Minimal phase spectral factorization
b = [2; 0.2; -0.3];
n = length( b );
cvx_begin sdp
variable X( n, n ) symmetric
dual variable y{n}
minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
for k = 1 : n,
sum( diag( X, k - 1 ) ) == b( k ) : y{k};
end
X >= 0;
cvx_end
y = [ y{:} ]';
disp( 'The optimal point, X:' );
disp( X )
disp( 'The diagonal sums:' );
disp( sum( spdiags( X, 0:n-1 ), 1 ) );
disp( 'min( eig( X ) ) (should be nonnegative):' );
disp( min( eig( X ) ) )
disp( 'The optimal weighted trace:' );
disp( ( n - 1 : -1 : 0 ) * diag( X ) );
Calling sedumi: 6 variables, 3 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 3, order n = 4, dim = 10, blocks = 2
nnz(A) = 6 + 0, nnz(ADA) = 9, nnz(L) = 6
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 7.04E+00 0.000
1 : 1.25E+00 1.56E+00 0.000 0.2213 0.9000 0.9000 0.99 1 1 1.1E+00
2 : 1.82E+00 7.85E-02 0.000 0.0504 0.9900 0.9900 1.23 1 1 5.1E-02
3 : 1.88E+00 2.33E-04 0.000 0.0030 0.9990 0.9990 0.99 1 1 1.5E-04
4 : 1.88E+00 4.91E-05 0.000 0.2110 0.9000 0.9000 1.00 1 1 3.1E-05
5 : 1.88E+00 2.25E-06 0.090 0.0459 0.9900 0.9900 1.00 1 1 1.4E-06
6 : 1.88E+00 1.18E-07 0.002 0.0522 0.9900 0.9901 1.00 1 1 7.1E-08
7 : 1.88E+00 9.90E-09 0.311 0.0841 0.9900 0.9900 1.00 1 1 5.9E-09
iter seconds digits c*x b*y
7 0.0 8.5 1.8772674419e+00 1.8772674359e+00
|Ax-b| = 1.8e-09, [Ay-c]_+ = 2.9E-09, |x|= 2.4e+00, |y|= 1.9e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 3.000E-02 1.000E-02
Max-norms: ||b||=1, ||c|| = 2,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.122733
The optimal point, X:
0.0468 -0.0369 -0.3000
-0.0369 0.0292 0.2369
-0.3000 0.2369 1.9240
The diagonal sums:
2.0000 0.2000 -0.3000
min( eig( X ) ) (should be nonnegative):
-2.9368e-09
The optimal weighted trace:
0.1227