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