Source code for galgebra.ga

"""
Geometric Algebra (inherits Metric)
"""
import warnings
import operator
import copy
from itertools import combinations
from functools import reduce
from typing import Tuple, TypeVar, Callable, Dict, Sequence, List, Optional, Union
from ._backports.typing import OrderedDict

from sympy import (
    diff, Rational, Symbol, S, Mul, Add, Expr,
    expand, simplify, eye, trigsimp,
    symbols, sqrt, Matrix,
)

from . import printer
from . import metric
from . import mv
from . import dop
from . import lt
from .atoms import (
    BasisBaseSymbol, BasisBladeSymbol, BasisBladeNoWedgeSymbol,
)
from ._utils import cached_property as _cached_property

half = Rational(1, 2)
one = S(1)
zero = S(0)


# Needed to avoid ambiguity with the methods of the same name, when used in
# type annotations.
_mv = mv
_dop = dop
_lt = lt


# template argument for functions which are Expr -> Expr and Mv -> Mv
_MaybeMv = TypeVar('_MaybeMv', Expr, _mv.Mv)


def is_bases_product(w):
    nc_w = w.args_cnc()
    nc = nc_w[1]
    return len(nc) == 2 or len(nc) == 1 and nc[0].is_Pow and nc[0].exp == 2


_K = TypeVar('_K')
_V = TypeVar('_V')


