Math Module

rmath.calculus

Automatic differentiation via dual numbers, numerical integration (trapezoidal and Simpson's rule), and Newton's method for root-finding.

autodiff.py

import rmath.calculus as rc

# f(x) = x² + 3x at x = 2
x = rc.Dual(2.0, 1.0)  # value=2, seed derivative=1
y = x * x + x * 3.0

print(y.value)       # 10.0
print(y.derivative)  # 7.0  → f'(2) = 2(2) + 3

Dual Numbers

Dual numbers a + bε (where ε² = 0) enable forward-mode automatic differentiation. Build an expression using Dual operands and the derivative is computed alongside the value.

Constructor & Properties

`Dual(val, der)`	Create a dual number with value and derivative seed
`.value`	float	The real part
`.derivative`	float	The derivative (epsilon coefficient)

Operators

`+ - * / **`	All work with Dual or float operands, propagating derivatives automatically
`-x`	Unary negation

Math Functions

`.sin()`	sin(x), derivative: cos(x)
`.cos()`	cos(x), derivative: -sin(x)
`.exp()`	eˣ, derivative: eˣ
`.log()`	ln(x), derivative: 1/x

Numerical Integration

Function	Description
`integrate_trapezoidal(x, y)`	Trapezoidal rule on sampled data (x, y vectors)
`integrate_simpson_array(y, dx)`	Simpson's rule on uniformly-spaced samples
`integrate_simpson(f, a, b, n)`	Simpson's rule on a callable f over [a, b] with n intervals

Root-Finding

Function	Description
`find_root_newton(f, x0, tol, max_iter)`	Newton's method using autodiff. `f` takes a Dual and returns a Dual. Derivatives are computed automatically.

root.py

import rmath.calculus as rc

# Find root of f(x) = x² - 2 → x = √2
def f(x):
    return x * x - 2.0

root = rc.find_root_newton(f, x0=1.0, tol=1e-12, max_iter=50)
# root ≈ 1.4142135623730951