Added a new routine to solve quadratic matrix equation (based on a Newton algorithm with line search).
Stéphane Adjemian (Charybdis)
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91499d7961
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54aff15b6a
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function X = fastgensylv(A, B, C, D, tol,maxit,X0)
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%@info:
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%! @deftypefn {Function File} {[@var{X1}, @var{info}] =} fastgensylv (@var{A},@var{B},@var{C},@var{tol},@var{maxit},@var{X0})
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%! @anchor{fastgensylv}
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%! @sp 1
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%! Solves the Sylvester equation A * X + B * X * C + D = 0 for X.
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%! @sp 2
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%! @strong{Inputs}
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%! @sp 1
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%! @table @ @var
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%! @item A
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%! Square matrix of doubles, n*n.
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%! @item B
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%! Square matrix of doubles, n*n.
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%! @item C
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%! Square matrix of doubles, n*n.
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%! @item tol
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%! Scalar double, tolerance parameter.
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%! @item maxit
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%! Integer scalar, maximum number of iterations.
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%! @item X0
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%! Square matrix of doubles, n*n, initial condition.
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%! @end table
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%! @sp 1
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%! @strong{Outputs}
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%! @sp 1
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%! @table @ @var
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%! @item X
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%! Square matrix of doubles, n*n, solution of the matrix equation.
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%! @item info
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%! Scalar integer, if nonzero the algorithm failed in finding the solution of the matrix equation.
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%! @end table
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%! @sp 2
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%! @strong{This function is called by:}
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%! @sp 2
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%! @strong{This function calls:}
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%! @sp 2
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%! @end deftypefn
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%@eod:
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% Copyright (C) 2012 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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if size(A,1)~=size(D,1) || size(A,1)~=size(B,1) || size(C,2)~=size(D,2)
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error('fastgensylv:: Dimension error!')
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end
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if nargin<7 || isempty(X0)
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X = zeros(size(A,2),size(C,1));
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elseif nargin==7 && ~isempty(X0)
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X = X0;
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end
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kk = 0;
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cc = 1+tol;
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iA = inv(A);
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Z = - (B * X * C + D);
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while kk<=maxit && cc>tol
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X = iA * Z;
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Z_old = Z;
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Z = - (B * X * C + D);
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cc = max(sum(abs(Z-Z_old)));
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kk = kk + 1;
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end
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if kk==maxit && cc>tol
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error(['fastgensylv:: Convergence not achieved in fixed point solution of Sylvester equation after ' int2str(maxit) ' iterations']);
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end
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% function X = fastgensylv(A, B, C, D)
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% Solve the Sylvester equation:
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% A * X + B * X * C + D = 0
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% INPUTS
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% A
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% B
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% C
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% D
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% block : block number (for storage purpose)
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% tol : convergence criteria
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% OUTPUTS
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% X solution
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%
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% ALGORITHM
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% fixed point method
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% MARLLINY MONSALVE (2008): "Block linear method for large scale
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% Sylvester equations", Computational & Applied Mathematics, Vol 27, n°1,
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% p47-59
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% ||A^-1||.||B||.||C|| < 1 is a suffisant condition:
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% - to get a unique solution for the Sylvester equation
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% - to get a convergent fixed-point algorithm
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%
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% SPECIAL REQUIREMENTS
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% none.
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% Copyright (C) 1996-2012 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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@ -0,0 +1,158 @@
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function [X,info] = quadratic_matrix_equation_solver(A,B,C,tol,maxit,line_search_flag,X)
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%@info:
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%! @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})
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%! @anchor{logarithmic_reduction}
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%! @sp 1
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%! Solves the quadratic matrix equation AX^2 + BX + C = 0 with a Newton algorithm.
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%! @sp 2
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%! @strong{Inputs}
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%! @sp 1
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%! @table @ @var
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%! @item A
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%! Square matrix of doubles, n*n.
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%! @item B
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%! Square matrix of doubles, n*n.
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%! @item C
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%! Square matrix of doubles, n*n.
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%! @item tol
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%! Scalar double, tolerance parameter.
