SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric


 tridiagonal form T by a orthogonal similarity transformation:


 Q**T * A * Q = T.

STAGE1

          STAGE1 is CHARACTER*1


          = 'N':  'No': to mention that the stage 1 of the reduction  


                  from dense to band using the ssytrd_sy2sb routine


                  was not called before this routine to reproduce AB. 


                  In other term this routine is called as standalone. 


          = 'Y':  'Yes': to mention that the stage 1 of the 


                  reduction from dense to band using the ssytrd_sy2sb 


                  routine has been called to produce AB (e.g., AB is


                  the output of ssytrd_sy2sb.

VECT

          VECT is CHARACTER*1


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


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


          = 'V':  the Householder representation is needed to 


                  either generate or to apply Q later on, 


                  then LHOUS 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.

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 REAL 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, the diagonal elements of AB are overwritten by the


          diagonal elements of the tridiagonal matrix T; if KD > 0, the


          elements on the first superdiagonal (if UPLO = 'U') or the


          first subdiagonal (if UPLO = 'L') are overwritten by the


          off-diagonal elements of T; the rest of AB is overwritten by


          values generated during the reduction.

LDAB

          LDAB is INTEGER


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

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:


          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

HOUS

          HOUS is REAL array, dimension LHOUS, that


          store the Householder representation.

LHOUS

          LHOUS is INTEGER


          The dimension of the array HOUS. LHOUS = MAX(1, dimension)


          If LWORK = -1, or LHOUS=-1,


          then a query is assumed; the routine


          only calculates the optimal size of the HOUS array, returns


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


          message related to LHOUS is issued by XERBLA.


          LHOUS = 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 LHOUS=-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   = (2KD+1)*N + KD*NTHREADS


          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