%% Problem 1.6.2
% Script: Prob1_6_2.m for Math 426/Cisc 410
% R.J. Braun
%%
format compact;  % get rid of white space
%% Part (a)
%
% The condition number is f''(x) = 1/sqrt(x^2-1).
% The conditioning appears to be fine as abs(x) increases. 

%% Part (b)
%
format long;
t = -4:-4:-16;
x = cosh(t);
tt = log(x-sqrt(x.^2 - 1));
errorb = abs(t-tt);
%
disp(' g(x) = log(x-sqrt(x.^2-1)) ')
disp(' t          x                g(x)           error  ')
disp('----------------------------------------------------')
for k = 1:4
      disp(sprintf('%3.0f    %7.4e    %16.12g  %8.5e',t(k),x(k),tt(k),errorb(k)))
end

%%  Part (c)
%
ttt = -2*log( sqrt((x+1)/2) + sqrt((x-1)/2) );
errorc = abs(t-ttt);
disp(' ')
disp(' h(x) = -2 log( sqrt((x+1)/2) + sqrt((x-1)/2) ) ')
disp(' t          x                h(x)           error  ')
disp('----------------------------------------------------')
for k = 1:4
      disp(sprintf('%3.0f    %7.4e    %16.12g  %8.5e',t(k),x(k),ttt(k),errorc(k)))
end

%% Part (d) 
% The formula of part (b) is unstable; the formula of part (d) is stable. 
% The argument of the log contains a difference of two closely spaced
% numbers causing subtractive cancelation! Part (c) avoids this.