PZLAHRD(3)    ScaLAPACK routine of NEC Numeric Library Collection   PZLAHRD(3)



NAME
       PZLAHRD - reduce the first NB columns of a complex general N-by-(N-K+1)
       distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-
       th subdiagonal are zero

SYNOPSIS
       SUBROUTINE PZLAHRD( N,  K,  NB,  A,  IA,  JA, DESCA, TAU, T, Y, IY, JY,
                           DESCY, WORK )

           INTEGER         IA, IY, JA, JY, K, N, NB

           INTEGER         DESCA( * ), DESCY( * )

           COMPLEX*16      A( * ), T( * ), TAU( * ), WORK( * ), Y( * )

PURPOSE
       PZLAHRD reduces the first NB columns of a complex general  N-by-(N-K+1)
       distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-
       th subdiagonal are zero. The reduction is performed by an unitary simi-
       larity  transformation  Q'  * A * Q. The routine returns the matrices V
       and T which determine Q as a block reflector I - V*T*V', and  also  the
       matrix Y = A * V * T.

       This  is  an auxiliary routine called by PZGEHRD. In the following com-
       ments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1).


ARGUMENTS
       N       (global input) INTEGER
               The number of rows and columns to  be  operated  on,  i.e.  the
               order of the distributed submatrix sub( A ).  N >= 0.

       K       (global input) INTEGER
               The offset for the reduction. Elements below the k-th subdiago-
               nal in the first NB columns are reduced to zero.

       NB      (global input) INTEGER
               The number of columns to be reduced.

       A       (local input/local output) COMPLEX*16 pointer into
               the local memory to an array of  dimension  (LLD_A,  LOCc(JA+N-
               K)).  On entry, this array contains the the local pieces of the
               N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K).
               On  exit, the elements on and above the k-th subdiagonal in the
               first NB columns are overwritten with  the  corresponding  ele-
               ments of the reduced distributed matrix; the elements below the
               k-th subdiagonal, with the array TAU, represent the matrix Q as
               a  product  of  elementary  reflectors.  The  other  columns of
               A(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details.   IA
               (global  input)  INTEGER  The  row  index in the global array A
               indicating the first row of sub( A ).

       JA      (global input) INTEGER
               The column index in the global array  A  indicating  the  first
               column of sub( A ).

       DESCA   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix A.

       TAU     (local output) COMPLEX*16 array, dimension LOCc(JA+N-2)
               The  scalar  factors  of the elementary reflectors (see Further
               Details). TAU is tied to the distributed matrix A.

       T       (local output) COMPLEX*16 array, dimension (NB_A,NB_A)
               The upper triangular matrix T.

       Y       (local output) COMPLEX*16 pointer into the local memory
               to an array of dimension (LLD_Y,NB_A). On exit, this array con-
               tains  the  local  pieces  of the N-by-NB distributed matrix Y.
               LLD_Y >= LOCr(IA+N-1).

       IY      (global input) INTEGER
               The row index in the global array Y indicating the first row of
               sub( Y ).

       JY      (global input) INTEGER
               The  column  index  in  the global array Y indicating the first
               column of sub( Y ).

       DESCY   (global and local input) INTEGER array of dimension DLEN_.
               The array descriptor for the distributed matrix Y.

       WORK    (local workspace) COMPLEX*16 array, dimension (NB)

FURTHER DETAILS
       The matrix Q is represented as a product of nb elementary reflectors

          Q = H(1) H(2) . . . H(nb).

       Each H(i) has the form

          H(i) = I - tau * v * v'

       where tau is  a  complex  scalar,  and  v  is  a  complex  vector  with
       v(1:i+k-1)   =  0,  v(i+k)  =  1;  v(i+k+1:n)  is  stored  on  exit  in
       A(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).

       The elements of the vectors v together form the (n-k+1)-by-nb matrix  V
       which is needed, with T and Y, to apply the transformation to the unre-
       duced  part  of  the   matrix,   using   an   update   of   the   form:
       A(ia:ia+n-1,ja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1,ja:ja+n-k)-Y*V').

       The  contents  of A(ia:ia+n-1,ja:ja+n-k) on exit are illustrated by the
       following example with n = 7, k = 3 and nb = 2:

          ( a   h   a   a   a )
          ( a   h   a   a   a )
          ( a   h   a   a   a )
          ( h   h   a   a   a )
          ( v1  h   a   a   a )
          ( v1  v2  a   a   a )
          ( v1  v2  a   a   a )

       where a denotes an element of the original matrix
       A(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper  Hes-
       senberg  matrix  H,  and  vi  denotes an element of the vector defining
       H(i).




ScaLAPACK routine               31 October 2017                     PZLAHRD(3)