[docs]class lazy_dict(Dict[_K, _V]): """ A dictionary that creates missing entries on the fly. When the dictionary is indexed and the key used is not one of the existing keys, ``self.f_value(key)`` is called to evaluate the key. The result is then added to the dictionary so that ``self.f_value`` is not used to evaluate the same key again. Parameters ---------- d : Arguments to pass on to the :class:`dict` constructor, typically a regular dictionary f_value : function The function to call to generate a value for a given key """ def __init__(self, d, f_value): dict.__init__(self, d) self.f_value = f_value def __missing__(self, key: _K) -> _V: value = self.f_value(key) self[key] = value return value def __repr__(self): return '{}({}, f_value={!r})'.format( type(self).__qualname__, dict.__repr__(self), self.f_value) def _repr_pretty_(self, p, cycle): # ipython support p_open, p_close = type(self).__qualname__ + '(', ')' with p.group(len(p_open), p_open, p_close): p.type_pprinters[dict](self, p, cycle) p.text(',') p.breakable() p.text('f_value={}'.format(self.f_value))
[docs]def update_and_substitute(expr1, expr2, mul_dict): """ Linear expand expr1 and expr2 to get (summation convention):: expr1 = coefs1[i] * bases1[i] expr2 = coefs2[j] * bases2[j] where ``coefs1`` and ``coefs2`` are lists of are commutative expressions and ``bases1`` and ``bases2`` are lists of bases for the geometric algebra. Then evaluate:: expr = coefs1[i] * coefs2[j] * mul_dict[bases1[i], bases2[j]] where ``mul_dict[bases1[i], bases2[j]]`` contains the appropriate product of ``bases1[i]*bases2[j]`` as a linear combination of scalars and bases of the geometric algebra. """ coefs1, bases1 = metric.linear_expand(expr1) coefs2, bases2 = metric.linear_expand(expr2) expr = S(0) for coef1, base1 in zip(coefs1, bases1): for coef2, base2 in zip(coefs2, bases2): expr += coef1 * coef2 * mul_dict[base1, base2] return expr
[docs]def nc_subs(expr, base_keys, base_values=None): """ See if expr contains nc (non-commutative) keys in base_keys and substitute corresponding value in base_values for nc key. This was written since standard sympy subs was very slow in performing this operation for non-commutative keys for long lists of keys. """ if base_values is None: [base_keys, base_values] = list(zip(*base_keys)) if expr.is_commutative: return expr if isinstance(expr, Add): args = expr.args else: args = [expr] s = zero for term in args: if term.is_commutative: s += term else: c, nc = term.args_cnc(split_1=False) key = Mul._from_args(nc) coef = Mul._from_args(c) if key in base_keys: base = base_values[base_keys.index(key)] s += coef * base else: s += term return s
_T = TypeVar('_T') _U = TypeVar('_U')
[docs]class GradedTuple(Tuple[Tuple[_T, ...], ...]): """ A nested tuple grouped by grade. ``self[i]`` refers to a the elements associated with grade ``i``. .. attribute:: flat :type: Tuple[T] The elements flattened out in order of grade. """ def __new__(cls, *args, **kwargs): # super does not work here in Python 3.5, as Tuple.__new__ is broken self = tuple.__new__(cls, *args, **kwargs) self.__dict__['flat'] = tuple(x for single_grade in self for x in single_grade) return self def __setattr__(self, attr, value): raise AttributeError("'GradedTuple' object has no attribute {!r}".format(attr)) def _map(self, func: Callable[[_T], _U]) -> 'GradedTuple[_U]': return GradedTuple( tuple( func(elem) for elem in elems ) for elems in self )
[docs]class OrderedBiMap(OrderedDict[_K, _V]): """ A dict with an ``.inverse`` attribute mapping in the other direction """ def __init__(self, items): # set up the inverse mapping, bypassing our __init__ self.inverse = OrderedBiMap.__new__(type(self)) # populate both super(OrderedBiMap, self).__init__(items) super(OrderedBiMap, self.inverse).__init__([(v, k) for k, v in items]) # and complete the inverse loop self.inverse.inverse = self
class ProductFunction: def __init__(self, ga): self._ga = ga def __call__(self, A: Expr, B: Expr) -> Expr: """ Perform the multiplication """ raise NotImplementedError # pragma: no cover
[docs]class BladeProductFunction(ProductFunction): """ Base class for implementations of products between blade representations .. automethod:: __call__ """
[docs] def of_basis_blades(self, blade1: Symbol, blade2: Symbol) -> Expr: """ Compute the product of two basis blades """ raise NotImplementedError # pragma: no cover
