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## NAME

```       PDPOEQU  -  compute  row  and column scalings intended to equilibrate a
distributed  symmetric  positive   definite   matrix   sub(   A   )   =
A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to
the two-norm)

```

## SYNOPSIS

```       SUBROUTINE PDPOEQU( N, A, IA, JA, DESCA, SR, SC, SCOND, AMAX, INFO )

INTEGER         IA, INFO, JA, N

DOUBLE          PRECISION AMAX, SCOND

INTEGER         DESCA( * )

DOUBLE          PRECISION A( * ), SC( * ), SR( * )

```

## PURPOSE

```       PDPOEQU computes row and column  scalings  intended  to  equilibrate  a
distributed   symmetric   positive   definite   matrix   sub(   A  )  =
A(IA:IA+N-1,JA:JA+N-1) and reduce its condition number (with respect to
the   two-norm).    SR  and  SC  contain  the  scale  factors,  S(i)  =
1/sqrt(A(i,i)), chosen so that the scaled distri- buted matrix  B  with
elements  B(i,j)  =  S(i)*A(i,j)*S(j)  has ones on the  diagonal.  This
choice of SR and SC puts the condition number of B within a factor N of
the  smallest  possible  condition  number  over  all possible diagonal
scalings.

The scaling factor are stored  along  process  rows  in  SR  and  along
process  columns  in  SC.  The  duplication  of  information simplifies
greatly the application of the factors.

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

```       N       (global input) INTEGER
The  number of rows and columns to be operated on i.e the order
of the distributed submatrix sub( A ). N >= 0.

A       (local input) DOUBLE PRECISION pointer into the local memory to
an
array of local dimension ( LLD_A, LOCc(JA+N-1)  ),  the  N-by-N
symmetric  positive  definite distributed matrix sub( A ) whose
scaling factors are to be computed.  Only the diagonal elements
of sub( A ) are referenced.

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.

SR      (local output) DOUBLE PRECISION array, dimension LOCr(M_A)
If  INFO  = 0, SR(IA:IA+N-1) contains the row scale factors for
sub( A ). SR is aligned with  the  distributed  matrix  A,  and
replicated  across  every  process  column.  SR  is tied to the
distributed matrix A.

SC      (local output) DOUBLE PRECISION array, dimension LOCc(N_A)
If INFO = 0, SC(JA:JA+N-1) contains the column scale factors
for A(IA:IA+M-1,JA:JA+N-1). SC is aligned  with  the  distribu-
ted matrix A, and replicated down every process row. SC is tied
to the distributed matrix A.

SCOND   (global output) DOUBLE PRECISION
If INFO = 0, SCOND contains the ratio of the smallest SR(i) (or
SC(j))  to the largest SR(i) (or SC(j)), with IA <= i <= IA+N-1
and JA <= j <= JA+N-1. If SCOND >= 0.1 and AMAX is neither  too
large nor too small, it is not worth scaling by SR (or SC).

AMAX    (global output) DOUBLE PRECISION
Absolute  value  of  largest  matrix  element.  If AMAX is very
close to overflow or very close to underflow, the matrix should
be scaled.

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.  > 0:  If
INFO = K, the K-th diagonal entry of sub( A ) is nonpositive.
```