nearestSPD
by
John D'Errico
30 Jul 2013
(Updated
31 Jul 2013)
Finding the nearest positive definite matrix

nearestSPD(A) 
function Ahat = nearestSPD(A)
% nearestSPD  the nearest (in Frobenius norm) Symmetric Positive Definite matrix to A
% usage: Ahat = nearestSPD(A)
%
% From Higham: "The nearest symmetric positive semidefinite matrix in the
% Frobenius norm to an arbitrary real matrix A is shown to be (B + H)/2,
% where H is the symmetric polar factor of B=(A + A')/2."
%
% http://www.sciencedirect.com/science/article/pii/0024379588902236
%
% arguments: (input)
% A  square matrix, which will be converted to the nearest Symmetric
% Positive Definite Matrix.
%
% Arguments: (output)
% Ahat  The matrix chosen as the nearest SPD matrix to A.
if nargin ~= 1
error('Exactly one argument must be provided.')
end
% test for a square matrix A
[r,c] = size(A);
if r ~= c
error('A must be a square matrix.')
elseif (r == 1) && (A <= 0)
% A was scalar and nonpositive, so just return eps
Ahat = eps;
return
end
% symmetrize A into B
B = (A + A')/2;
% Compute the symmetric polar factor of B. Call it H.
% Clearly H is itself SPD.
[U,Sigma,V] = svd(B);
H = V*Sigma*V';
% get Ahat in the above formula
Ahat = (B+H)/2;
% ensure symmetry
Ahat = (Ahat + Ahat')/2;
% test that Ahat is in fact PD. if it is not so, then tweak it just a bit.
p = 1;
k = 0;
while p ~= 0
[R,p] = chol(Ahat);
k = k + 1;
if p ~= 0
% Ahat failed the chol test. It must have been just a hair off,
% due to floating point trash, so it is simplest now just to
% tweak by adding a tiny multiple of an identity matrix.
mineig = min(eig(Ahat));
Ahat = Ahat + (mineig*k.^2 + eps(mineig))*eye(size(A));
end
end


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