 AB to tridiagonal form.


          Eigenvalues will be computed most accurately when ABSTOL is


          set to twice the underflow threshold 2*DLAMCH('S'), not zero.


          If this routine returns with INFO>0, indicating that some


          eigenvectors did not converge, try setting ABSTOL to


          2*DLAMCH('S').


          See "Computing Small Singular Values of Bidiagonal Matrices


          with Guaranteed High Relative Accuracy," by Demmel and


          Kahan, LAPACK Working Note #3.

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 DOUBLE PRECISION array, dimension (N)


          The first M elements contain the selected eigenvalues in


          ascending order.

Z

          Z is DOUBLE PRECISION 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 an eigenvector fails to converge, then that column of Z


          contains the latest approximation to the eigenvector, and the


          index of the eigenvector is returned in IFAIL.


          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).

WORK

          WORK is DOUBLE PRECISION array, dimension (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, 7*N, dimension) where


                                   dimension = (2KD+1)*N + KD*NTHREADS + 2*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.

IWORK

          IWORK is INTEGER array, dimension (5*N)

IFAIL

          IFAIL is INTEGER array, dimension (N)


          If JOBZ = 'V', then if INFO = 0, the first M elements of


          IFAIL are zero.  If INFO > 0, then IFAIL contains the


          indices of the eigenvectors that failed to converge.


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

INFO

          INFO is INTEGER


          = 0:  successful exit.


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


          > 0:  if INFO = i, then i eigenvectors failed to converge.


                Their indices are stored in array IFAIL.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

  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