.TH PDPOCON l "12 May 1997" "LAPACK version 1.5" "LAPACK routine (version 1.5)"
.SH NAME
PDPOCON - estimate the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite distributed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by PDPOTRF
.SH SYNOPSIS
.TP 20
SUBROUTINE PDPOCON(
UPLO, N, A, IA, JA, DESCA, ANORM, RCOND, WORK,
LWORK, IWORK, LIWORK, INFO )
.TP 20
.ti +4
CHARACTER
UPLO
.TP 20
.ti +4
INTEGER
IA, INFO, JA, LIWORK, LWORK, N
.TP 20
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DOUBLE
PRECISION ANORM, RCOND
.TP 20
.ti +4
INTEGER
DESCA( * ), IWORK( * )
.TP 20
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DOUBLE
PRECISION A( * ), WORK( * )
.SH PURPOSE
PDPOCON estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric positive definite distributed matrix
using the Cholesky factorization A = U**T*U or A = L*L**T computed by
PDPOTRF.

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
.br
           RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1)      ) *
                         norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).

Notes
.br
=====
.br

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.
.br

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".
.br

NOTATION        STORED IN      EXPLANATION
.br
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type.  In this case,
                               DTYPE_A = 1.
.br
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.
.br
M_A    (global) DESCA( M_ )    The number of rows in the global
                               array A.
.br
N_A    (global) DESCA( N_ )    The number of columns in the global
                               array A.
.br
MB_A   (global) DESCA( MB_ )   The blocking factor used to distribute
                               the rows of the array.
.br
NB_A   (global) DESCA( NB_ )   The blocking factor used to distribute
                               the columns of the array.
.br
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.
.br
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.
.br
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.
.br
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.
.br
The values of LOCr() and LOCc() may be determined via a call to the
ScaLAPACK tool function, NUMROC:
.br
        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:
.br
        LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
.br
        LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
.br

.SH ARGUMENTS
.TP 8
UPLO    (global input) CHARACTER
Specifies whether the factor stored in
A(IA:IA+N-1,JA:JA+N-1) is upper or lower triangular.
.br
= 'U':  Upper triangular
.br
= 'L':  Lower triangular
.TP 8
N       (global input) INTEGER
.br
The order of the distributed matrix A(IA:IA+N-1,JA:JA+N-1).
N >= 0.
.TP 8
A       (local input) DOUBLE PRECISION 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 or U from
the Cholesky factorization A(IA:IA+N-1,JA:JA+N-1) = U'*U or
L*L', as computed by PDPOTRF.
.TP 8
IA      (global input) INTEGER
The row index in the global array A indicating the first
row of sub( A ).
.TP 8
JA      (global input) INTEGER
The column index in the global array A indicating the
first column of sub( A ).
.TP 8
DESCA   (global and local input) INTEGER array of dimension DLEN_.
The array descriptor for the distributed matrix A.
.TP 8
ANORM   (global input) DOUBLE PRECISION
The 1-norm (or infinity-norm) of the symmetric distributed
matrix A(IA:IA+N-1,JA:JA+N-1).
.TP 8
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
.br
RCOND = 1 / ( norm( A(IA:IA+N-1,JA:JA+N-1)      ) *
.br
norm( inv(A(IA:IA+N-1,JA:JA+N-1)) ) ).
.TP 8
WORK    (local workspace/local output) DOUBLE PRECISION array,
dimension (LWORK)
On exit, WORK(1) returns the minimal and optimal LWORK.
.TP 8
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)) + 2*LOCc(N+MOD(JA-1,NB_A)) +
MAX( 2, MAX(NB_A*CEIL(NPROW-1,NPCOL),LOCc(N+MOD(JA-1,NB_A)) +
NB_A*CEIL(NPCOL-1,NPROW)) ).

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.
.TP 8
IWORK   (local workspace/local output) INTEGER array,
dimension (LIWORK)
On exit, IWORK(1) returns the minimal and optimal LIWORK.
.TP 8
LIWORK  (local or global input) INTEGER
The dimension of the array IWORK.
LIWORK is local input and must be at least
LIWORK >= LOCr(N+MOD(IA-1,MB_A)).

If LIWORK = -1, then LIWORK 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.
.TP 8
INFO    (global output) INTEGER
= 0:  successful exit
.br
< 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.
