### Added a new routine to solve quadratic matrix equation (based on a Newton...

`Added a new routine to solve quadratic matrix equation (based on a Newton algorithm with line search).`
parent 91499d79
 function X = fastgensylv(A, B, C, D, tol,maxit,X0) %@info: %! @deftypefn {Function File} {[@var{X1}, @var{info}] =} fastgensylv (@var{A},@var{B},@var{C},@var{tol},@var{maxit},@var{X0}) %! @anchor{fastgensylv} %! @sp 1 %! Solves the Sylvester equation A * X + B * X * C + D = 0 for X. %! @sp 2 %! @strong{Inputs} %! @sp 1 %! @table @ @var %! @item A %! Square matrix of doubles, n*n. %! @item B %! Square matrix of doubles, n*n. %! @item C %! Square matrix of doubles, n*n. %! @item tol %! Scalar double, tolerance parameter. %! @item maxit %! Integer scalar, maximum number of iterations. %! @item X0 %! Square matrix of doubles, n*n, initial condition. %! @end table %! @sp 1 %! @strong{Outputs} %! @sp 1 %! @table @ @var %! @item X %! Square matrix of doubles, n*n, solution of the matrix equation. %! @item info %! Scalar integer, if nonzero the algorithm failed in finding the solution of the matrix equation. %! @end table %! @sp 2 %! @strong{This function is called by:} %! @sp 2 %! @strong{This function calls:} %! @sp 2 %! @end deftypefn %@eod: % Copyright (C) 2012 Dynare Team % % This file is part of Dynare. % % Dynare is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % % Dynare is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with Dynare. If not, see . if size(A,1)~=size(D,1) || size(A,1)~=size(B,1) || size(C,2)~=size(D,2) error('fastgensylv:: Dimension error!') end if nargin<7 || isempty(X0) X = zeros(size(A,2),size(C,1)); elseif nargin==7 && ~isempty(X0) X = X0; end kk = 0; cc = 1+tol; iA = inv(A); Z = - (B * X * C + D); while kk<=maxit && cc>tol X = iA * Z; Z_old = Z; Z = - (B * X * C + D); cc = max(sum(abs(Z-Z_old))); kk = kk + 1; end if kk==maxit && cc>tol error(['fastgensylv:: Convergence not achieved in fixed point solution of Sylvester equation after ' int2str(maxit) ' iterations']); end % function X = fastgensylv(A, B, C, D) % Solve the Sylvester equation: % A * X + B * X * C + D = 0 % INPUTS % A % B % C % D % block : block number (for storage purpose) % tol : convergence criteria % OUTPUTS % X solution % % ALGORITHM % fixed point method % MARLLINY MONSALVE (2008): "Block linear method for large scale % Sylvester equations", Computational & Applied Mathematics, Vol 27, n°1, % p47-59 % ||A^-1||.||B||.||C|| < 1 is a suffisant condition: % - to get a unique solution for the Sylvester equation % - to get a convergent fixed-point algorithm % % SPECIAL REQUIREMENTS % none. % Copyright (C) 1996-2012 Dynare Team % % This file is part of Dynare. % % Dynare is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % % Dynare is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with Dynare. If not, see .
