function [fnoise,level,inform] = ECNoise(nf,fval)
%
% Determines the noise of a function from the function values
%
% [fnoise,level,inform] = ECnoise(nf,fval)
%
% The user must provide the function value at nf equally-spaced points.
% For example, if nf = 7, the user could provide
%
% f(x-3h), f(x-2h), f(x-h), f(x), f(x+h), f(x+2h), f(x+3h)
%
% in the array fval. Although nf >= 4 is allowed, the use of at least
% nf = 7 function evaluations is recommended.
%
% Noise will not be detected by this code if the function values differ
% in the first digit.
%
% If noise is not detected, the user should increase or decrease the
% spacing h according to the ouput value of inform. In most cases,
% the subroutine detects noise with the initial value of h.
%
% On exit:
% fnoise is set to an estimate of the function noise;
% fnoise is set to zero if noise is not detected.
%
% level is set to estimates for the noise. The k-th entry is an
% estimate from the k-th difference.
%
% inform is set as follows:
% inform = 1 Noise has been detected.
% inform = 2 Noise has not been detected; h is too small.
% Try 100*h for the next value of h.
% inform = 3 Noise has not been detected; h is too large.
% Try h/100 for the next value of h.
%
% Argonne National Laboratory
% Jorge More' and Stefan Wild. November 2009.
level = zeros(nf-1,1);
dsgn = zeros(nf-1,1);
fnoise = 0.0;
gamma = 1.0; % = gamma(0)
% Compute the range of function values.
fmin = min(fval); fmax = max(fval);
if (fmax-fmin)/max(abs(fmax),abs(fmin))>.1;
inform = 3; return;
end
% Construct the difference table.
for j = 1:nf-1
for i = 1:nf-j
fval(i) = fval(i+1) - fval(i);
end
% h is too small only when half the function values are equal.
if (j==1 && sum(fval(1:nf-1)==0)>=nf/2)
inform = 2; return;
end
gamma = 0.5*(j/(2*j-1))*gamma;
% Compute the estimates for the noise level.
level(j) = sqrt(gamma*mean(fval(1:nf-j).^2));
% Determine differences in sign.
emin = min(fval(1:nf-j));
emax = max(fval(1:nf-j));
if (emin*emax < 0.0); dsgn(j) = 1; end
end
% Determine the noise level.
for k = 1:nf-3
emin = min(level(k:k+2));
emax = max(level(k:k+2));
if (emax<=4*emin && dsgn(k))
fnoise = level(k);
inform = 1;
return;
end
end
% If noise not detected then h is too large.
inform = 3;
return;