PDGEQL2(l)

NAME
PDGEQL2 - compute a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L

SYNOPSIS

SUBROUTINE PDGEQL2(
    M, N, A, IA, JA, DESCA, TAU, WORK, LWORK, INFO )

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

INTEGER
    DESCA( * )

DOUBLE
    PRECISION A( * ), TAU( * ), WORK( * )

PURPOSE
PDGEQL2 computes a QL factorization of a real distributed M-by-N matrix sub( A ) = A(IA:IA+M-1,JA:JA+N-1) = Q * L.

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) DOUBLE PRECISION 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, if M >= N, the lower triangle of the distributed submatrix A( IA+M-N:IA+M-1, JA:JA+N-1 ) contains the N-by-N lower triangular matrix L; if M <= N, the elements on and below the (N-M)-th superdiagonal contain the M by N lower trapezoidal matrix L; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors (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) DOUBLE PRECISION, array, dimension LOCc(JA+N-1)
    This array contains the scalar factors of the elementary reflectors. TAU is tied to the distributed matrix A.

WORK    (local workspace/local output) DOUBLE PRECISION 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 >= Mp0 + MAX( 1, Nq0 ), 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    (local 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 matrix Q is represented as a product of elementary reflectors

   Q = H(ja+k-1) . . . H(ja+1) H(ja), where k = min(m,n).

Each H(i) has the form

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

where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2,ja+n-k+i-1), and tau in TAU(ja+n-k+i-1).