HTN: Hyper Triple Numbers Library

(! Broken English)

Automatic forward-mode differential library.
We can calculate gradient vector and hessian matrix automatically.
This library uses Eigen and ViennaCL library.

The library is MIT Lisence but Lisences of Eigen and ViennaCL library take over each one.

HTN's calculation is based on maltivariate Taylor expansion version of dual numbers.

Example Code

//#define VIENNACL_WITH_OPENMP
//#define VIENNACL_WITH_OPENCL
//#define VIENNACL_WITH_CUDA
#include "HTN.hpp"

#include "Eigen/Core"
#include "Eigen/Geometry"
#include "Eigen/StdVector"
#include "Eigen/SVD"
#include "Eigen/Dense"
#include "Eigen/LU"

#include <iostream> 

void main() {

	using f_type = double;
	const static unsigned int dim = 3;

	/* scalar */
	using htn0 = HTN::htn_r0<f_type, dim>;

	/* scalar & gradient vector */
	using htn1 = HTN::htn_r1<f_type, dim>;

	/* scalar & gradient vector & hessian matrix */
	using htn2 = HTN::htn_r2<f_type, dim>;

	/* x_0 <= 3 */
	auto x0 = htn2::x(3); /* or htn2::x<0>(3); or htn2::x(3, 0); */
	/* x_1 <= -pi */
	auto x1 = htn2::x<1>(-HTN::PI()); /* or htn2::x(-HTN::PI(), 1) */

	/* htn is initialized to 0(constant). */
	/* x2 <= 0 (not variable) */
	htn2 x2; 

	/* !warning: One variable can represent multiple states.
	   example; auto x0 = htn2::x(3); and x0 = htn2::x(-2);  */

	/* function: 
	inverse, pow, sqrt, exp, log, 
	sin, cos, tan, asin, acos, atan, 
	sinh, cosh, tanh, asinh, acosh, atanh  */
	auto y = HTN::exp(htn2::x<1>(0.5));
	auto z = htn2::x<2>(0.5).acos();
	float a = 3.2;
	auto f = HTN::pow(2.5 * htn2::x(3), a) / y;
	f -= z;

	/* convert to scalar, Eigen vector(Dynamic size) and Eigen matrix(Dynamic size) */
	std::cout << f.X0_to_type() << std::endl << std::endl;
	std::cout << f.X1_to_eig() << std::endl << std::endl;
	std::cout << f.X2_to_eig() << std::endl << std::endl;

	/*-----------------------------------------------------------------------------*/


	auto Rosenbrock = [](const auto& data) {
		htn2 f;
		for (int i = 0; i <= htn2::get_dim() - 2; ++i) {
			f += 100 * HTN::pow(htn2::x(data[i + 1], i + 1) - htn2::x(data[i], i).pow(2), 2) + HTN::pow(1 - htn2::x(data[i], i), 2);
		}
		return f;
	};

	/*Newton Method*/
	std::cout << "---Minimization of Rosenbrock function by Newton method---" << std::endl;

	Eigen::Matrix<f_type, dim, 1> x(4, 3, -4);
	for (int i = 0; i < 15; ++i) {
		auto rb = Rosenbrock(x);
		x -= rb.X2_to_eig().inverse() * rb.X1_to_eig();

		std::cout << x.transpose() << std::endl;
	}
	std::cout << std::endl;


	std::cout << std::endl;
}

Base Math

values

All HTN values are as follows;

, where

and dual units;

.

If HTN value is a i-th base variable, .

Else, if constant, .

functions

All functions are based on

, where .

In particular,

.