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function [Db, S] = williamson(V, tol)
% Williamson decomposition of positive-definite (real) symmetric matrix.
%
% Args:
% V (matrix): positive definite symmetric (real) matrix
% tol (scalar): tolerance for symmetry check (default: 1e-11)
%
% Returns:
% Db (matrix): diagonal matrix
% S (matrix): symplectic matrix such that V = S' * Db * S
if nargin < 2
tol = 1e-11;
end
[n, m] = size(V);
if n ~= m
error('The input matrix is not square');
end
diffn = norm(V - V');
if diffn >= tol
error('The input matrix is not symmetric');
end
if mod(n, 2) ~= 0
error('The input matrix must have an even number of rows/columns');
end
n = n / 2;
omega = sympmat(n);
vals = eig(V);
if any(vals <= 0)
error('Input matrix is not positive definite');
end
Mm12 = real(sqrtm(inv(V)));
r1 = Mm12 * omega * Mm12;
[K, s1] = schur(r1);
X = [0 1; 1 0];
I = eye(2);
seq = cell(1, n);
% Construct a permutation matrix p
for i = 1:n
if s1(2*i-1, 2*i) > 0
seq{i} = I;
else
seq{i} = X;
end
end
p = blkdiag(seq{:});
Kt = K * p;
s1t = p * s1 * p;
s1t
dd = xpxp_to_xxpp(s1t);
perm_indices = xpxp_to_xxpp(1:2*n);
Ktt = Kt(:, perm_indices);
Db_diag = [1 ./ diag(dd(1:n, n+1:end)); 1 ./ diag(dd(1:n, n+1:end))];
Db = diag(Db_diag);
S = Mm12 * Ktt * sqrtm(Db);
S = inv(S)';
end
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