8 Automatic Differentiation

8.1 Dual number system

Dual is structure for AD
- value(&self) -> f64 : Value
- slope(&self) -> f64 : Slope (value of derivatives)
- extract(&self) -> (f64,f64) : Extract both
Constructor for Dual
- Dual::new(T, T)
- dual(T, T)

Implemented Operations (Real trait is implemented)

Add, Sub, Mul, Div
sin, cos, tan
asin, acos, atan
sinh, cosh, tanh
asinh, acosh, atanh
sin_cos
exp, ln, log, log2, log10
powi, powf, sqrt

fn main() {
    let x = dual(1, 1); // x at x = 1
    (x.clone() + x.clone()).print(); // dual(2, 2)
    (x.clone() - x.clone()).print(); // dual(0, 0)
    (x.clone() * x.clone()).print(); // dual(1, 2)
    (x.clone() / x.clone()).print(); // dual(1, 0)
    x.sin().print();
    x.cos().print();
    x.exp().print();
    x.ln().print();
    x.powi(2).print(); // same as x * x
}

After ver 0.10.1, you can use reference operations

fn main() {
    let x = dual(1,1);
    (&x + &x).print();
    (&x - &x).print();
    // and etc.
}

8.2 Hyper dual number system

HyperDual is structure for 2nd order AD
- value(&self) -> f64
- slope(&self) -> f64
- accel(&self) -> f64

There are two constructors

HyperDual::new(T, T, T)
hyper_dual(T, T, T)

fn main() {
    let x = hyper_dual(1, 1, 0); // x at x = 1
    (x.clone() + x.clone()).print(); // hyper_dual(2, 2, 0)
    (x.clone() * x.clone()).print(); // hyper_dual(1, 2, 2)
    // and etc.
}

Also, after 0.10.1, you can use reference ops.

fn main() {
    let x = hyper_dual(1,1,0);
    (&x + &x).print();
    (&x * &x).print();
    // and etc.
}

8.3 `Real` trait

Real is a trait for binding f64, Dual, HyperDual

Real requires PowOps, TrigOps, ExpLogOps & std::Ops<Self> & std::Ops<f64>

fn main() {
    let x_f64 = 2f64;
    let x_dual = dual(2,1);
    let x_hyper = hyper_dual(2, 1, 0);

    f(x_f64).print();
    f(x_dual).print();
    f(x_hyper).print();
}

fn f<T: Real>(x: T) -> T {
    return x.powi(2)
}