ShaneLogic/chebfun-analysis

Chebfun Analysis Operations

Global Extrema

Finding Minimum and Maximum Values

% Create chebfun
f = chebfun(@(x) sin(x) + 0.1*x.^2, [-5, 5]);

% Get global minimum value
min_val = min(f);

% Get global maximum value  
max_val = max(f);

Finding Both Value and Location

% Get minimum value and its location
[min_val, min_pos] = min(f);

% Get maximum value and its location
[max_val, max_pos] = max(f);

How It Works

Chebfun computes extrema by:

1. Evaluating f at all interval endpoints
2. Finding all zeros of the derivative f'
3. Evaluating f at these critical points
4. Selecting the global minimum/maximum

Important Notes

• Only one position is returned even if extrema occur at multiple points
• Computing min(f) and max(f) separately is inefficient for large chebfuns
• Each call independently computes derivative and finds its zeros

Efficiency Tip

For large chebfuns, use combined computation:

% Faster alternative (if available)
[min_val, max_val, min_pos, max_pos] = minandmax(f);

Vector Norms

1-Norm (Integral of Absolute Value)

% Compute 1-norm
one_norm = norm(f, 1);

• Definition: ||f||_1 = ∫ |f(x)| dx
• Chebfun computes by summing segments between zeros of f
• At zeros of f, |f(x)| typically has discontinuous slope

Infinity-Norm (Maximum Absolute Value)

% Compute infinity-norm
inf_norm = norm(f, inf);

• Definition: ||f||_inf = max |f(x)|
• Computed as: max(max(f), -min(f))

Comparison with Other Norms

% For single chebfuns, 2-norm equals Frobenius norm
two_norm = norm(f, 2);
fro_norm = norm(f, 'fro');
% two_norm == fro_norm for single chebfuns

Note: For quasimatrices, 2-norm and Frobenius norm differ.

When to Use

• Finding global bounds of a function
• Locating extreme points of a function
• Computing function magnitude measures
• Analyzing function size for error estimation
• Comparing function sizes in optimization