Numerical Differentiation

Functions for numerical differentiation using finite difference and complex step methods.

Finite Difference Derivative: Calculates a finite-difference approximation to the derivative of a function \(f\) at point \(x\). This problem is ill-conditioned: truncation error (\(O(h^k)\)) decreases with \(h\), but roundoff error increases. The function balances these errors automatically. The default order is 6. Requires the function to be differentiable (up to the order requested).

Complex Step Derivative: Computes the derivative of a real-valued function \(f(x)\) using the complex step approximation: $$f'(x) \approx \frac{\Im(f(x + ih))}{h}$$ This method avoids the subtractive cancellation error inherent in finite differences and is extremely accurate. However, it requires \(f\) to be a holomorphic function (complex-differentiable) that takes real values at real arguments. Ideally, the function f should be able to accept a complex argument.

Usage

finite_difference_derivative(f, x, order = 1)

complex_step_derivative(f, x)

Arguments

f

A function to differentiate. It should accept a single numeric/complex value and return a single numeric/complex value.

x

The point at which to evaluate the derivative.

order

The order of accuracy of the finite difference method. Can be 1, 2, 4, 6, or 8. Default is 1.

Value

The approximate value of the derivative at the point x.

See also

numerical_integration

Examples

# Finite difference derivative of sin(x) at pi/4
finite_difference_derivative(sin, pi / 4)
#> [1] 0.7071068

# Complex step derivative of exp(x) at 1.7 (Requires f to handle complex input ideally)
# Note: In pure R, `exp` handles complex numbers automatically.
complex_step_derivative(exp, 1.7)
#> [1] 5.473947