`fminbnd`

Finds a local minimum of a function of one variable inside a fixed interval.

Syntax

x = fminbnd(fun, x1, x2)

Returns the point in [x1, x2] where fun reaches a local minimum.

x = fminbnd(fun, x1, x2, options)

Same, with settings created by optimset.

[x, fval, exitflag, output] = fminbnd(___)

Also returns the minimum value, a status flag and a struct with run statistics.

Description

The method combines golden section search, which always shrinks the interval by a fixed ratio, with parabolic interpolation, which converges faster when the function is smooth near the minimum. It needs no derivatives.

Only a local minimum is guaranteed, and it can lie on an end point if the function keeps decreasing toward it. To maximize, minimize the negative of the function.

Input arguments

fun: Function handle that takes a scalar and returns a scalar.
x1, x2: Finite end points of the search interval, with x1 < x2.
options: Struct from optimset; TolX (default 1e-4) and MaxIter are the options that matter.

Output arguments

x: The location of the minimum found inside the interval.
fval: Value of fun at x.
exitflag: 1 when the method converged, 0 when the iteration limit was reached.
output: Struct with the fields iterations, funcCount, algorithm and message.

Examples

Minimum of a parabola

[x, fval, flag] = fminbnd(@(x) (x-2)^2 + 1, 0, 5);
fprintf('x = %.4f, f = %.4f, flag = %d\n', x, fval, flag)

Output

x = 2.0000, f = 1.0000, flag = 1

Trough of the sine

x = fminbnd(@sin, 3, 6);
fprintf('%.4f (3*pi/2 = %.4f)\n', x, 3*pi/2)