echo off;
% NUMERICAL METHODS: MATLAB Programs, (c) John H. Mathews 1994
% To accompany the text:
% NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
% Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.
% This free software is complements of the author.

% Algorithm 8.3 (Local Minimum Search: Quadratic Interpolation).
% Section 8.1, Minimization of a Function, Page 416
echo on; clc; format long; hold off; clear

% This program finds the minimum of a function f(x).

% The method is quadratic interpolation.

% It is assumed that f(x) is unimodal over [a,b].

% The function f(x) is to be placed in the M-file f.m
% function z = f(x)
% z = x.^2 - sin(x);

delete f.m
diary f.m; disp('function z = f(x)');...
disp('z = x.^2 - sin(x);');...
diary off;

f(0); % Remark. f.m and quadmin.m are used for Algorithm 8.3

pause % Press any key to continue.

clc;

% Place the endpoints interval [a,b] in a and b.

a = 0;

b = 1;

pause % Press any key to graph the function.

clc; clg;
hs = (b-a)/150;
Xs = a:hs:b;
Ys = f(Xs); % User defined function.
axis([-0.05 1.05 -0.25 0.16]);...
plot(Xs,Ys,'g');...
hold on;...
plot([-0.05 1.05],[0 0],'b',[0 0],[-0.25 0.16],'b');...
xlabel('x');...
ylabel('y');...
title('To search for the minimum of y = f(x).');...
grid;...
axis;...
hold off;...
shg; pause % Press any key to continue.

clc;

% Place the endpoints interval [a,b] in a and b.

% Place the tolerance for the abscissas in delta.

% Place the tolerance for the ordinates in epsilon.

a = 0;

b = 1;

delta = 1e-14;

epsilon = 1E-15;

[p,yp,dp,dy,P] = quadmin('f',a,b,delta,epsilon);

pause % Press any key to find the minimum of f(x).

clc; clg;
hs = (b-a)/150;
Xs = a:hs:b;
Ys = f(Xs);
YP = f(P);
axis([a b -0.25 0.175]);...
plot(Xs,Ys,'g',P,YP,'or');...
hold on;...
plot([a b],[0 0],'b',[0 0],[-0.25 0.175],'b');...
xlabel('x');...
ylabel('y');...
title('Searching for the minimum of y = f(x).');...
grid;...
axis;...
hold off;...
shg; pause % Press any key to continue.

Mx1 = 'The results for a quadratic minimization search.';
Mx2 = 'The solution is [p f(p)];';
Mx3 = 'The abscissa p ';
Mx4 = 'error bound for p';
Mx5 = 'The minimum value f(p)';
Mx6 = 'error bound for f(p)';
Mx7 = 'The list of iterations is:';
Mx8 = ' p(k) f(p(k))';
clc,echo off,diary output,...
disp(''),disp(Mx1),disp(''),disp(Mx2),disp(''),...
disp(Mx3),disp(p),disp(Mx4),disp(['± ',num2str(abs(dp))]),disp(''),...
disp(Mx5),disp(yp),disp(Mx4),disp(['± ',num2str(abs(dy))]),...
disp(''),disp(Mx7),disp(Mx8),disp([P;YP]'),diary off,echo on