Vector (Eigen) Module¶

The Eigen integration module enables using TruncatedTaylorExpansionT types inside Eigen matrices, vectors, and tensors. This provides seamless interoperability between the tax library and the Eigen linear algebra ecosystem, so you can build vector-valued Taylor maps, extract Jacobians and Hessians, evaluate polynomial flows, and invert nonlinear maps -- all with natural Eigen syntax.

Requirements¶

Eigen 3.4+ must be available (found via find_package(Eigen3) or CMAKE_PREFIX_PATH).
The unsupported Eigen Tensor module is used for higher-order derivative tensors (\(K \ge 3\)).

Headers¶

The module is composed of seven headers, all under include/tax/eigen/:

Header	Purpose
`tax/eigen/num_traits.hpp`	`Eigen::NumTraits` specialization for `TruncatedTaylorExpansionT`, allowing Eigen to treat TTE as a scalar type
`tax/eigen/types.hpp`	Convenience type aliases (`Mat`, `VecT`, `TEVec`, `TEnVec`, etc.)
`tax/eigen/variables.hpp`	Create TTE variables from Eigen vectors: `tax::vector` and `tax::variables`
`tax/eigen/value.hpp`	Extract the scalar constant term from TTE containers: `tax::value`
`tax/eigen/eval.hpp`	Evaluate TTE polynomials at displacements: `tax::eval` (including vectorized univariate path)
`tax/eigen/derivative.hpp`	Compute gradients, Jacobians, Hessians, and higher-order derivative tensors: `tax::gradient`, `tax::jacobian`, `tax::derivative`
`tax/eigen/invert_map.hpp`	Invert a square Taylor map via Picard iteration: `tax::invert`

All public symbols live in namespace tax. Implementation details are in namespace tax::detail.

Quick Start¶

#include <tax/tax.hpp>

// Expansion point
Eigen::Vector3d x0{1.0, 2.0, 3.0};

// Create multivariate TTE variables (order 4, 3 variables)
auto [x, y, z] = tax::variables<tax::TEn<4, 3>>(x0);

// Build a vector-valued function F : R^3 -> R^2
Eigen::Vector2<tax::TEn<4, 3>> F;
F(0) = sin(x) * y + z;
F(1) = exp(x - y) * cos(z);

// Extract values and derivatives at the expansion point
Eigen::Vector2d vals = tax::value(F);          // F(x0)
Eigen::Matrix<double, 2, 3> J = tax::jacobian(F);  // 2x3 Jacobian

// Evaluate at a displaced point
Eigen::Vector3d dx{0.01, -0.02, 0.03};
Eigen::Vector2d result = tax::eval(F, dx);     // F(x0 + dx)

Module Pages¶

Mathematical Foundations -- theory behind vector Taylor maps, gradients, Jacobians, Hessians, higher-order tensors, and map inversion.
API Reference -- complete function signatures and parameter descriptions.
Examples -- practical, self-contained code examples.