Find minimum in one variable only

I have a complicated two-variable function f(x,t), for 0 < x, t < 1.

My goal is to estimate the integral from 0 to 1 of min_{0<t<1} |f(x,t)|.

I have tried using fminbnd, but I'm having problems since the input is a function of two variables.

My initial definition of f looked like

syms x t

f = cos(x)*exp(t^2);  % actual function is much more complicated than this.

Note that I do not expect to be able to locate extrema analytically.

The best I can do so far is compute the minimum at a number of x-values, like this:

d = []; % vector to hold minima
N = 100; % number of x-values to check

for j = 0 : N
    f = @(t)cos(j/N)*exp(t^2); 
    d = [d, [fminbnd(f,0,1)]];
end;

I can then use the vector d to approximate the integral I want. However, this seems slow and inefficient. Any suggestions would be welcome.

Answer 1

[Rik]

You can use a trick to make it a single input function: use a vector.

f = @(x,t)cos(x)*exp(t^2); 
B=fminbnd(@(b)f(b(1),b(2)),[0;1],[0;1]);

I use this trick often with fminsearch if I want to fit a function, but don't know if the end user has the curve fitting toolbox:

%needed input: x and yx

% Objective function (Exponential)
y = @(b,x) b(1)*exp(b(2)*x);

% Ordinary Least Squares cost function
OLS = @(b) sum((y(b,x) - yx).^2);
opts = optimset('MaxFunEvals',50000, 'MaxIter',10000);
% Use 'fminsearch' to minimise the 'OLS' function
fit_output = fminsearch(OLS, intial_b_vals, opts);