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## NAME

```       PDTRCON  -  estimate  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

```

## SYNOPSIS

```       SUBROUTINE PDTRCON( NORM, UPLO, DIAG, N, A, IA, JA, DESCA, RCOND, WORK,
LWORK, IWORK, LIWORK, INFO )

CHARACTER       DIAG, NORM, UPLO

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

DOUBLE          PRECISION RCOND

INTEGER         DESCA( * ), IWORK( * )

DOUBLE          PRECISION A( * ), WORK( * )

```

## PURPOSE

```       PDTRCON  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) DOUBLE PRECISION 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) 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))    +
LOCc(N+MOD(JA-1,NB_A))    +   MAX(   2,   MAX(   NB_A*MAX(   1,
CEIL(NPROW-1,NPCOL) ), LOCc(N+MOD(JA-1,NB_A))  +  NB_A*MAX(  1,
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.
```