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## ssytrd_sb2st()

subroutine ssytrd_sb2st | ( | character | stage1,
|  | character | vect,
|  | character | uplo,
|  | integer | n,
|  | integer | kd,
|  | real, dimension( ldab, * ) | ab,
|  | integer | ldab,
|  | real, dimension( * ) | d,
|  | real, dimension( * ) | e,
|  | real, dimension( * ) | hous,
|  | integer | lhous,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer | info )

SSYTRD_SB2ST reduces a real symmetric band matrix A to real symmetric tridiagonal form T

Download SSYTRD_SB2ST + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> 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.
      !>

Parameters

  [in]

  STAGE1

      !>          STAGE1 is CHARACTER*1
      !>          = 'N':  : 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':  : 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.
      !>

  [in]

  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).
      !>

  [in]

  UPLO

      !>          UPLO is CHARACTER*1
      !>          = 'U':  Upper triangle of A is stored;
      !>          = 'L':  Lower triangle of A is stored.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrix A.  N >= 0.
      !>

  [in]

  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.
      !>

  [in,out]

  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.
      !>

  [in]

  LDAB

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

  [out]

  D

      !>          D is REAL array, dimension (N)
      !>          The diagonal elements of the tridiagonal matrix T.
      !>

  [out]

  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'.
      !>

  [out]

  HOUS

      !>          HOUS is REAL array, dimension (MAX(1,LHOUS))
      !>          Stores the Householder representation.
      !>

  [in]

  LHOUS

      !>          LHOUS is INTEGER
      !>          The dimension of the array HOUS.
      !>          If N = 0 or KD <= 1, LHOUS >= 1, else 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'
      !>

  [out]

  WORK

      !>          WORK is REAL array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO = 0, WORK(1) returns optimal LWORK.
      !>

  [in]

  LWORK

      !>          LWORK is INTEGER
      !>          The dimension of the array WORK.
      !>          IF N = 0 or KD <= 1, LWORK >= 1, else 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.
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>          = 0:  successful exit
      !>          < 0:  if INFO = -i, the i-th argument had an illegal value
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

Further Details:

      !>
      !>  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
      !>
      !>