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 (0 free), 3 equality constraints
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SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
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 : 1.69E+001 0.000
1 : -2.44E-001 3.91E+000 0.000 0.2312 0.9000 0.9000 1.39 1 1 2.0E+000
2 : 1.86E-001 3.22E-001 0.000 0.0824 0.9900 0.9900 1.31 1 1 4.0E-001
3 : 1.23E-001 1.12E-004 0.000 0.0003 0.9999 0.9999 0.99 1 1 2.6E-004
4 : 1.23E-001 5.18E-006 0.000 0.0463 0.9900 0.9900 1.00 1 1 1.2E-005
5 : 1.23E-001 2.36E-007 0.000 0.0455 0.9900 0.9900 1.00 1 1 6.8E-007
6 : 1.23E-001 1.90E-008 0.327 0.0806 0.9900 0.9900 1.00 2 2 5.5E-008
7 : 1.23E-001 8.16E-010 0.000 0.0429 0.9906 0.9900 0.99 2 2 2.2E-009
iter seconds digits c*x b*y
7 0.0 8.5 1.2273256559e-001 1.2273256524e-001
|Ax-b| = 1.1e-010, [Ay-c]_+ = 1.4E-010, |x|= 2.0e+000, |y|= 7.6e-001
Detailed timing (sec)
Pre IPM Post
1.001E-002 1.001E-002 0.000E+000
Max-norms: ||b||=2, ||c|| = 2,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 9.99993.
------------------------------------------------------------------------
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):
4.4503e-011
The optimal weighted trace:
0.1227