GESVJ1 - 5.2 English - 68552

GSVJ1 pre-processor for the routine gesvj, applies Jacobi rotations targeting only particular pivots.

Purpose:

   GSVJ1 is called from SGESVJ as a pre-processor and that is its main
   purpose. It applies Jacobi rotations in the same way as SGESVJ does, but
   it targets only particular pivots and it does not check convergence
   (stopping criterion). Few tunning parameters (marked by [TP]) are
   available for the implementer.

   Further Details
   ~~~~~~~~~~~~~~~
   SGSVJ1 applies few sweeps of Jacobi rotations in the column space of
   the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
   off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
   block-entries (tiles) of the (1,2) off-diagonal block are marked by the
   [x]'s in the following scheme:

      | *  *  * [x] [x] [x]|
      | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
      | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
      |[x] [x] [x] *  *  * |
      |[x] [x] [x] *  *  * |
      |[x] [x] [x] *  *  * |

   In terms of the columns of A, the first N1 columns are rotated 'against'
   the remaining N-N1 columns, trying to increase the angle between the
   corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
   tiled using quadratic tiles of side KBL. Here, KBL is a tunning parameter.
   The number of sweeps is given in NSWEEP and the orthogonality threshold
   is given in TOL.

Parameters:

• JOBV – [in]

  JOBV is CHARACTER*1

  Specifies whether the output from this procedure is used to compute the matrix V:

  = ‘V’: the product of the Jacobi rotations is accumulated by postmulyiplying the N-by-N array V. (See the description of V.)

  = ‘A’: the product of the Jacobi rotations is accumulated by postmulyiplying the MV-by-N array V. (See the descriptions of MV and V.)

  = ‘N’: the Jacobi rotations are not accumulated.

• M – [in]

  M is INTEGER

  The number of rows of the input matrix A. M >= 0.

• N – [in]

  N is INTEGER

  The number of columns of the input matrix A. M >= N >= 0.

• N1 – [in]

  N1 is INTEGER

  N1 specifies the 2 x 2 block partition, the first N1 columns are rotated ‘against’ the remaining N-N1 columns of A.

• A – [inout]

  A is REAL array, dimension (LDA,N)

  On entry, M-by-N matrix A, such that A*diag(D) represents the input matrix.

  On exit, A_onexit * D_onexit represents the input matrix A*diag(D) post-multiplied by a sequence of Jacobi rotations, where the rotation threshold and the total number of sweeps are given in TOL and NSWEEP, respectively.

  (See the descriptions of N1, D, TOL and NSWEEP.)

• LDA – [in]

  LDA is INTEGER

  The leading dimension of the array A. LDA >= fla_max(1,M).

• D – [inout]

  D is REAL array, dimension (N)

  The array D accumulates the scaling factors from the fast scaled Jacobi rotations.

  On entry, A*diag(D) represents the input matrix.

  On exit, A_onexit*diag(D_onexit) represents the input matrix post-multiplied by a sequence of Jacobi rotations, where the rotation threshold and the total number of sweeps are given in TOL and NSWEEP, respectively.

  (See the descriptions of N1, A, TOL and NSWEEP.)

• SVA – [inout]

  SVA is REAL array, dimension (N)

  On entry, SVA contains the Euclidean norms of the columns of the matrix A*diag(D).

  On exit, SVA contains the Euclidean norms of the columns of the matrix onexit*diag(D_onexit).

• MV – [in]

  MV is INTEGER

  If JOBV = ‘A’, then MV rows of V are post-multipled by a sequence of Jacobi rotations.

  If JOBV = ‘N’, then MV is not referenced.

• V – [inout]

  V is REAL array, dimension (LDV,N)

  If JOBV = ‘V’ then N rows of V are post-multipled by a sequence of Jacobi rotations.

  If JOBV = ‘A’ then MV rows of V are post-multipled by a sequence of Jacobi rotations.

  If JOBV = ‘N’, then V is not referenced.

• LDV – [in]

  LDV is INTEGER

  The leading dimension of the array V, LDV >= 1.

  If JOBV = ‘V’, LDV >= N.

  If JOBV = ‘A’, LDV >= MV.

• EPS – [in]

  EPS is REAL

  EPS = SLAMCH(‘Epsilon’)

• SFMIN – [in]

  SFMIN is REAL

  SFMIN = SLAMCH(‘Safe Minimum’)

• TOL – [in]

  TOL is REAL

  TOL is the threshold for Jacobi rotations. For a pair A(:,p), A(:,q) of pivot columns, the Jacobi rotation is applied only if ABS(COS(angle(A(:,p),A(:,q)))) > TOL.

• NSWEEP – [in]

  NSWEEP is INTEGER

  NSWEEP is the number of sweeps of Jacobi rotations to be performed.

• WORK – [out] WORK is REAL array, dimension (LWORK)

• LWORK – [in]

  LWORK is INTEGER

  LWORK is the dimension of WORK. LWORK >= M.

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit.

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