A numerical and automatic math library for scientific research and graphical applications
Theoretica is a header-only mathematical library which provides algorithms for systems simulation, statistical analysis of lab data and numerical approximation, using a functional oriented paradigm to mimic mathematical notation and formulas. The aim of the library is to provide simple access to powerful algorithms while keeping an elegant and transparent interface, enabling the user to focus on the problem at hand.
Given a Hamiltonian function H(q, p) and a function f(q, p) defined on its phase space, you can compute its exact time derivative at a position eta = (q, p) like this:
Where J is the symplectic matrix. The equation can be translated into code as:
mat<N, N> J = mat<N, N>::symplectic();
real df_dt = gradient(f, eta) * J * gradient(H, eta);The library includes real and complex analysis functions optimized for the x86 architecture, linear algebra, quaternions, roots and extrema search, numerical approximation of derivatives, integrals and differential equations, as well as more advanced features like dual numbers for automatic differentiation, statistical functions including distribution sampling, pseudorandom and quasirandom number generation for Monte Carlo methods and simulations.
This is an overview of the library's functionalities. For a more detailed list see FEATURES.md
- Real and complex analysis
- Linear algebra with common vector and matrix operations
- Complex numbers in algebraic and exponential form
- Quaternions
- Dual numbers, Multivariable Automatic Differentiation and Differential Operators
- Pseudorandom and Quasirandom number generation (LCG, Xoshiro256++, Splitmix64, Wyrand, Weyl)
- Statistical functions, including Least Squares Linearization
- Random distribution sampling and Monte Carlo
- Approximation of roots, extrema, derivatives and integrals of real functions
- Numerical integration of Ordinary Differential Equations (Euler, Heun, RK4, K38, multistep)
- Polynomial interpolation with Chebyshev nodes, Bezier curves and spline interpolation
The library has no dependencies. Only the C++ Standard Library with C++11 capabilities is needed to use it. You can include it in your project straight away!
You don't need anything other than your compiler to use the library. You can run make all in the root directory of the library to make sure it works.
Define THEORETICA_INCLUDE_BASE if you intend to use only basic functionalities (linear algebra, real functions, complex numbers), as by default theoretica.h includes all headers.
All library functions and objects are implemented in the theoretica namespace (th is a shorter namespace alias).
The documentation for the library is available here.
Introductory examples can be found in EXAMPLES.md and more advanced examples can be found inside the examples/ folder.
The bibliography used for researching the algorithms used in the library is available here.
You can try to compile this simple code to check if you set up the library correctly:
#include "theoretica.h"
using namespace th;
int main() {
vec3 v = {1, 2, 3};
mat3 A = mat3::identity();
vec3 w = A * v;
return 0;
}- Statistics
- Chaotic Attractors
- Automatic differentiation
- Sampling distributions
- Monte Carlo integration
- Fitting data
- Automatic error propagation
- Histogram usage
Contributions are welcome and very appreciated! Make sure to read the Contributing Guide to know more about how you can help. If you participate, you are expected to follow the Code of Conduct.
The library uses the custom built Chebyshev testing framework to estimate the precision of functions and test their performance. Tests are automatically run on Windows, Linux and MacOS on every commit to ensure stability. Test units are placed inside the test folder while benchmarks are placed inside the benchmark folder.
Where the current stable version of the library is developed.
The physics module implementing quantum mechanics features.
Sparse matrix and vector algebra classes.
Machine learning development branch, with cost functions, models and training algorithms.
This project is currently under the GNU Lesser General Public License 3.0.
These are common macros that can be defined to change the library's behaviour:
| Macro | Description |
|---|---|
| THEORETICA_INCLUDE_BASE | Including theoretica.h will only include base headers |
| THEORETICA_THROW_EXCEPTIONS | Exceptions will be thrown and errno set on error (by default errno is set and NaN is returned) |
| THEORETICA_ONLY_EXCEPTIONS | Exceptions will be thrown on error (without modifying errno) |
| THEORETICA_X86 | Assembly x86 implementations will be used whenever possible (automatically defined on most compilers by the library) |
| THEORETICA_FLOAT_PREC | Floating point precision (float) will be used for the real type (by default double is used) |
| THEORETICA_LONG_DOUBLE_PREC | Long double precision (long double) will be used |
| THEORETICA_ROW_FIRST | The mat<N, K> class will use row-first storage of matrix data instead of column-first. |
See src/core/constants.h for more specific macros.
The library uses errno and th::math_exception (if enabled) to report errors. The behaviour of the library may be modified using the THEORETICA_THROW_EXCEPTIONS and THEORETICA_ONLY_EXCEPTIONS. See Macros to learn more.
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