 function [X,info] = quadratic_matrix_equation_solver(A,B,C,tol,maxit,line_search_flag,X) %@info: %! @deftypefn {Function File} {[@var{X1}, @var{info}] =} quadratic_matrix_equation_solver (@var{A},@var{B},@var{C},@var{tol},@var{maxit},@var{line_search_flag},@var{X0}) %! @anchor{logarithmic_reduction} %! @sp 1 %! Solves the quadratic matrix equation AX^2 + BX + C = 0 with a Newton algorithm. %! @sp 2 %! @strong{Inputs} %! @sp 1 %! @table @ @var %! @item A %! Square matrix of doubles, n*n. %! @item B %! Square matrix of doubles, n*n. %! @item C %! Square matrix of doubles, n*n. %! @item tol %! Scalar double, tolerance parameter. %! @item maxit %! Scalar integer, maximum number of iterations. %! @item line_search_flag %! Scalar integer, if nonzero an exact line search algorithm is used. %! @item X %! Square matrix of doubles, n*n, initial condition. %! @end table %! @sp 1 %! @strong{Outputs} %! @sp 1 %! @table @ @var %! @item X %! Square matrix of doubles, n*n, solution of the matrix equation. %! @item info %! Scalar integer, if nonzero the algorithm failed in finding the solution of the matrix equation. %! @end table %! @sp 2 %! @strong{This function is called by:} %! @sp 2 %! @strong{This function calls:} %! @sp 1 %! @ref{fastgensylv} %! @sp 2 %! @strong{References:} %! @sp 1 %! N.J. Higham and H.-M. Kim (2001), "Solving a quadratic matrix equation by Newton's method with exact line searches.", in SIAM J. Matrix Anal. Appl., Vol. 23, No. 3, pp. 303-316. %! @sp 2 %! @end deftypefn %@eod: % Copyright (C) 2012 Dynare Team % % This file is part of Dynare. % % Dynare is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % % Dynare is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with Dynare. If not, see . provide_initial_condition_to_fastgensylv = 0; info = 0; F = eval_quadratic_matrix_equation(A,B,C,X); if max(max(abs(F)))tol if provide_initial_condition_to_fastgensylv && exist('H','var') H = fastgensylv(A*X+B,A,X,F,tol,maxit,H); else try H = fastgensylv(A*X+B,A,X,F,tol,maxit); catch X = zeros(length(X)); H = fastgensylv(A*X+B,A,X,F,tol,maxit); end end if line_search_flag step_length = line_search(A,H,F); end X = X + step_length*H; F = eval_quadratic_matrix_equation(A,B,C,X); cc = max(max(abs(F))); kk = kk +1; end if cc>tol X = NaN(size(X)); info = 1; end function f = eval_quadratic_matrix_equation(A,B,C,X) f = C + (B + A*X)*X; function [p0,p1] = merit_polynomial(A,H,F) AHH = A*H*H; gamma = norm(AHH,'fro')^2; alpha = norm(F,'fro')^2; beta = trace(F*AHH*AHH*F); p0 = [gamma, -beta, alpha+beta, -2*alpha, alpha]; p1 = [4*gamma, -3*beta, 2*(alpha+beta), -2*alpha]; function t = line_search(A,H,F) [p0,p1] = merit_polynomial(A,H,F); if any(isnan(p0)) || any(isinf(p0)) t = 1.0; return end r = roots(p1); s = [Inf(3,1),r]; for i = 1:3 if isreal(r(i)) s(i,1) = p0(1)*r(i)^4 + p0(2)*r(i)^3 + p0(3)*r(i)^2 + p0(4)*r(i) + p0(5); end end s = sortrows(s,1); t = s(1,2); if t<=1e-12 || t>=2 t = 1; end %@test:1 %\$ addpath ../matlab %\$ %\$ % Set the dimension of the problem to be solved %\$ n = 200; %\$ % Set the equation to be solved %\$ A = eye(n); %\$ B = diag(30*ones(n,1)); B(1,1) = 20; B(end,end) = 20; B = B - diag(10*ones(n-1,1),-1); B = B - diag(10*ones(n-1,1),1); %\$ C = diag(15*ones(n,1)); C = C - diag(5*ones(n-1,1),-1); C = C - diag(5*ones(n-1,1),1); %\$ %\$ % Solve the equation with the cycle reduction algorithm %\$ tic, X1 = cycle_reduction(C,B,A,1e-7); toc %\$ %\$ % Solve the equation with the logarithmic reduction algorithm %\$ tic, X2 = quadratic_matrix_equation_solver(A,B,C,1e-16,100,1,zeros(n)); toc %\$ %\$ % Check the results. %\$ t(1) = dyn_assert(X1,X2,1e-12); %\$ %\$ T = all(t); %@eof:1 \ No newline at end of file
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