SYNOPSIS

SUBROUTINE PCTRCON(

NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK, LWORK, RWORK, LRWORK, INFO )

    

CHARACTER DIAG, NORM, UPLO

    

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

    

REAL RCOND

    

INTEGER DESCA( * )

    

REAL RWORK( * )

    

COMPLEX A( * ), WORK( * )

PURPOSE

PCTRCON estimates the reciprocal of the condition number of a triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1), in either the 1-norm or the infinity-norm.

The norm of A(IA:IA+N-1,JA:JA+N-1) is computed and an estimate is obtained for norm(inv(A(IA:IA+N-1,JA:JA+N-1))), then 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.

UPLO (global input) CHARACTER

= 'U': A(IA:IA+N-1,JA:JA+N-1) is upper triangular;
= 'L': A(IA:IA+N-1,JA:JA+N-1) is lower triangular.

DIAG (global input) CHARACTER

= 'N': A(IA:IA+N-1,JA:JA+N-1) is non-unit triangular;
= 'U': A(IA:IA+N-1,JA:JA+N-1) is unit triangular.

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 pointer into the local memory

to an array of dimension ( LLD_A, LOCc(JA+N-1) ). This array contains the local pieces of the triangular distributed matrix A(IA:IA+N-1,JA:JA+N-1). If UPLO = 'U', the leading N-by-N upper triangular part of this distributed matrix con- tains the upper triangular matrix, and its strictly lower triangular part is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of this ditributed matrix contains the lower triangular matrix, and the strictly upper triangular part is not referenced. If DIAG = 'U', the diagonal elements of A(IA:IA+N-1,JA:JA+N-1) are also not referenced and are assumed to be 1.

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.

RCOND (global output) REAL

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 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(P-1,Q),LOCc(N+MOD(JA-1,NB_A)) + NB_A*CEIL(Q-1,P)) ).

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) REAL 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 >= 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.