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%! @item maxit
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%! Scalar integer, maximum number of iterations.
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%! @item line_search_flag
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%! Scalar integer, if nonzero an exact line search algorithm is used.
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%! @item X
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%! Square matrix of doubles, n*n, initial condition.
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%! @end table
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%! @sp 1
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%! @strong{Outputs}
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%! @sp 1
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%! @table @ @var
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%! @item X
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%! Square matrix of doubles, n*n, solution of the matrix equation.
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%! @item info
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%! Scalar integer, if nonzero the algorithm failed in finding the solution of the matrix equation.
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%! @end table
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%! @sp 2
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%! @strong{This function is called by:}
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%! @sp 2
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%! @strong{This function calls:}
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%! @sp 1
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%! @ref{fastgensylv}
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%! @sp 2
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%! @strong{References:}
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%! @sp 1
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%! 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.
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%! @sp 2
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%! @end deftypefn
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%@eod:
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% Copyright (C) 2012 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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provide_initial_condition_to_fastgensylv = 0;
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info = 0;
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F = eval_quadratic_matrix_equation(A,B,C,X);
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if max(max(abs(F)))<tol
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return
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end
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kk = 0.0;
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cc = 1+tol;
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step_length = 1.0;
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while kk<maxit && cc>tol
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if provide_initial_condition_to_fastgensylv && exist('H','var')
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H = fastgensylv(A*X+B,A,X,F,tol,maxit,H);
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else
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try
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H = fastgensylv(A*X+B,A,X,F,tol,maxit);
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catch
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X = zeros(length(X));
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H = fastgensylv(A*X+B,A,X,F,tol,maxit);
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end
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end
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if line_search_flag
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step_length = line_search(A,H,F);
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end
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X = X + step_length*H;
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F = eval_quadratic_matrix_equation(A,B,C,X);
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cc = max(max(abs(F)));
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kk = kk +1;
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end
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if cc>tol
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X = NaN(size(X));
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info = 1;
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end
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function f = eval_quadratic_matrix_equation(A,B,C,X)
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f = C + (B + A*X)*X;
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function [p0,p1] = merit_polynomial(A,H,F)
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AHH = A*H*H;
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gamma = norm(AHH,'fro')^2;
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alpha = norm(F,'fro')^2;
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beta = trace(F*AHH*AHH*F);
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p0 = [gamma, -beta, alpha+beta, -2*alpha, alpha];
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p1 = [4*gamma, -3*beta, 2*(alpha+beta), -2*alpha];
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function t = line_search(A,H,F)
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[p0,p1] = merit_polynomial(A,H,F);
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if any(isnan(p0)) || any(isinf(p0))
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t = 1.0;
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return
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end
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r = roots(p1);
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s = [Inf(3,1),r];
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for i = 1:3
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if isreal(r(i))
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s(i,1) = p0(1)*r(i)^4 + p0(2)*r(i)^3 + p0(3)*r(i)^2 + p0(4)*r(i) + p0(5);
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end
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end
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s = sortrows(s,1);
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t = s(1,2);
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if t<=1e-12 || t>=2
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t = 1;
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end
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%@test:1
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%$ addpath ../matlab
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%$
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%$ % Set the dimension of the problem to be solved
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%$ n = 200;
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%$ % Set the equation to be solved
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%$ A = eye(n);
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%$ 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);
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%$ C = diag(15*ones(n,1)); C = C - diag(5*ones(n-1,1),-1); C = C - diag(5*ones(n-1,1),1);
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%$
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%$ % Solve the equation with the cycle reduction algorithm
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%$ tic, X1 = cycle_reduction(C,B,A,1e-7); toc
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%$
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%$ % Solve the equation with the logarithmic reduction algorithm
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%$ tic, X2 = quadratic_matrix_equation_solver(A,B,C,1e-16,100,1,zeros(n)); toc
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%$
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%$ % Check the results.
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%$ t(1) = dyn_assert(X1,X2,1e-12);
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%$
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%$ T = all(t);
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%@eof:1
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