description:
[X,fX,eig_val,eig_vec,A,fA]=NCopt(f,AW) computes minimizers and the minimum of the polynomial f.
arguments:
f is an NCpoly representing a polynomial.
AW: true/false whether AWbd is called.
output:
X: from GNS - a matrix where each of its rows represents a square matrix
fX: f(X) where X is from GNS
eig_val: eigenvalues of fX
eig_vec: corresponding eigenvectors
A: from AWbd
fA: f(A) where A is from AWbd - actually on each block of A
possible usage:
NCopt(f), NCopt(f,AW)
see also:
example:
>> f = (1 - y + x*y + y*x)'*(1 - y + x*y + y*x) + (-2 + y^2)^2 + (-x + x^2)^2;
>> [X,fX,eig_val,eig_vec]=NCopt(f)
*** Minimum eigenvalue for f is 0.000000. ***
X = Columns 1 through 8
0.1492 0.1176 0.3287 0.0611 0.1176 0.9387 -0.0007 -0.2507
1.4113 -0.0868 0.0193 -0.0090 -0.0868 -1.3218 0.0909 -0.4394
Columns 9 through 16
0.3287 -0.0007 0.7765 0.0740 0.0611 -0.2507 0.0740 -0.6993
0.0193 0.0909 -1.3201 -0.4347 -0.0090 -0.4394 -0.4347 0.9773
fX = 0.0108 -0.0222 0.0066 -0.1102
-0.0222 0.2500 0.1858 -0.2756
0.0066 0.1858 0.2098 -0.4285
-0.1102 -0.2756 -0.4285 3.8200
eig_val = 0.0000 0 0 0
0 0.0427 0 0
0 0 0.3507 0
0 0 0 3.8971
eig_vec = 0.9304 0.3562 -0.0825 0.0278
0.2804 -0.5590 0.7761 0.0806
-0.2353 0.7468 0.6105 0.1191
0.0207 0.0543 0.1344 -0.9892