dual-numbers

A small library of functions implemneting dual arithmetic. The dual numbers make it easy to compute derivatives of functions and do so accurately (up to machine precision). Derivatives are not approximated; they are computed exactly. The only limitation is the precision of the machine on which the code is being executed.

The supported functions are:

• arithmetic operations Add, Sub, Mul, Div
• Sin
• Cos
• Exp
• Log
• Power
• Abs

Example

Say we wanted to compute the derivative of the function f(x) = (x + 1)*(x - 2)/(x + 3) at x = 3. We convert this function into code by using the primitives listed above:

DualNumber Func(double x) {
	DualNumber d = MakeVariable(x);
	return Div(Mul(Add(d, MakeConstant(1)), Sub(d, MakeConstant(2))),
		Add(d, MakeConstant(3)));
}

The above function operates on real numbers and returns a dual number. DualNumber has a real part and a dual part. The real part stores the value of the function at the given point and the dual part stores the value of the first order derivative of the function at the given point. A call to Func(3) returns a dual number whose real part is equal ot 0.666667 and whose dual part is equal to 0.722222.

We can compute the derivatives of complicated functions by using this technique. The functions are assumed to be differentiable at the input points.