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funm_atom.m: rename fun_atom to funm_atom * funm_atom.m: rename fun_atom to funm_atom.
author Antonio Pino Robles <data.script93@gmail.com>
date Fri, 29 May 2015 09:48:36 +0200
parents funm_files/fun_atom.m@e0817ccb2834
children
comparison
equal deleted inserted replaced
7:e0817ccb2834 8:a587712dcf5f
1 function [F,n_terms] = fun_atom(T,fun,tol,prnt)
2 %FUNM_ATOM Function of triangular matrix with nearly constant diagonal.
3 % [F, N_TERMS] = FUNM_ATOM(T, FUN, TOL, PRNT) evaluates function
4 % FUN at the upper triangular matrix T, where T has nearly constant
5 % diagonal. A Taylor series is used.
6 % FUN(X,K) must return the K'th derivative of
7 % the function represented by FUN evaluated at the vector X.
8 % TOL is a convergence tolerance for the Taylor series,
9 % defaulting to EPS.
10 % If PRNT ~= 0 trace information is printed.
11 % N_TERMS is the number of terms taken in the Taylor series.
12 % N_TERMS = -1 signals lack of convergence.
13
14 if nargin < 3 | isempty(tol), tol = eps; end
15 if nargin < 4, prnt = 0; end
16
17 if isequal(fun,@fun_log) % LOG is special case.
18 [F,n_terms] = logm_isst(T,prnt);
19 return
20 end
21
22 itmax = 500;
23
24 n = length(T);
25 if n == 1, F = feval(fun,T,0); n_terms = 1; return, end
26
27 lambda = sum(diag(T))/n;
28 F = eye(n)*feval(fun,lambda,0);
29 f_deriv_max = zeros(itmax+n-1,1);
30 N = T - lambda*eye(n);
31 mu = norm( (eye(n)-abs(triu(T,1)))\ones(n,1),inf );
32
33 P = N;
34 max_d = 1;
35
36 for k = 1:itmax
37
38 f = feval(fun,lambda,k);
39 F_old = F;
40 F = F + P*f;
41 rel_diff = norm(F - F_old,inf)/(tol+norm(F_old,inf));
42 if prnt
43 fprintf('%3.0f: coef = %5.0e', k, abs(f)/factorial(k));
44 fprintf(' N^k/k! = %7.1e', norm(P,inf));
45 fprintf(' rel_d = %5.0e',rel_diff);
46 fprintf(' abs_d = %5.0e',norm(F - F_old,inf));
47 end
48 P = P*N/(k+1);
49
50 if rel_diff <= tol
51
52 % Approximate the maximum of derivatives in convex set containing
53 % eigenvalues by maximum of derivatives at eigenvalues.
54 for j = max_d:k+n-1
55 f_deriv_max(j) = norm(feval(fun,diag(T),j),inf);
56 end
57 max_d = k+n;
58 omega = 0;
59 for j = 0:n-1
60 omega = max(omega,f_deriv_max(k+j)/factorial(j));
61 end
62
63 trunc = norm(P,inf)*mu*omega; % norm(F) moved to RHS to avoid / 0.
64 if prnt
65 fprintf(' [trunc,test] = [%5.0e %5.0e]', ...
66 trunc, tol*norm(F,inf))
67 end
68 if prnt == 5, trunc = 0; end % Force simple stopping test.
69 if trunc <= tol*norm(F,inf)
70 n_terms = k;
71 if prnt, fprintf('\n'), end, return
72 end
73 end
74
75 if prnt, fprintf('\n'), end
76
77 end
78 n_terms = -1;