PZLATRZ(l)

NAME
PZLATRZ - reduce the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)]

SYNOPSIS

SUBROUTINE PZLATRZ(
    M, N, L, A, IA, JA, DESCA, TAU, WORK )

INTEGER
    IA, JA, L, M, N

INTEGER
    DESCA( * )

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

PURPOSE
PZLATRZ reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = [A(IA:IA+M-1,JA:JA+M-1) A(IA:IA+M-1,JA+N-L:JA+N-1)] to upper triangular form by means of unitary transformations.

The upper trapezoidal matrix sub( A ) is factored as

   sub( A ) = ( R 0 ) * Z,

where Z is an N-by-N unitary matrix and R is an M-by-M upper triangular matrix.

Notes
=====

Each global data object is described by an associated description vector. This vector stores the information required to establish the mapping between an object element and its corresponding process and memory location.

Let A be a generic term for any 2D block cyclicly distributed array. Such a global array has an associated description vector DESCA. In the following comments, the character _ should be read as "of the global array".

NOTATION STORED IN EXPLANATION
--------------- -------------- -------------------------------------- DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
                        DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
                        the BLACS process grid A is distribu-
                        ted over. The context itself is glo-
                        bal, but the handle (the integer
                        value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
                        array A.
N_A (global) DESCA( N_ ) The number of columns in the global
                        array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
                        the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
                        the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
                        row of the array A is distributed. CSRC_A (global) DESCA( CSRC_ ) The process column over which the
                        first column of the array A is
                        distributed.
LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
                        array. LLD_A >= MAX(1,LOCr(M_A)).

Let K be the number of rows or columns of a distributed matrix, and assume that its process grid has dimension p x q.
LOCr( K ) denotes the number of elements of K that a process would receive if K were distributed over the p processes of its process column.
Similarly, LOCc( K ) denotes the number of elements of K that a process would receive if K were distributed over the q processes of its process row.
The values of LOCr() and LOCc() may be determined via a call to the ScaLAPACK tool function, NUMROC:
        LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
        LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper bound for these quantities may be computed by:
        LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
        LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A

ARGUMENTS

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

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

L       (global input) INTEGER
    The columns of the distributed submatrix sub( A ) containing the meaningful part of the Householder reflectors. L > 0.

A       (local input/local output) COMPLEX*16 pointer into the
    local memory to an array of dimension (LLD_A, LOCc(JA+N-1)). On entry, the local pieces of the M-by-N distributed matrix sub( A ) which is to be factored. On exit, the leading M-by-M upper triangular part of sub( A ) contains the upper trian- gular matrix R, and elements N-L+1 to N of the first M rows of sub( A ), with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors.

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 LOCr(IA+M-1)
    This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A.

WORK    (local workspace) COMPLEX*16 array, dimension (LWORK)
    LWORK >= Nq0 + MAX( 1, Mp0 ), where

    IROFF = MOD( IA-1, MB_A ), ICOFF = MOD( JA-1, NB_A ), IAROW = INDXG2P( IA, MB_A, MYROW, RSRC_A, NPROW ), IACOL = INDXG2P( JA, NB_A, MYCOL, CSRC_A, NPCOL ), Mp0 = NUMROC( M+IROFF, MB_A, MYROW, IAROW, NPROW ), Nq0 = NUMROC( N+ICOFF, NB_A, MYCOL, IACOL, NPCOL ),

    and NUMROC, INDXG2P are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL can be determined by calling the subroutine BLACS_GRIDINFO.

FURTHER DETAILS
The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), whose conjugate transpose is used to introduce zeros into the (m - k + 1)th row of sub( A ), is given in the form

   Z( k ) = ( I 0 ),
            ( 0 T( k ) )

where

   T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
                        ( 0 )
                        ( z( k ) )

tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of sub( A ).

The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of sub( A ), such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of sub( A ).

Z is given by

   Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).