PZGECON(l)

NAME
PZGECON - estimate the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PZGETRF

SYNOPSIS

SUBROUTINE PZGECON(
    NORM, N, A, IA, JA, DESCA, ANORM, RCOND, WORK, LWORK, RWORK, LRWORK, INFO )

CHARACTER
    NORM

INTEGER
    IA, INFO, JA, LRWORK, LWORK, N

DOUBLE
    PRECISION ANORM, RCOND

INTEGER
    DESCA( * )

DOUBLE
    PRECISION RWORK( * )

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

PURPOSE
PZGECON estimates the reciprocal of the condition number of a general distributed complex matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm, using the LU factorization computed by PZGETRF.

An estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), and the reciprocal of the condition number is computed as
           RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) *
                        norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).

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

NORM    (global input) CHARACTER
    Specifies whether the 1-norm condition number or the infinity-norm condition number is required:
    = '1' or 'O': 1-norm
    = 'I': Infinity-norm

N       (global input) INTEGER
    The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1). N >= 0.

A       (local input) COMPLEX*16 pointer into the local memory
    to an array of dimension ( LLD_A, LOCc(JA+N-1) ). On entry, this array contains the local pieces of the factors L and U from the factorization A(IA:IA+N-1,JA:JA+N-1) = P*L*U; the unit diagonal elements of L are not stored.

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.

ANORM   (global input) DOUBLE PRECISION
    If NORM = '1' or 'O', the 1-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1). If NORM = 'I', the infinity-norm of the original distributed matrix A(IA:IA+N-1,JA:JA+N-1).

RCOND   (global output) DOUBLE PRECISION
    The reciprocal of the condition number of the distributed matrix A(IA:IA+N-1,JA:JA+N-1), computed as
    RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1) ) *
    norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).

WORK    (local workspace/local output) COMPLEX*16 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 >= 2*LOCr(N+MOD(IA-1,MB_A)) + MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(NPCOL-1,NPROW)) ).

    LOCr and LOCc values can be computed using the ScaLAPACK tool function NUMROC; 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.

RWORK   (local workspace/local output) DOUBLE PRECISION array,
    dimension (LRWORK) On exit, RWORK(1) returns the minimal and optimal LRWORK.

LRWORK  (local or global input) INTEGER
    The dimension of the array RWORK. LRWORK is local input and must be at least LRWORK >= 2*LOCc(N+MOD(JA-1,NB_A)).

    If LRWORK = -1, then LRWORK 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.