%**************************************************************************
% Sum of hermitian squares (SOHS) related *
%**************************************************************************
% -------------------------------------------------------------------------
% First we construct some symbolic noncommuting variables and some
% polynomials
% -------------------------------------------------------------------------
>> NCvars x y z
>> f = y*x^2*y - y*x*z + 4*y*z^2*y - z*x*y + z^2;
>> g = 2*x + 2*x^2 + x*y + 2*y + y*x + y^2;
>> h = x^2 + x*y + y*x + 2*y^2 + z^2 - 1;
% -------------------------------------------------------------------------
% We can check whether the polynomial is a sum of hermitian squares and
% compute the desired decomposition.
% -------------------------------------------------------------------------
>> [IsSohs,X,base,sohs,fsos] = NCsos(f)
***** NCSOStools: module NCSos started *****
Input polynomial has (max) degree 4 and min degree 2.
Detected 5 monomials in 3 variables.
There are 121 monomials in 3 variables of degree at most 4.
There are 117 monomials in 3 variables of degree at most 4 and at least 2.
After Newton Chip Method keeping 3 monomials.
Number of linear constraints: 6.
Starting SDP solver ...
...
Residual norm: 1.1896e-014
Computing SOHS decomposition ... done.
Found SOHS decomposition with 3 factors.
*************** Polynomial is SOHS ***************
IsSohs = 1
X = 1.0000 0.0000 -1.0000
0.0000 4.0000 -0.0000
-1.0000 -0.0000 1.0000
base = 'x*y'
'z*y'
'z'
sohs = x*y-z
2*z*y
5e-008*z
fsos = y*x^2*y-y*x*z+4*y*z^2*y-z*x*y+1*z^2
% -------------------------------------------------------------------------
% NCsos has many optional parameters. For example we can set the smallest
% value that is considered to be nonzero in numerical calculations.
% -------------------------------------------------------------------------
>> params.precision = 1e-4;
>> [IsSohs,X,base,sohs,fsos] = NCsos(f,params)
***** NCSOStools: module NCSos started *****
Input polynomial has (max) degree 4 and min degree 2.
Detected 5 monomials in 3 variables.
There are 121 monomials in 3 variables of degree at most 4.
There are 117 monomials in 3 variables of degree at most 4 and at least 2.
After Newton Chip Method keeping 3 monomials.
Number of linear constraints: 6.
Starting SDP solver ...
...
Residual norm: 1.1896e-014
Computing SOHS decomposition ... done.
Found SOHS decomposition with 2 factors.
*************** Polynomial is SOHS ***************
IsSohs = 1
X = 1.0000 0.0000 -1.0000
0.0000 4.0000 -0.0000
-1.0000 -0.0000 1.0000
base = 'x*y'
'z*y'
'z'
sohs = x*y-z
2*z*y
fsos = y*x^2*y-y*x*z+4*y*z^2*y-z*x*y+z^2
% -------------------------------------------------------------------------
% We can also computes the maximal epsilon such that the polynomial
% g-epsilon is a sum of hermitian squares.
% -------------------------------------------------------------------------
>> g
g = 2*x+2*x^2+x*y+2*y+y*x+y^2
>> [epsilon,X,base,sohs,gmin] = NCmin(g)
Input polynomial has no constant term ... we add it for computing the base of monomials.
Input polynomial has (max) degree 2 and min degree 0.
Detected 7 monomials in 2 variables.
There are 7 monomials in 2 variables of degree at most 2.
After Newton Chip Method keeping 3 monomials.
Number of linear constraints: 5.
Starting SDP solver ...
...
Residual norm: 2.7182e-010
Computing SOHS decomposition ... done.
Found SOHS decomposition with 3 factors.
epsilon = -1.0000
X = 1.0000 1.0000 1.0000
1.0000 2.0000 1.0000
1.0000 1.0000 1.0000
base = ''
'x'
'y'
sohs = 1+x+y
x
4.1e-007*y
gmin = 1+2*x+2*x^2+x*y+2*y+y*x+1*y^2
% -------------------------------------------------------------------------
% NCmin has many optional parameters. For example we can set the smallest
% value that is considered to be nonzero in numerical calculations.
% -------------------------------------------------------------------------
>> params.precision = 1e-4;
>> [epsilon,X,base,sohs,gmin] = NCmin(g,params)
Input polynomial has no constant term ... we add it for computing the base of monomials.
Input polynomial has (max) degree 2 and min degree 0.
Detected 7 monomials in 2 variables.
There are 7 monomials in 2 variables of degree at most 2.
After Newton Chip Method keeping 3 monomials.
Number of linear constraints: 5.
Starting SDP solver ...
...
Residual norm: 2.7182e-010
Computing SOHS decomposition ... done.
Found SOHS decomposition with 2 factors.
epsilon = -1.0000
X = 1.0000 1.0000 1.0000
1.0000 2.0000 1.0000
1.0000 1.0000 1.0000
base = ''
'x'
'y'
sohs = 1+x+y
x
gmin = 1+2*x+2*x^2+x*y+2*y+y*x+y^2
% -------------------------------------------------------------------------
% For given polynomials f and g we can compute the maximal epsilon such
% that the polynomial f-epsilon*g is a sum of hermitian squares.
% -------------------------------------------------------------------------
>> [epsilon,X,base,sohs,poly] = NCdiff(x^2-x^2*y-y*x^2+y*x^2*y+4*y^2,x*y+y*x-2*y*x*y)
Input polynomial has (max) degree 4 and min degree 2.
Detected 8 monomials in 2 variables.
There are 31 monomials in 2 variables of degree at most 4.
There are 28 monomials in 2 variables of degree at most 4 and at least 2.
After Newton Chip Method keeping 3 monomials.
Number of linear constraints: 6.
Starting SDP solver ...
...
Residual norm: 1.285e-011
Computing SOHS decomposition ... done.
Found SOHS decomposition with 3 factors.
***** Optimal solution found: epsilon_max = 2.000000 *****
epsilon = 2.0000
X = 1.0000 -2.0000 -1.0000
-2.0000 4.0000 2.0000
-1.0000 2.0000 1.0000
base = 'x'
'y'
'x*y'
sohs = x-x*y-2*y
3.95e-006*x*y+5.58e-006*y
2.54e-006*x*y
poly = x^2-x^2*y-2*x*y-2*y*x-y*x^2+1*y*x^2*y+4*y*x*y+4*y^2
% -------------------------------------------------------------------------
% We can check if the polynomial is convex, i.e., if its second directional
% derivative is a sum of hermitian squares.
% -------------------------------------------------------------------------
>> g
g = 2*x+2*x^2+x*y+2*y+y*x+y^2
>> [isConvex,g,sohs] = NCisConvex0(g)
isConvex = 1
g = 4*h1^2+2*h1*h2+2*h2*h1+2*h2^2
sohs = 2*h1+h2
h2
% -------------------------------------------------------------------------
% We can also check if the polynomial is convex without SDP solvers
% -------------------------------------------------------------------------
>> isConvex = NCisConvex(g)
isConvex = 1