--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/funm_files/funm_atom.m	Fri May 29 09:48:36 2015 +0200
@@ -0,0 +1,78 @@
+function [F,n_terms] = fun_atom(T,fun,tol,prnt)
+%FUNM_ATOM  Function of triangular matrix with nearly constant diagonal.
+%           [F, N_TERMS] = FUNM_ATOM(T, FUN, TOL, PRNT) evaluates function
+%           FUN at the upper triangular matrix T, where T has nearly constant
+%           diagonal.  A Taylor series is used.
+%           FUN(X,K) must return the K'th derivative of
+%           the function represented by FUN evaluated at the vector X.
+%           TOL is a convergence tolerance for the Taylor series,
+%           defaulting to EPS.
+%           If PRNT ~= 0 trace information is printed.
+%           N_TERMS is the number of terms taken in the Taylor series.
+%           N_TERMS  = -1 signals lack of convergence.
+
+if nargin < 3 | isempty(tol), tol = eps; end
+if nargin < 4, prnt = 0; end
+
+if isequal(fun,@fun_log)   % LOG is special case.
+   [F,n_terms]  = logm_isst(T,prnt);
+   return
+end
+
+itmax = 500;
+
+n = length(T);
+if n == 1, F = feval(fun,T,0); n_terms = 1; return, end
+
+lambda = sum(diag(T))/n;
+F = eye(n)*feval(fun,lambda,0);
+f_deriv_max = zeros(itmax+n-1,1);
+N = T - lambda*eye(n);
+mu = norm( (eye(n)-abs(triu(T,1)))\ones(n,1),inf );
+
+P = N;
+max_d = 1;
+
+for k = 1:itmax
+
+    f = feval(fun,lambda,k);
+    F_old = F;
+    F = F + P*f;
+    rel_diff = norm(F - F_old,inf)/(tol+norm(F_old,inf));
+    if prnt
+        fprintf('%3.0f: coef = %5.0e', k, abs(f)/factorial(k));
+        fprintf('  N^k/k! = %7.1e', norm(P,inf));
+        fprintf('  rel_d = %5.0e',rel_diff);
+        fprintf('  abs_d = %5.0e',norm(F - F_old,inf));
+    end
+    P = P*N/(k+1);
+
+    if rel_diff <= tol
+
+      % Approximate the maximum of derivatives in convex set containing
+      % eigenvalues by maximum of derivatives at eigenvalues.
+      for j = max_d:k+n-1
+          f_deriv_max(j) = norm(feval(fun,diag(T),j),inf);
+      end
+      max_d = k+n;
+      omega = 0;
+      for j = 0:n-1
+          omega = max(omega,f_deriv_max(k+j)/factorial(j));
+      end
+
+      trunc = norm(P,inf)*mu*omega;  % norm(F) moved to RHS to avoid / 0.
+      if prnt
+          fprintf('  [trunc,test] = [%5.0e %5.0e]', ...
+                   trunc, tol*norm(F,inf))
+      end
+      if prnt == 5, trunc = 0; end % Force simple stopping test.
+      if trunc <= tol*norm(F,inf)
+         n_terms = k;
+         if prnt, fprintf('\n'), end, return
+      end
+    end
+
+    if prnt, fprintf('\n'), end
+
+end
+n_terms = -1;