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## NAME

```       PCPOCON  -  estimate  the  reciprocal  of  the condition number (in the
1-norm) of a complex Hermitian  positive  definite  distributed  matrix
using  the  Cholesky factorization A = U**H*U or A = L*L**H computed by
PCPOTRF

```

## SYNOPSIS

```       SUBROUTINE PCPOCON( UPLO, N, A, IA,  JA,  DESCA,  ANORM,  RCOND,  WORK,
LWORK, RWORK, LRWORK, INFO )

CHARACTER       UPLO

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

REAL            ANORM, RCOND

INTEGER         DESCA( * )

REAL            RWORK( * )

COMPLEX         A( * ), WORK( * )

```

## PURPOSE

```       PCPOCON  estimates  the  reciprocal  of  the  condition  number (in the
1-norm) of a complex Hermitian  positive  definite  distributed  matrix
using  the  Cholesky factorization A = U**H*U or A = L*L**H computed by
PCPOTRF.

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

```       UPLO    (global input) CHARACTER
Specifies whether the factor stored  in  A(IA:IA+N-1,JA:JA+N-1)
is upper or lower triangular.
= ’U’:  Upper triangular
= ’L’:  Lower 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) ). 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 PCPOTRF.

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) REAL
The  1-norm  (or  infinity-norm)  of  the hermitian distributed
matrix A(IA:IA+N-1,JA:JA+N-1).

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