133 lines
3.9 KiB
Matlab
133 lines
3.9 KiB
Matlab
function AA = generalized_cholesky2(A)
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%function AA = generalized_cholesky2(A)
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%
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% This procedure produces:
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%
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% y = chol(A+E), where E is a diagonal matrix with each element as small
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% as possible, and A and E are the same size. E diagonal values are
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% constrained by iteravely updated Gerschgorin bounds.
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%
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% REFERENCES:
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%
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% Jeff Gill and Gary King. 1998. "`Hessian not Invertable.' Help!"
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% manuscript in progress, Harvard University.
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%
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% Robert B. Schnabel and Elizabeth Eskow. 1990. "A New Modified Cholesky
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% Factorization," SIAM Journal of Scientific Statistical Computating,
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% 11, 6: 1136-58.
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% Copyright (C) 2003 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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n = size(A,1);
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L = zeros(n,n);
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deltaprev = 0;
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gamm = max(abs(diag(A)));
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tau = eps^(1/3);
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if min(eig(A)) > 0;
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tau = -1000000;
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end;
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norm_A = max(sum(abs(A))');
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gamm = max(abs(diag(A)));
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delta = max([eps*norm_A;eps]);
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Pprod = eye(n);
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if n > 2;
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for k = 1,n-2;
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if min((diag(A(k+1:n,k+1:n))' - A(k,k+1:n).^2/A(k,k))') < tau*gamm ...
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& min(eig(A((k+1):n,(k+1):n))) < 0;
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[tmp,dmax] = max(diag(A(k:n,k:n)));
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if A(k+dmax-1,k+dmax-1) > A(k,k);
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P = eye(n);
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Ptemp = P(k,:);
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P(k,:) = P(k+dmax-1,:);
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P(k+dmax-1,:) = Ptemp;
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A = P*A*P;
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L = P*L*P;
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Pprod = P*Pprod;
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end;
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g = zeros(n-(k-1),1);
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for i=k:n;
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if i == 1;
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sum1 = 0;
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else;
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sum1 = sum(abs(A(i,k:(i-1)))');
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end;
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if i == n;
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sum2 = 0;
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else;
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sum2 = sum(abs(A((i+1):n,i)));
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end;
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g(i-k+1) = A(i,i) - sum1 - sum2;
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end;
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[tmp,gmax] = max(g);
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if gmax ~= k;
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P = eye(n);
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Ptemp = P(k,:);
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P(k,:) = P(k+dmax-1,:);
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P(k+dmax-1,:) = Ptemp;
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A = P*A*P;
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L = P*L*P;
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Pprod = P*Pprod;
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end;
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normj = sum(abs(A(k+1:n,k)));
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delta = max([0;deltaprev;-A(k,k)+normj;-A(k,k)+tau*gamm]);
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if delta > 0;
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A(k,k) = A(k,k) + delta;
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deltaprev = delta;
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end;
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end;
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A(k,k) = sqrt(A(k,k));
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L(k,k) = A(k,k);
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for i=k+1:n;
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if L(k,k) > eps;
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A(i,k) = A(i,k)/L(k,k);
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end;
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L(i,k) = A(i,k);
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A(i,k+1:i) = A(i,k+1:i) - L(i,k)*L(k+1:i,k)';
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if A(i,i) < 0;
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A(i,i) = 0;
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end;
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end;
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end;
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end;
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A(n-1,n) = A(n,n-1);
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eigvals = eig(A(n-1:n,n-1:n));
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dlist = [ 0 ; deltaprev ;...
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-min(eigvals)+tau*max((inv(1-tau)*max(eigvals)-min(eigvals))|gamm) ];
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if dlist(1) > dlist(2);
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delta = dlist(1);
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else;
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delta = dlist(2);
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end;
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if delta < dlist(3);
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delta = dlist(3);
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end;
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if delta > 0;
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A(n-1,n-1) = A(n-1,n-1) + delta;
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A(n,n) = A(n,n) + delta;
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deltaprev = delta;
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end;
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A(n-1,n-1) = sqrt(A(n-1,n-1));
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L(n-1,n-1) = A(n-1,n-1);
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A(n,n-1) = A(n,n-1)/L(n-1,n-1);
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L(n,n-1) = A(n,n-1);
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A(n,n) = sqrt(A(n,n) - L(n,(n-1))^2);
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L(n,n) = A(n,n);
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AA = Pprod'*L'*Pprod'; |