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   Invert the Pei test matrix

   Erwin Kalvelagen, july 2008

   Command line arguments:
      --n=integer   order of the matrix
      --alpha=real  scalar defining the Pei matrix
      --method=1    invert using equations A AINV = I
      --method=2    invert using equations A AINV = I using advanced basis
      --method=3    invert using external solver


   Reference:

    Erwin Kalvelagen
    SOLVING SYSTEMS OF LINEAR EQUATIONS WITH GAMS
    http://www.amsterdamoptimization.com/pdf/lineq.pdf  

    ML Pei,
    A test matrix for inversion procedures,
    Communications of the ACM,
    Volume 5, 1962, page 508.

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*
* defaults
*
$if not set n $set n 100
$if not set method $set method 1
$if not set alpha $set alpha 1

scalar alpha /%alpha%/;

set i /i1*i%n%/;
alias (i,j,k);

parameter a(i,j);
a(i,j)=1;
a(i,i)=1+alpha;

parameter ident(i,j);
ident(i,i)=1;

variables
   ainv(i,j)
   dummy
;

equations
   e(i,j)
   edummy
;

edummy.. dummy =e= 0;
e(i,j)..  sum(k, a(i,k)*ainv(k,j)) =e= ident(i,j);

model m /all/;

option iterlim=1000000;



$if "%method%"==1 solve m minimizing dummy using lp;

$ifthen "%method%"==2

edummy.m=1;
e.m(i,j)=1;
dummy.m=0;
ainv.m(i,j)=0;
solve m minimizing dummy using lp;

$endif

$ifthen "%method%"==3

execute_unload 'a.gdx',i,a;
execute '=invert2.exe a.gdx i a b.gdx pinva';
parameter pinva(i,j);
execute_load 'b.gdx',pinva;

$endif