@_cached_property def table_dict(self) -> lazy_dict[Tuple[Symbol, Symbol], Expr]: """ A cache of the result of :meth:`of_basis_blades` """ return lazy_dict({}, f_value=lambda b: self.of_basis_blades(*b))
[docs] def __call__(self, A: Expr, B: Expr) -> Expr: return update_and_substitute(A, B, self.table_dict)
class _SingleGradeProductFunction(BladeProductFunction): r""" Base class for all product functions :math:`\circ` which select a single grade from the geometric product, :math:`A_r \circ B_s = \left<A_rB_s\right>_{f(r, s)}. """ def _result_grade(self, grade1: int, grade2: int) -> Optional[int]: """ Get the grade to select from the geometric product, for a given dot product """ raise NotImplementedError def _of_basis_blades_ortho(self, blade1: Symbol, blade2: Symbol): # dot (|), left (<), and right (>) products # dot product for orthogonal basis index1 = self._ga.indexes_to_blades_dict.inverse[blade1] index2 = self._ga.indexes_to_blades_dict.inverse[blade2] index = list(index1 + index2) grade = self._result_grade(len(index1), len(index2)) if grade is None: return zero n = len(index) sgn = S(1) result = S(1) ordered = False while n > grade: ordered = True i2 = 1 while i2 < n: i1 = i2 - 1 index1 = index[i1] index2 = index[i2] if index1 == index2: n -= 2 if n < grade: return zero result *= self._ga.g[index1, index1] index = index[:i1] + index[i2 + 1:] elif index1 > index2: ordered = False index[i1] = index2 index[i2] = index1 sgn = -sgn i2 += 1 else: i2 += 1 if ordered: break if n > grade: return zero else: if index == []: return sgn * result else: return sgn * result * self._ga.indexes_to_blades_dict[tuple(index)] def _of_basis_blades_non_ortho(self, blade1: Symbol, blade2: Symbol) -> Expr: # dot product of basis blades if basis vectors are non-orthogonal # inner (|), left (<), and right (>) products of basis blades # grades of input blades grade1 = self._ga.blades_to_grades_dict[blade1] grade2 = self._ga.blades_to_grades_dict[blade2] grade = self._result_grade(grade1, grade2) if grade is None: return zero # Need base rep for blades since that is all we can multiply base1 = self._ga.blade_expansion_dict[blade1] base2 = self._ga.blade_expansion_dict[blade2] # geometric product of basis blades base12 = self._ga.mul(base1, base2) # blade rep of geometric product blade12 = self._ga.base_to_blade_rep(base12) # decompose geometric product by grades grade_dict = self._ga.grade_decomposition(blade12) return grade_dict.get(grade, zero) def of_basis_blades(self, blade1: Symbol, blade2: Symbol) -> Expr: if self._ga.is_ortho: return self._of_basis_blades_ortho(blade1, blade2) else: return self._of_basis_blades_non_ortho(blade1, blade2) def __call__(self, A: Expr, B: Expr) -> Expr: return update_and_substitute(A, B, self.table_dict) class _HestenesDotFunction(_SingleGradeProductFunction): def _result_grade(self, grade1: int, grade2: int) -> Optional[int]: if grade1 == 0 or grade2 == 0: return None return abs(grade1 - grade2) class _ScalarProductFunction(_SingleGradeProductFunction): def _result_grade(self, grade1: int, grade2: int) -> Optional[int]: return 0 class _LeftContractFunction(_SingleGradeProductFunction): def _result_grade(self, grade1: int, grade2: int) -> Optional[int]: grade = grade2 - grade1 if grade < 0: return None return grade class _RightContractFunction(_SingleGradeProductFunction): def _result_grade(self, grade1: int, grade2: int) -> Optional[int]: grade = grade1 - grade2 if grade < 0: return None return grade class _WedgeProductFunction(_SingleGradeProductFunction): def _result_grade(self, grade1: int, grade2: int) -> Optional[int]: grade = grade1 + grade2 if grade > self._ga.n: return None return grade # override the base class method with a faster approach def of_basis_blades(self, blade1: Symbol, blade2: Symbol) -> Expr: # outer (^) product of basis blades # this method works for both orthogonal and non-orthogonal basis index1 = self._ga.indexes_to_blades_dict.inverse[blade1] index2 = self._ga.indexes_to_blades_dict.inverse[blade2] index12 = list(index1 + index2) grade = self._result_grade(len(index1), len(index2)) if grade is None: return zero sgn, wedge12 = Ga.blade_reduce(index12) if sgn != 0: return sgn * self._ga.indexes_to_blades_dict[tuple(wedge12)] else: return S(0) class _GeometricProductFunction(BladeProductFunction): def of_basis_blades(self, blade1: Symbol, blade2: Symbol) -> Expr: # geometric (*) product for orthogonal basis if