How to pass derivatives to dual number? How to combine analytical/numerical derivative computation with autodiff?

[ghost]

Hello all,
I'm not able to find an example through this tutorial for this scenario:
Consider, I have two functions of A(x,y) and B(x,y). The definition for A and B are not given, but point-wise value and derivatives of A(x,y) and B(x,y) w.r.t x and y are known. I have a third function C(x,y)= A(x,y)**2+ B(x,y) and I want to calculate derivatives of C(x,y) with the notion that derivatives of A(x,y) and B(x,y) are known from other parts of the code. How can I construct my dual number?

[allanleal]

Hi @Sajedeh1996 , please check possible solution below. As you see, you'll need to define a function wrapper Adual and Bdual as shown below, which inspect if incoming x and y arguments have been "seeded" in their grad member:

#include <iostream>

#include <autodiff/forward/dual.hpp>
using namespace autodiff;

auto  A(double x, double y) -> double { return x*y; }
auto Ax(double x, double y) -> double { return x; }
auto Ay(double x, double y) -> double { return y; }

auto  B(double x, double y) -> double { return x + y; }
auto Bx(double x, double y) -> double { return 1.0; }
auto By(double x, double y) -> double { return 1.0; }

auto Adual(dual const& x, dual const& y) -> dual
{
    dual res = A(x.val, y.val);

    if(x.grad != 0.0)
        res.grad += x.grad * Ax(x.val, y.val);

    if(y.grad != 0.0)
        res.grad += y.grad * Ay(x.val, y.val);

    return res;
}

auto Bdual(dual const& x, dual const& y) -> dual
{
    dual res = B(x.val, y.val);

    if(x.grad != 0.0)
        res.grad += x.grad * Bx(x.val, y.val);

    if(y.grad != 0.0)
        res.grad += y.grad * By(x.val, y.val);

    return res;
}

auto C(dual const& x, dual const& y) -> dual
{
    const auto A = Adual(x, y);
    const auto B = Bdual(x, y);
    return A*A + B;
}

int main()
{
    dual x = 1.0;
    dual y = 2.0;


    auto C0 = C(x, y);
    auto Cx = derivative(C, wrt(x), at(x, y));
    auto Cy = derivative(C, wrt(y), at(x, y));

    auto expectedCx = 2.0*A(x.val, y.val)*Ax(x.val, y.val) + Bx(x.val, y.val);
    auto expectedCy = 2.0*A(x.val, y.val)*Ay(x.val, y.val) + By(x.val, y.val);

    std::cout << "C0 = " << C0 << "\n";

    std::cout << "Cx(computed) = " << Cx << "\n";
    std::cout << "Cx(expected) = " << expectedCx << "\n";

    std::cout << "Cy(computed) = " << Cy << "\n";
    std::cout << "Cy(expected) = " << expectedCy << "\n";
}

// Output:
// C0 = 7
// Cx(computed) = 5
// Cx(expected) = 5
// Cy(computed) = 9
// Cy(expected) = 9

This example has now been added to the list of examples and it's available on the website:
https://autodiff.github.io/tutorials/#derivatives-of-a-multi-variable-function-that-also-relies-on-analytical-derivatives