AutoDiff examples

This page shows basic uses of the Auto Diff module.

Tracking first derivatives

AutoDiffScalar stores a scalar value together with its derivatives. The derivative vector size is the number of independent variables. A common pattern is to seed each independent variable with one unit vector:

Example:

Output:

// SPDX-FileCopyrightText: The Eigen Authors // SPDX-License-Identifier: MPL-2.0 #include <Eigen/Dense> #include <cmath> #include <iostream> #include <contrib/Eigen/AutoDiff> template <typename Scalar> Scalar f(const Scalar& x, const Scalar& y) { using std::sin; return x * x * y + sin(y); } int main() { typedef Eigen::AutoDiffScalar<Eigen::Vector2d> ADScalar; ADScalar x(2.0, Eigen::Vector2d::UnitX()); ADScalar y(0.5, Eigen::Vector2d::UnitY()); ADScalar z = f(x, y); std::cout << "f(x,y) = " << z.value() << "\n"; std::cout << "df/dx = " << z.derivatives()[0] << "\n"; std::cout << "df/dy = " << z.derivatives()[1] << "\n"; }

f(x,y) = 2.47943 df/dx = 2 df/dy = 4.87758

Computing a Jacobian

AutoDiffJacobian wraps a functor whose operator() is templated on the scalar type. Calling it with a Jacobian pointer evaluates the functor with active AutoDiffScalar inputs and fills both the value and Jacobian:

Example:

Output:

// SPDX-FileCopyrightText: The Eigen Authors // SPDX-License-Identifier: MPL-2.0 #include <Eigen/Dense> #include <cmath> #include <iostream> #include <contrib/Eigen/AutoDiff> struct ExampleFunctor { typedef Eigen::Vector2d InputType; typedef Eigen::Vector2d ValueType; template <typename Scalar> void operator()(const Eigen::Matrix<Scalar, 2, 1>& x, Eigen::Matrix<Scalar, 2, 1>* y) const { using std::sin; (*y)[0] = x[0] * x[0] + x[1]; (*y)[1] = x[0] * sin(x[1]); } }; int main() { Eigen::AutoDiffJacobian<ExampleFunctor> functor; Eigen::Vector2d x; Eigen::Vector2d y; Eigen::AutoDiffJacobian<ExampleFunctor>::JacobianType jacobian; x << 2.0, 0.5; functor(x, &y, &jacobian); std::cout << "value:\n" << y << "\n\n"; std::cout << "jacobian:\n" << jacobian << "\n"; }

value: 4.5 0.958851 jacobian: 4 1 0.479426 1.75517