self._ga.is_ortho: index1 = self._ga.indexes_to_blades_dict.inverse[blade1] index2 = self._ga.indexes_to_blades_dict.inverse[blade2] blade_index = list(index1 + index2) repeats = [] sgn = 1 for i in range(1, len(blade_index)): save = blade_index[i] j = i while j > 0 and blade_index[j - 1] > save: sgn = -sgn blade_index[j] = blade_index[j - 1] j -= 1 blade_index[j] = save if blade_index[j] == blade_index[j - 1]: repeats.append(save) result = S(sgn) for i in repeats: blade_index.remove(i) blade_index.remove(i) result *= self._ga.g[i, i] if len(blade_index) > 0: result *= self._ga.indexes_to_blades_dict[tuple(blade_index)] return result else: base1 = self._ga.blade_to_base_rep(blade1) base2 = self._ga.blade_to_base_rep(blade2) base12 = self._ga.basic_mul(base1, base2) return self._ga.base_to_blade_rep(base12)
[docs]class BaseProductFunction(ProductFunction): """ Base class for implementations of products between base blade representations """
[docs] def of_basis_bases(self, base1: Symbol, base2: Symbol) -> Expr: """ Compute the product of two basis bases """ raise NotImplementedError
class _BaseGeometricProductFunction(BaseProductFunction): def of_basis_bases(self, base1: Symbol, base2: Symbol) -> Expr: # geometric product of bases for non-orthogonal basis vectors index = self._ga.indexes_to_bases_dict.inverse[base1] + self._ga.indexes_to_bases_dict.inverse[base2] coefs, indexes = self._ga.reduce_basis(index) return sum(( coef * self._ga.indexes_to_bases_dict[tuple(index)] for coef, index in zip(coefs, indexes) ), S(0)) @_cached_property def table_dict(self) -> OrderedDict[Mul, Expr]: return OrderedDict( (base1 * base2, self.of_basis_bases(base1, base2)) for base1 in self._ga.bases.flat for base2 in self._ga.bases.flat ) def __call__(self, A: Expr, B: Expr) -> Expr: # geometric product (*) of base representations # only multiplicative operation to assume A and B are in base representation AxB = expand(A * B) AxB = nc_subs(AxB, self.table_dict.items()) return expand(AxB)
[docs]class Ga(metric.Metric): r""" The vector space (basis, metric, derivatives of basis vectors) is defined by the base class :class:`~galgebra.metric.Metric`. The instanciating the class :class:`Ga` constructs the geometric algebra of the vector space defined by the metric. The construction includes the multivector bases, multiplication tables or functions for the geometric (``*``), inner (``|``), outer (``^``) products, plus the left (``<``) and right (``>``) contractions. The geometric derivative operator and any required connections for the derivative are also calculated. Except for the geometric product in the case of a non-orthogonal set of basis vectors all products and connections (if needed) are calculated when needed and place in dictionaries (lists of tuples) to be used when needed. This greatly speeds up evaluations of multivector expressions over previous versions of this code since the products of multivector bases and connection are not calculated unless they are actually needed in the current calculation. Only instantiate the :class:`Ga` class via the :class:`~galgebra.mv.Mv` class or any use of enhanced printing (text or latex) will cause the bases and multiplication table entries to be incorrectly labeled . .. rubric:: Inherited from Metric class .. autosummary:: ~galgebra.metric.Metric.g ~galgebra.metric.Metric.g_inv ~galgebra.metric.Metric.norm ~galgebra.metric.Metric.coords ~galgebra.metric.Metric.is_ortho ~galgebra.metric.Metric.connect_flg ~galgebra.metric.Metric.basis ~galgebra.metric.Metric.r_symbols ~galgebra.metric.Metric.n ~galgebra.metric.Metric.n_range ~galgebra.metric.Metric.de .. rubric:: Basis, basis bases, and basis blades data structures .. autosummary:: ~galgebra.ga.Ga.indexes ~galgebra.ga.Ga.bases ~galgebra.ga.Ga.blades ~galgebra.ga.Ga.mv_blades ~galgebra.ga.Ga.coord_vec ~galgebra.ga.Ga.indexes_to_blades_dict ~galgebra.ga.Ga.indexes_to_bases_dict .. rubric:: Multiplication data structures The following properties contain implementations of the operators ``*``, ``^``, ``|``, ``<``, and ``>``: .. autosummary:: ~galgebra.ga.Ga.mul ~galgebra.ga.Ga.wedge ~galgebra.ga.Ga.hestenes_dot ~galgebra.ga.Ga.left_contract ~galgebra.ga.Ga.right_contract While behaving like functions, each of these also has a :attr:`BladeProductFunction.table_dict` attribute, which contains a lazy lookup table of the products of basis blades. For non-orthogonal