A Python Library for Applied Mathematics in Physical Sciences.
Project description
SigmaEpsilon.Math - A Python Library for Applied Mathematics in Physical Sciences
SigmaEpsilon.Math
is a Python library that provides tools to formulate and solve problems related to all kinds of scientific disciplines. It is a part of the SigmaEpsilon ecosystem, which is designed mainly to solve problems related to computational solid mechanics, but if something is general enough, it ends up here. A good example is the included vector and tensor algebra modules, or the various optimizers, which are applicable in a much broader context than they were originally designed for.
The most important features:
-
Linear Algebra
- A mechanism that guarantees to maintain the property of objectivity of tensorial quantities.
- A
ReferenceFrame
class for all kinds of frames, and dedicatedRectangularFrame
andCartesianFrame
classes as special cases, all NumPy compliant. - NumPy compliant classes like
Tensor
andVector
to handle various kinds of tensorial quantities efficiently. - A
JaggedArray
and a Numba-jittablecsr_matrix
to handle sparse data.
-
Operations Research
- Classes to define and solve linear and nonlinear optimization problems.
- A
LinearProgrammingProblem
class to define and solve any kind of linear optimization problem. - A
BinaryGeneticAlgorithm
class to tackle more complicated optimization problems.
- A
- Classes to define and solve linear and nonlinear optimization problems.
-
Graph Theory
- Algorithms to calculate rooted level structures and pseudo peripheral nodes of a
networkx
graph, which are useful if you want to minimize the bandwidth of sparse symmetrix matrices.
- Algorithms to calculate rooted level structures and pseudo peripheral nodes of a
Note Be aware, that the library uses JIT-compilation through Numba, and as a result, first calls to these functions may take longer, but it pays off big time in the long run.
Documentation
The documentation is hosted on ReadTheDocs.
Installation
sigmaepsilon.math
can be installed (either in a virtual enviroment or globally) from PyPI using pip
on Python >= 3.7:
>>> pip install sigmaepsilon.math
or chechkout with the following command using GitHub CLI
gh repo clone sigma-epsilon/sigmaepsilon.math
and install from source by typing
>>> python install setup.py
Motivating Examples
Linear Algebra
Define a reference frame $\mathbf{B}$ relative to the frame $\mathbf{A}$:
>>> from sigmaepsilon.math.linalg import ReferenceFrame, Vector, Tensor
>>> A = ReferenceFrame(name='A', axes=np.eye(3))
>>> B = A.orient_new('Body', [0, 0, 90*np.pi/180], 'XYZ', name='B')
Get the DCM matrix of the transformation between two frames:
>>> B.dcm(target=A)
Define a vector $\mathbf{v}$ in frame $\mathbf{A}$ and show the components of it in frame $\mathbf{B}$:
>>> v = Vector([0.0, 1.0, 0.0], frame=A)
>>> v.show(B)
Define the same vector in frame $\mathbf{B}$:
>>> v = Vector(v.show(B), frame=B)
>>> v.show(A)
Linear Programming
Solve the following Linear Programming Problem (LPP) with one unique solution:
>>> from sigmaepsilon.math.optimize import LinearProgrammingProblem as LPP
>>> from sigmaepsilon.math.function import Function, Equality
>>> import sympy as sy
>>> variables = ['x1', 'x2', 'x3', 'x4']
>>> x1, x2, x3, x4 = syms = sy.symbols(variables, positive=True)
>>> obj1 = Function(3*x1 + 9*x3 + x2 + x4, variables=syms)
>>> eq11 = Equality(x1 + 2*x3 + x4 - 4, variables=syms)
>>> eq12 = Equality(x2 + x3 - x4 - 2, variables=syms)
>>> problem = LPP(cost=obj1, constraints=[eq11, eq12], variables=syms)
>>> problem.solve()['x']
array([0., 6., 0., 4.])
NonLinear Programming
Find the minimizer of the Rosenbrock function:
>>> from sigmaepsilon.math.optimize import BinaryGeneticAlgorithm
>>>
>>> def Rosenbrock(x):
... a, b = 1, 100
... return (a-x[0])**2 + b*(x[1]-x[0]**2)**2
>>>
>>> ranges = [[-10, 10], [-10, 10]]
>>> BGA = BinaryGeneticAlgorithm(Rosenbrock, ranges, length=12, nPop=200)
>>> BGA.solve()
...
License
This package is licensed under the MIT license.
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