mplusa 0.0.2

A library for calculations in tropical and arctic semirings.

MPlusA

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MPlusA is a small Python library for tropical algebra (also known as $(\min, +)$ and $(\max, +)$ algebra). It provides the definitions of basic operations on numbers and NumPy arrays, as well as a basic implementation of tropical polynomials.

Any improvements or fixes are always welcome.

How to use

After having installed the library one can import one of the two modules the package consists of (minplus and maxplus) and use the full array of its capabilities. The functions are essentially the same between the modules. The list below is a full list of the library's capabilities.

add(*args) -> Real Tropical addition. Essentially an alias for Python's min function.

mult(*args) -> Real Tropical multiplication. Essentially an alias for Python's sum function.

add_matrices(A : np.ndarray, B : np.ndarray) -> np.ndarray Tropical addition of NumPy arrays. The summed matrices have to be of the same shape.

mult_matrices(A : np.ndarray, B : np.ndarray) -> np.ndarray Tropical multiplication of NumPy arrays. The multiplied matrices have to be of sizes MxN and NxP and their order matters. The result is of shape MxP.

modulo(a : Real, t : int) -> Real Tropical modulo operator. It can be understood as the difference between the number $a$ and $t^k$ where $k$ is the largest integer that satisfies $a \geq t^k$.

modulo_matrices(A : np.ndarray, b : np.ndarray) -> np.ndarray Tropical modulo operator for NumPy arrays. The input matrices should be of size MxN and Mx1. The result is an MxN matrix.

power(a : real, k : int) -> Real Tropical power operator. Applies the multiplication k times.

power_matrix(A : np.ndarray, k : int) -> np.ndarray Tropical power operator for NumPy arrays. It multiplies the matrix k times.

unit_matrix(width : int, height : int) -> np.ndarray Creates a tropical unit matrix of given width and height.

star(A : np.ndarray) -> np.ndarray Definition of a unique operator of tropical algebra, usually denoted as $\mathbf{A}^*$. It returns the value to which an infinite recursive sum of matrices converges. The input matrix has to be square and the series created in the process of calculating the value needs to be convergent.

Polynomial(*coefficients) This is a class that implements basic single-variable tropical polynomials. Calling an object of this class allows to take a value the polynomial takes at a given point, it also implements function get_hypersurface which returns a list of its roots.

Example code

import numpy as np
from mplusa import minplus

# Basic operators
s = minplus.add(10, 6, 4, 13)  # -> 4
p = minplus.mult(10, 5, 8)  # -> 23
mod = minplus.modulo(p, s)  # -> 3

# NumPy arrays
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
B = np.array([[9, 8, 7], [6, 5, 4], [3, 2, 1]])
S = minplus.add_matrices(A, B)  # -> [[1, 2, 3], [4, 5, 4], [3, 2, 1]]
P = minplus.mult_matrices(A, B)  # -> [[6, 5, 4], [9, 8, 7], [12, 11, 10]]

Bibliography

1. A. Obuchowicz, K. A. D'Souza, Z. A. Banaszak. An Algebraic Approach to Modelling and Performance Optimisation of a Traffic Route System. International Journal of Applied Mathematics and Computer Science, vol. 8, no. 2, pp. 335-365, June 1998.
2. D. Speyer, B. Sturmfels. "Tropical Mathematics". Mathematics Magazine, vol. 82, no. 3, pp. 163-173, June 2009, doi: https://doi.org/10.4169%2F193009809x468760.