gauss.gms : Matrix Inversion with Full Pivoting
This example demonstrates the use of Loops and Dynamic definition
of sets in elementary transformations using Gaussian Elimination with
full pivot selection.
Reference:
- GAMS Development Corporation, Formulation and Language Example.
Small Model of Type: GAMS
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$Title Matrix Inversion with full Pivoting (GAUSS,SEQ=71)
$Ontext
This example demonstrates the use of Loops and Dynamic definition
of sets in elementary transformations using Gaussian Elimination with
full pivot selection.
GAMS Development Corporation, Formulation and Language Example.
$Offtext
Sets i row labels / row-1 * row-3 /
j column labels / col-1 * col-3 / ; alias (i,ip), (j,jp)
Table a(i,j) original matrix
col-1 col-2 col-3
row-1 1 2 3
row-2 1 3 4
row-3 1 4 3
Parameters b(j,i) inverse of a
bp(i,j) permuted and transposed inverse of a
pair(i,j) pivoting sequence and permutation
rank rank of matrix a
adet absolute value of determinant of matrix a
piv, big, tol ;
Sets r(i) pivot row candidates , npr(i) non pivot rows
s(j) pivot column candidates, nps(j) non pivot columns ;
r(i) = yes; s(j) = yes; bp(i,j) = a(i,j); rank = 0;
adet = 1; tol = 1e-5;
Loop(j, big = smax((r,s), abs(bp(r,s))); big$(big lt tol) = 0;
npr(i) = yes; nps(jp) = yes;
Loop((r,s)$(big and big eq abs(bp(r,s))),
rank = rank+1; pair(r,s) = rank; piv = 1/bp(r,s);
big = 0; adet = adet/piv; npr(r) = no; nps(s) = no;
bp( r,nps) = bp(r,nps)*piv;
bp(npr,nps) = bp(npr,nps) - bp(r,nps)*bp(npr,s);
bp(npr, s) = -bp(npr,s)*piv;
bp( r, s) = piv );
r(r) = npr(r); s(s) = nps(s) );
b(j,i) = sum((ip,jp)$(pair(i,jp) and pair(ip,j)), bp(ip,jp));
adet = abs(adet);
Display a, rank, adet, pair, bp, b;
Parameters ir(i,ip) a times b, ic(j,jp) b times a;
ir(i,ip) = sum(j, a(i,j)*b(j,ip));
ic(j,jp) = sum(i, b(j,i)*a(i,jp));
Display ir, ic;
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