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Getting Started

Installation

Building from Source

cmake -S . -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build
ctest --test-dir build --output-on-failure

CMake Options

Option Default Description
TAX_BUILD_TEST ON Build Google Test suite
TAX_BUILD_BENCHMARK OFF Build Google Benchmark suite
TAX_USE_DACE OFF Enable DACE comparison tests

Using tax in Your Project

cmake --install build --prefix /your/install/prefix

find_package(tax CONFIG REQUIRED)
target_link_libraries(your_target PRIVATE tax::tax)

Include

A single umbrella header pulls in everything:

#include <tax/tax.hpp>

Core Type

The central type is TruncatedTaylorExpansionT<T, N, M>:

Parameter Meaning
T Scalar coefficient type (e.g. double)
N Maximum total polynomial order
M Number of independent variables (default 1)

Two convenience aliases:

tax::TE<N>        // univariate (M = 1)
tax::TEn<N, M>    // multivariate

Creating Variables

auto x = tax::TE<5>::variable(1.0);   // x = 1 + δx

auto [x, y] = tax::TEn<3, 2>::variables({1.0, 2.0});

auto z = tax::TEn<2, 3>::variable<2>({1.0, 2.0, 3.0});

Building Expressions

Arithmetic and math functions work naturally:

auto x = tax::TE<6>::variable(0.0);
tax::TE<6> f = tax::sin(x) + tax::square(x) / 2.0;

The right-hand side builds a lazy expression tree. Evaluation happens only on assignment to a TruncatedTaylorExpansionT object.

Extracting Results

f.value();              // f(x₀) — the constant term
f.coeff({k});           // coefficient of δxᵏ
f.derivative({k});      // k-th derivative (= k! · coeff)
f.eval(0.3);            // evaluate polynomial at x₀ + 0.3

For multivariate DA objects:

auto [x, y] = tax::TEn<3, 2>::variables({0.0, 0.0});
tax::TEn<3, 2> g = x*x + x*y + y*y;

g.derivative({2, 0});   // ∂²g/∂x²   = 2
g.derivative({1, 1});   // ∂²g/∂x∂y  = 1
g.coeff<1, 1>();         // compile-time access

Coefficients vs Derivatives

A DA polynomial stores coefficients of the monomial basis:

\[ f(\mathbf{x}_0 + \delta\mathbf{x}) = \sum_{|\alpha| \le N} f_\alpha \, \delta\mathbf{x}^\alpha \]

The relationship to partial derivatives is:

\[ f_\alpha = \frac{1}{\alpha!} \frac{\partial^{|\alpha|} f}{\partial x_1^{\alpha_1} \cdots \partial x_M^{\alpha_M}} \bigg|_{\mathbf{x}_0} \]

The derivative() method returns the partial derivative (multiplies the coefficient by \(\alpha!\)), while coeff() returns the raw coefficient.

Next Steps