SYNOPSIS

SUBROUTINE PCTZRZF(

M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

    

INTEGER IA, INFO, JA, LWORK, M, N

    

INTEGER DESCA( * )

    

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

PURPOSE

PCTZRZF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix sub( A ) = A(IA:IA+M-1,JA: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.

A (local input/local output) COMPLEX 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 M+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, 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/local output) COMPLEX array,

dimension (LWORK) On exit, WORK(1) returns the minimal and optimal LWORK.

LWORK (local or global input) INTEGER

The dimension of the array WORK. LWORK is local input and must be at least LWORK >= MB_A * ( Mp0 + Nq0 + MB_A ), 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.

If LWORK = -1, then LWORK is global input and a workspace query is assumed; the routine only calculates the minimum and optimal size for all work arrays. Each of these values is returned in the first entry of the corresponding work array, and no error message is issued by PXERBLA.

INFO (global output) INTEGER

= 0: successful exit
< 0: If the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j), if the i-th argument is a scalar and had an illegal value, then INFO = -i.

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 ).