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## 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

```

## SYNOPSIS

```       SUBROUTINE PDPOCON( UPLO, N, A, IA,  JA,  DESCA,  ANORM,  RCOND,  WORK,
LWORK, IWORK, LIWORK, INFO )

CHARACTER       UPLO

INTEGER         IA, INFO, JA, LIWORK, LWORK, N

DOUBLE          PRECISION ANORM, RCOND

INTEGER         DESCA( * ), IWORK( * )

DOUBLE          PRECISION A( * ), WORK( * )

```

## 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
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) 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.

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
The  1-norm  (or  infinity-norm)  of  the symmetric 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) 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    >=   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.

IWORK   (local workspace/local output) INTEGER array,
dimension (LIWORK) On exit, IWORK(1) returns  the  minimal  and
optimal LIWORK.

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.

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.
```