ShaneLogic/chebfun-integration-quadrature

Chebfun Integration and Quadrature

Definite Integration via sum Command

Use sum(f) to compute the definite integral of a chebfun over its domain.

Basic Usage

integral_value = sum(f)

Algorithm

• Uses FFT-based version of Clenshaw-Curtis quadrature
• Applied to each smooth piece (fun) of the chebfun
• Results summed across all pieces

Handling Special Cases

Piecewise smooth functions:

f = chebfun('abs(x)', [-1, 1], 'splitting', 'on')
integral_value = sum(f)

Functions with narrow spikes:

% Use minSamples flag to avoid missing narrow features
f = chebfun(@(x) spike_function(x), 'minSamples', 1000)
integral_value = sum(f)

Infinite intervals:

f = chebfun(@(x) 1./x.^4, [1, inf])
integral_value = sum(f)  % May lose several digits of accuracy

Divergent functions:

f = chebfun(@(x) 1./x.^0.9, [-1, 1], 'exps')
integral_value = sum(f)

2D Integration over Rectangles

Basic Method

% Construct chebfun with vectorize flag for 2D function
f = chebfun(@(x,y) function_handle(x,y), 'vectorize')

% Compute double integral
result = sum(sum(f))

Recommendation

For better performance and plotting capabilities, use Chebfun2 instead of nested 1D chebfuns.

Specialized Quadrature Rules

Legendre Quadrature

% Get quadrature nodes and weights
[s, w] = legpts(n)

% Compute integral
I = w * f(s)

Chebyshev Quadrature

% Get Chebyshev points
x = chebpts(n)

% Compute integral using built-in sum
I = sum(f)

Jacobi Quadrature (for singularities at endpoints)

% Get Jacobi quadrature nodes and weights
[s, w] = jacpts(n, alpha, beta)

% Compute integral
I = w * f(s)

General Intervals

All quadrature operators work on general intervals [a, b], not just [-1, 1].

When to Use Each Method

• sum(f): Standard definite integration of chebfun objects
• 2D integration: Use Chebfun2 for best results
• Gauss quadrature: High-precision integration with known polynomial basis
• Jacobi quadrature: Functions with endpoint singularities