SGSVJ1 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 tuning 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 tuning parameter.


 The number of sweeps is given in NSWEEP and the orthogonality threshold


 is given in TOL.

JOBV

          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 postmultiplying the N-by-N array V.


                (See the description of V.)


          = 'A': the product of the Jacobi rotations is accumulated


                 by postmultiplying the MV-by-N array V.


                (See the descriptions of MV and V.)


          = 'N': the Jacobi rotations are not accumulated.

M

          M is INTEGER


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

N

          N is INTEGER


          The number of columns of the input matrix A.


          M >= N >= 0.

N1

          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

          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

          LDA is INTEGER


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

D

          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

          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

          MV is INTEGER


          If JOBV = 'A', then MV rows of V are post-multiplied by a


                           sequence of Jacobi rotations.


          If JOBV = 'N',   then MV is not referenced.

V

          V is REAL array, dimension (LDV,N)


          If JOBV = 'V' then N rows of V are post-multiplied by a


                           sequence of Jacobi rotations.


          If JOBV = 'A' then MV rows of V are post-multiplied by a


                           sequence of Jacobi rotations.


          If JOBV = 'N',   then V is not referenced.

LDV

          LDV is INTEGER


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


          If JOBV = 'V', LDV >= N.


          If JOBV = 'A', LDV >= MV.

EPS

          EPS is REAL


          EPS = SLAMCH('Epsilon')

SFMIN

          SFMIN is REAL


          SFMIN = SLAMCH('Safe Minimum')

TOL

          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

          NSWEEP is INTEGER


          NSWEEP is the number of sweeps of Jacobi rotations to be


          performed.

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


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

INFO

          INFO is INTEGER


          = 0:  successful exit.


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

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Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)