ShaneLogic/chebfun-root-finding-extrema

Chebfun Root Finding and Extrema

Global Root Finding

Find all zeros of a chebfun in its region of definition.

Basic Usage

roots_vector = roots(f)

Algorithm

• Uses method due to Boyd and Battles
• For chebfun degree > 50, recursively breaks into smaller pieces
• On each piece, zeros found as eigenvalues of colleague matrix (analogue of companion matrix for Chebyshev polynomials)

Handling Discontinuities

Default behavior: Includes roots at jumps for piecewise functions.

Omit jump roots:

genuine_roots = roots(f, 'nojump')

Applications

Find intersections of curves:

g = chebfun('cos(x)')
intersection_points = roots(f - g)

Local Extrema Finding

Find local minima and maxima without explicitly computing derivatives.

Basic Usage

% Find local minima
[min_points, min_values] = min(f, 'local')

% Find local maxima
[max_points, max_values] = max(f, 'local')

Return Values

The command returns:

• Interior local extrema points
• Endpoints of the function domain

Smooth Functions

Extrema located by finding zeros of the derivative.

Non-smooth Functions

Non-smooth extrema identified at points where:

• Derivative "jumps from one sign to the other"
• Corresponds to "zeros" of the derivative at discontinuities

Example

f = chebfun('sin(x) + 0.5*cos(3*x)', [0, 2*pi])
[local_mins, min_vals] = min(f, 'local')
[local_maxs, max_vals] = max(f, 'local')
plot(f)
hold on
plot(local_mins, min_vals, 'ro')
plot(local_maxs, max_vals, 'go')

When to Use

• roots(f): Find all zeros of a function in its domain
• roots(f, 'nojump'): Find only genuine roots, excluding jump discontinuities
• min/max(f, 'local'): Find local extrema without derivative computation
• Curve intersections: Compute roots of f - g