Functions for differentiation

proc initdual(x: real)

Initializes the input to the appropriate dual number to evalute the derivative.

Arguments:: x : real or [dom] real – point where to evaluate the derivative
Returns:: If x is a real number, then it is initialized to \(x+\epsilon\). If x is a vector of reals, it is initialized to the vector of multiduals \(\begin{bmatrix}x_1+\epsilon_1\\\vdots\\x_n+\epsilon_n\end{bmatrix}\).
Return type:: dual if x is real or [dom] multidual if x is [dom] real.

proc initdual(x: [?D] ?t, v: [D] ?s)

Given a vector x and a vector v, creates a vector of duals \([x_i + \epsilon v_i]\). Used to compute directional derivative and Jacobian-vector product (JVP).

Arguments:

x : [D] real – point where to evaluate the directional derivative / JVP
v : [D] real – direction

Returns:

Vector of dual numbers

Return type:

[D] dual

proc derivative(f, x: real)

Evaluates the derivative of f at x.

Arguments:

f : Function – Function, note that this must be a concrete function.
x : real – point at which the derivative is evaluated

Returns:

value of f’(x)

Return type:

real

Note that f must be a concrete function, if it’s written as a generic function, you can pass derivative a lambda as follows

proc f(x) {
    return x**2 + 2*x + 1;
}

var dfx = derivative(proc(x : dual){return f(x);}, 1.0);
//outputs
//4.0

proc derivative(x: dual): Extracts the derivative from a dual number.

proc gradient(f, x: [?D] real)

Evaluates the gradient of f at x.

Arguments:

f : Function – Function, note that this must be a concrete function.
x : [ ] real – point at which the gradient is evaluated

Returns:

value of \(\nabla f(x)\)

Return type:

[ ] real

Note that f must be a concrete function, if it’s written as a generic function, you can pass gradient a lambda as follows

proc h(x) {
    return x[0] ** 2 + 3 * x[0] * x[1];
}

type D = [0..#2] multidual; // domain for the lambda function

var dh = gradient(proc(x : D){return h(x);}, [1.0, 2.0]);
//outputs
//8.0 3.0

proc gradient(x: multidual): Extracts the gradient from a multidual number.

proc jacobian(f, x: [?D])

Evaluates the jacobian of f at x.

Arguments:

f : Function – Function, note that this must be a concrete function.
x : [ ] real – point at which the jacobian is evaluated

Returns:

value of \(J_f\)

Return type:

[Dout, Din] real

Note that f must be a concrete function, if it’s written as a generic function, you can pass jacobian a lambda as follows

proc F(x) {
    return [x[0] ** 2 + x[1] + 1, x[0] + x[1] ** 2 + x[0] * x[1]];
}

type D = [0..#2] multidual; // domain for the lambda function

var Jf = jacobian(proc(x : D){return F(x);}, [1.0, 2.0]);

writeln(Jf, "\n");
//outputs
//2.0 1.0
//3.0 5.0

proc jacobian(x: [?D] multidual): Extracts the Jacobian from an array of multidual numbers.

proc value(x)

Extracts the function value.

Arguments:: x : dual, multidual or [] multidual. – result of computations using dual numbers.

proc directionalDerivative(x: dual): Extracts the directional derivative from a dual number.

proc directionalDerivative(f, x: [?D], v: [D]): Computes the directional derivative of f at x in the direction of v.

proc jvp(x: [] dual): Extracts the Jacobian-vector product from a vector of dual numbers.

proc jvp(f, x: [?D], v: [D]): Computes the Jacobian-vector product of the Jacobian of f at x and vector v.