function oppgave_1_2_6
f =  @(x) exp(x-2)+x.^3 - x;

% fixed point funciton and its derivative
g1 = @(x) exp(x-2)+x.^3;
dg1 = @(x) exp(x-2) + 3*x.^2;

% fixed point funciton and its derivative
g2 = @(x) (x-exp(x-2)).^(1/3);
dg2 = @(x) 1/3*(x-exp(x-2)).^(-2/3).*(1-exp(x-2));

% fixed point funciton and its derivative
% this one is derived from exp(x-2) + x*(x-1)*(x+1) = 0
g3 = @(x) -1 -exp(x-2)./(x.^2-x);
dg3 = @(x) -(exp(x-2).*(x.^2-x)-exp(x-2).*(2.*x-1))./(x.^2-x).^2;

% plot the function to visually find the roots
X = linspace(-1.5,2,100);
figure(1);
plot(X,f(X)); grid on;
xlabel('x'); ylabel('f(x)');

figure(2);
plot(X,X,X,g1(X)); grid on;
xlabel('x'); ylabel('g1(x)');
legend('x','g1(x)');

figure(3);
plot(X,X,X,g2(X)); grid on;
xlabel('x'); ylabel('g2(x)');
legend('x','g2(x)');

figure(4);
plot(X,X,X,g3(X)); grid on;
xlabel('x'); ylabel('g3(x)');
legend('x','g3(x)');


fprintf('\n\n\nTrying g1\n\n\n');

run_fpi(f,g1,dg1);


fprintf('\n\n\nTrying g2\n\n\n');

run_fpi(f,g2,dg2);


fprintf('\n\n\nTrying g3\n\n\n');

run_fpi(f,g3,dg3);

function run_fpi(f,g,dg)
% try fpi from different starting points
x = 0.75;
x = fpi(f,g,x);
fprintf('x = %e, f(x) = %e, dg(x)=%e\n', x,f(x),dg(x));

x = 0.1;
x = fpi(f,g,x);
fprintf('x = %e, f(x) = %e, dg(x)=%e\n', x,f(x),dg(x));

x = -1.5;
x = fpi(f,g,x);
fprintf('x = %e, f(x) = %e, dg(x)=%e\n', x,f(x),dg(x));


function x=fpi(f,g,x)
e0 = f(x);
for i = 1:100,
    x1=g(x);
    if ~isfinite(x1)
        break;
    end
    e1 = abs(f(x1));
    %if e1 < 1.0E-10,
    %    break;
    %end
    if abs(x1-x)/max(abs(x),1.0) < 1.0E-06,
        break;
    end
    fprintf('e1/e0 = %e\n', e1/e0);
    x  = x1;
    e0 = e1;
end