Source code for geomstats.geometry.lie_algebra

"""Module providing an implementation of MatrixLieAlgebras.

There are two main forms of representation for elements of a MatrixLieAlgebra
implemented here. The first one is as a matrix, as elements of R^(n x n).
The second is by choosing a base and remembering the coefficients of an element
in that base. This base will be provided in child classes
(e.g. SkewSymmetricMatrices).

Lead author: Stefan Heyder.

import abc

import geomstats.backend as gs
import geomstats.errors
from geomstats.geometry.base import VectorSpace
from geomstats.geometry.matrices import Matrices

from ._bch_coefficients import BCH_COEFFICIENTS

[docs]class MatrixLieAlgebra(VectorSpace, abc.ABC): """Class implementing matrix Lie algebra related functions. Parameters ---------- dim : int Dimension of the Lie algebra as a real vector space. n : int Amount of rows and columns in the matrix representation of the Lie algebra. """ def __init__(self, dim, n, **kwargs): super(MatrixLieAlgebra, self).__init__(shape=(n, n), **kwargs) geomstats.errors.check_integer(dim, "dim") geomstats.errors.check_integer(n, "n") self.dim = dim self.n = n bracket = Matrices.bracket
[docs] def baker_campbell_hausdorff(self, matrix_a, matrix_b, order=2): """Calculate the Baker-Campbell-Hausdorff approximation of given order. The implementation is based on [CM2009a]_ with the pre-computed constants taken from [CM2009b]_. Our coefficients are truncated to enable us to calculate BCH up to order 15. This represents Z = log(exp(X)exp(Y)) as an infinite linear combination of the form Z = sum z_i e_i where z_i are rational numbers and e_i are iterated Lie brackets starting with e_1 = X, e_2 = Y, each e_i is given by some i',i'': e_i = [e_i', e_i'']. Parameters ---------- matrix_a, matrix_b : array-like, shape=[..., n, n] Matrices. order : int The order to which the approximation is calculated. Note that this is NOT the same as using only e_i with i < order. Optional, default 2. References ---------- .. [CM2009a] F. Casas and A. Murua. An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications. Journal of Mathematical Physics 50, 2009 .. [CM2009b] """ if order > 15: raise NotImplementedError("BCH is not implemented for order > 15.") number_of_hom_degree = gs.array( [2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182] ) n_terms = gs.sum(number_of_hom_degree[:order]) el = [matrix_a, matrix_b] result = matrix_a + matrix_b for i in gs.arange(2, n_terms): i_p = BCH_COEFFICIENTS[i, 1] - 1 i_pp = BCH_COEFFICIENTS[i, 2] - 1 el.append(self.bracket(el[i_p], el[i_pp])) result += ( float(BCH_COEFFICIENTS[i, 3]) / float(BCH_COEFFICIENTS[i, 4]) * el[i] ) return result
[docs] @abc.abstractmethod def basis_representation(self, matrix_representation): """Compute the coefficients of matrices in the given basis. Parameters ---------- matrix_representation : array-like, shape=[..., n, n] Matrix. Returns ------- basis_representation : array-like, shape=[..., dim] Coefficients in the basis. """ raise NotImplementedError("basis_representation not implemented.")
[docs] def matrix_representation(self, basis_representation): """Compute the matrix representation for the given basis coefficients. Sums the basis elements according to the coefficients given in basis_representation. Parameters ---------- basis_representation : array-like, shape=[..., dim] Coefficients in the basis. Returns ------- matrix_representation : array-like, shape=[..., n, n] Matrix. """ if self.basis is None: raise NotImplementedError("basis not implemented") return gs.einsum("...i,ijk ->...jk", basis_representation, self.basis)