1function [X,info] = quadratic_matrix_equation_solver(A,B,C,tol,maxit,line_search_flag,X)
2
3%@info:
4%! @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})
5%! @anchor{logarithmic_reduction}
6%! @sp 1
7%! Solves the quadratic matrix equation AX^2 + BX + C = 0 with a Newton algorithm.
8%! @sp 2
9%! @strong{Inputs}
10%! @sp 1
11%! @table @ @var
12%! @item A
13%! Square matrix of doubles, n*n.
14%! @item B
15%! Square matrix of doubles, n*n.
16%! @item C
17%! Square matrix of doubles, n*n.
18%! @item tol
19%! Scalar double, tolerance parameter.
20%! @item maxit
21%! Scalar integer, maximum number of iterations.
22%! @item line_search_flag
23%! Scalar integer, if nonzero an exact line search algorithm is used.
24%! @item X
25%! Square matrix of doubles, n*n, initial condition.
26%! @end table
27%! @sp 1
28%! @strong{Outputs}
29%! @sp 1
30%! @table @ @var
31%! @item X
32%! Square matrix of doubles, n*n, solution of the matrix equation.
33%! @item info
34%! Scalar integer, if nonzero the algorithm failed in finding the solution of the matrix equation.
35%! @end table
36%! @sp 2
37%! @strong{This function is called by:}
38%! @sp 2
39%! @strong{This function calls:}
40%! @sp 1
41%! @ref{fastgensylv}
42%! @sp 2
43%! @strong{References:}
44%! @sp 1
45%! 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.
46%! @sp 2
47%! @end deftypefn
48%@eod:
49
50% Copyright (C) 2012-2017 Dynare Team
51%
52% This file is part of Dynare.
53%
54% Dynare is free software: you can redistribute it and/or modify
55% it under the terms of the GNU General Public License as published by
56% the Free Software Foundation, either version 3 of the License, or
57% (at your option) any later version.
58%
59% Dynare is distributed in the hope that it will be useful,
60% but WITHOUT ANY WARRANTY; without even the implied warranty of
61% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
62% GNU General Public License for more details.
63%
64% You should have received a copy of the GNU General Public License
65% along with Dynare.  If not, see <http://www.gnu.org/licenses/>.
66
67provide_initial_condition_to_fastgensylv = 0;
68
69info = 0;
70
71F = eval_quadratic_matrix_equation(A,B,C,X);
72
73if max(max(abs(F)))<tol
74    return
75end
76
77kk = 0.0;
78cc = 1+tol;
79
80step_length = 1.0;
81
82while kk<maxit && cc>tol
83    if provide_initial_condition_to_fastgensylv && exist('H','var')
84        H = fastgensylv(A*X+B,A,X,F,tol,maxit,H);
85    else
86        try
87            H = fastgensylv(A*X+B,A,X,F,tol,maxit);
88        catch
89            X = zeros(length(X));
90            H = fastgensylv(A*X+B,A,X,F,tol,maxit);
91        end
92    end
93    if line_search_flag
94        step_length = line_search(A,H,F);
95    end
96    X = X + step_length*H;
97    F = eval_quadratic_matrix_equation(A,B,C,X);
98    cc = max(max(abs(F)));
99    kk = kk +1;
100end
101
102if cc>tol
103    X = NaN(size(X));
104    info = 1;
105end
106
107
108function f = eval_quadratic_matrix_equation(A,B,C,X)
109f = C + (B + A*X)*X;
110
111function [p0,p1] = merit_polynomial(A,H,F)
112AHH = A*H*H;
113gamma = norm(AHH,'fro')^2;
114alpha = norm(F,'fro')^2;
115beta  = trace(F*AHH*AHH*F);
116p0 = [gamma, -beta, alpha+beta, -2*alpha, alpha];
117p1 = [4*gamma, -3*beta, 2*(alpha+beta), -2*alpha];
118
119function t = line_search(A,H,F)
120[p0,p1] = merit_polynomial(A,H,F);
121if any(isnan(p0)) || any(isinf(p0))
122    t = 1.0;
123    return
124end
125r = roots(p1);
126s = [Inf(3,1),r];
127for i = 1:3
128    if isreal(r(i))
129        s(i,1) = p0(1)*r(i)^4 + p0(2)*r(i)^3 + p0(3)*r(i)^2 + p0(4)*r(i) + p0(5);
130    end
131end
132s = sortrows(s,1);
133t = s(1,2);
134if t<=1e-12 || t>=2
135    t = 1;
136end
137
138%@test:1
139%$ addpath ../matlab
140%$
141%$ % Set the dimension of the problem to be solved
142%$ n = 200;
143%$ % Set the equation to be solved
144%$ A = eye(n);
145%$ 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);
146%$ C = diag(15*ones(n,1)); C = C - diag(5*ones(n-1,1),-1); C = C - diag(5*ones(n-1,1),1);
147%$
148%$ % Solve the equation with the cycle reduction algorithm
149%$ tic, X1 = cycle_reduction(C,B,A,1e-7); toc
150%$
151%$ % Solve the equation with the logarithmic reduction algorithm
152%$ tic, X2 = quadratic_matrix_equation_solver(A,B,C,1e-16,100,1,zeros(n)); toc
153%$
154%$ % Check the results.
155%$ t(1) = dassert(X1,X2,1e-12);
156%$
157%$ T = all(t);
158%@eof:1