ShaneLogic/chebfun-calculus-operations

Chebfun Calculus Operations

Differentiation via diff

Compute the derivative of a chebfun using continuous analogue of finite differences.

Basic Usage

derivative = diff(f)

Higher Order Derivatives

% Second derivative
derivative_2 = diff(f, 2)

% Fourth derivative
derivative_4 = diff(f, 4)

Handling Discontinuities

If the function has a jump, the derivative introduces a delta function with amplitude equal to the jump size.

Inverse Relationships

% Differentiating indefinite integral returns original function
diff(cumsum(f)) ≈ f

% Integrating derivative returns original plus constant
cumsum(diff(f)) ≈ f + C
% Add value at left endpoint to recover f exactly

Stability Warning

Differentiation is an ill-posed problem:

• Errors in stable operations accumulate exponentially
• Successive derivatives lose information
• Differentiating a low-degree polynomial many times may eliminate the function entirely

Example

f = chebfun('sin(pi*x)')
f_prime = diff(f)
f_double_prime = diff(f, 2)

plot(f)
hold on
plot(f_prime, 'r--')
plot(f_double_prime, 'g:')
legend('f', "f'", "f''")

Non-smooth Operations

Operations that introduce breakpoints/discontinuities.

Absolute Value

g = abs(f)
% Introduces breakpoints where f = 0

Minimum and Maximum

g = min(f, h)  % or max(f, h)
% Introduces breakpoints where f - h = 0

Sign Function

g = sign(f)
% Introduces breaks where f = 0

Rounding Operations

g = round(f)   % or floor(f), ceil(f)
% Introduces breaks based on rounding logic

Mechanism

These commands use root finding to determine exactly where to place discontinuities:

• abs(f): zeros of f
• min/max(f,g): zeros of f - g
• sign(f): zeros of f
• round/floor/ceil: based on function value transitions

Example

f = chebfun('sin(pi*x)', [-1, 1])
g = abs(f)

% g has breakpoints at x = -1, 0, 1
plot(g)
% Notice the corners at the zeros of sin(pi*x)

Piecewise Function Construction

% Create piecewise function using min/max
f = chebfun('x.^2')
g = chebfun('1')
h = min(f, g)

% h equals x^2 for |x| <= 1, equals 1 for |x| > 1
plot(h)

When to Use

• diff(f): Compute first or higher-order derivatives
• diff(f, n): Compute nth derivative
• abs(f): Create function with absolute value, introduces corners at zeros
• min/max(f,g): Create piecewise functions, introduces breaks at intersections
• sign(f): Create sign function, introduces jumps at zeros
• round/floor/ceil(f): Apply rounding operations, creates step functions

Important Considerations

1. Stability: Repeated differentiation is ill-posed
2. Delta functions: Derivatives of discontinuous functions include delta functions
3. Breakpoints: Non-smooth operations automatically detect and place breakpoints
4. Root finding: Essential for determining where discontinuities occur