description:
[iscConvex,g,sohs,h] = NCisCycConvex(f,precision) checks if the second directional derivative of a polynomial is cyclically equivalent to a sum of hermitian squares.
arguments:
f is an NCpoly representing a polynomial.
With precision we can set the smallest value that is considered to be nonzero in numerical calculations; if the command is called without it, we assume the precision set with the command NCsetPrecision or the value set in NCparam.m.
output:
iscConvex equals 1 if the second directional derivative of a polynomial f is cyclically equivalent to a sum of hermitian squares and 0 otherwise
g is an NCpoly representing the second derivative of the polynomial f
sohs is the SOHS decomposition of the polynomial cyclically equivalent to the second derivative of the polynomial f
h is the NCpoly representing sum_i m_i^*m_i cyclically equivalent to the second derivative of the polynomial f
possible usage:
NCisCycConvex(f), NCisCycConvex(f,precision)
example:
>>f = 1+2*x^2+x*y+2*y^2;
>> [iscConvex,g,sohs,h] = NCisCycConvex(f,10e-5)
iscConvex = 1
g = 4*h1^2+2*h1*h2+4*h2^2
sohs = 0.5*h1+2*h2
1.937*h1
h = 4.00003225*h1^2+h1*h2+h2*h1+4*h2^2