algebras, there is one additional operation, this one mapping bases instead of blades. Unlike the others, the ``table_dict`` attribute is pre-computed: .. autosummary:: ~galgebra.ga.Ga.basic_mul .. rubric:: Reciprocal basis data structures .. autosummary:: ~galgebra.metric.Metric.r_symbols ~galgebra.ga.Ga.r_basis ~galgebra.ga.Ga.r_basis_dict ~galgebra.ga.Ga.r_basis_mv .. rubric:: Derivative data structures .. attribute:: de Derivatives of basis functions. Two dimensional list. First entry is differentiating coordinate index. Second entry is basis vector index. Quantities are linear combinations of basis vector symbols. .. attribute:: grad Geometric derivative operator from left. ``grad*F`` returns multivector derivative, ``F*grad`` returns differential operator. .. attribute:: rgrad Geometric derivative operator from right. ``rgrad*F`` returns differential operator, ``F*rgrad`` returns multivector derivative. .. Sphinx adds all the other members below this docstring .. rubric:: Other members .. attribute:: dot_mode Controls the behavior of :meth:`dot` ======= ====================== value ``dot`` aliases ======= ====================== ``'|'`` :meth:`hestenes_dot` ``'<'`` :meth:`left_contract` ``'>'`` :meth:`right_contract` ======= ====================== """ dual_mode_value = 'I+' dual_mode_lst = ['+I', 'I+', '-I', 'I-', '+Iinv', 'Iinv+', '-Iinv', 'Iinv-'] presets = {'o3d': 'x,y,z:[1,1,1]:[1,1,0]', 'cyl3d': 'r,theta,z:[1,r**2,1]:[1,1,0]:norm=True', 'sph3d': 'r,theta,phi:[1,X[0]**2,X[0]**2*cos(X[1])**2]:[1,1,0]:norm=True', 'para3d': 'u,v,z:[u**2+v**2,u**2+v**2,1]:[1,1,0]:norm=True'}
[docs] @staticmethod def dual_mode(mode='I+'): """ Sets mode of multivector dual function for all geometric algebras in users program. If Ga.dual_mode(mode) not called the default mode is ``'I+'``. ===== ============ mode return value ===== ============ +I I*self -I -I*self I+ self*I I- -self*I +Iinv Iinv*self -Iinv -Iinv*self Iinv+ self*Iinv Iinv- -self*Iinv ===== ============ """ if mode not in Ga.dual_mode_lst: raise ValueError('mode = ' + mode + ' not allowed for Ga.dual_mode.') Ga.dual_mode_value = mode
[docs] @staticmethod def com(A, B): r""" Calculate commutator of multivectors :math:`A` and :math:`B`. Returns :math:`(AB-BA)/2`. Additionally, commutator and anti-commutator operators are defined by .. math:: \begin{aligned} \texttt{A >> B} \equiv & {\displaystyle\frac{AB - BA}{2}} \\ \texttt{A << B} \equiv & {\displaystyle\frac{AB + BA}{2}}. \end{aligned} """ return half * (A * B - B * A)
[docs] @staticmethod def build(*args, **kwargs): """ Static method to instantiate geometric algebra and return geometric algebra, basis vectors, and grad operator as a tuple. """ GA = Ga(*args, **kwargs) basis = list(GA.mv()) return tuple([GA] + basis)
@staticmethod def preset(setting, root='e', debug=False): if setting not in Ga.presets: raise ValueError(str(setting) + 'not in Ga.presets.') set_lst = Ga.presets[setting].split(':') X = symbols(set_lst[0], real=True) g = eval(set_lst[1]) simps = eval(set_lst[2]) args = [root] kwargs = {'g': g, 'coords': X, 'debug': debug, 'I': True, 'gsym': False} if len(set_lst) > 3: args_lst = set_lst[-1].split(';') for arg in args_lst: [name, value] = arg.split('=') kwargs[name] = eval(value) Ga.set_simp(*simps) return Ga(*args, **kwargs) def __hash__(self): return hash(self.name) def __eq__(self, ga): return self.name == ga.name def __init__(self, bases, *, wedge=True, **kwargs): """ Parameters ---------- bases : Passed as ``basis`` to ``Metric``. wedge : Use ``^`` symbol to print basis blades **kwargs : See :class:`galgebra.metric.Metric`. """ self.wedge_print = wedge metric.Metric.__init__(self, bases, **kwargs) self.par_coords = None if self.debug: self._print_basis_and_blade_debug() self.dot_mode = '|' if self.coords is not None: self.coords = list(self.coords) self.e = mv.Mv(self.blades.flat[-1], ga=self) # Pseudo-scalar for geometric algebra if self.coords is not None: self._update_de_from_rbasis() self._build_grads() if self.connect_flg: self._build_connection() # Calculate normalized pseudo scalar (I**2 = +/-1) self.sing_flg = False if self.e_sq.is_number: if self.e_sq == S(0): self.sing_flg = True print('!!!!If I**2 = 0, I cannot be normalized!!!!') # raise ValueError('!!!!If I**2 = 0, I cannot be normalized!!!!') if self.e_sq > S(0): self.i = self.e/sqrt(self.e_sq) self.i_inv = self.i else: # I**2 = -1 self.i = self.e/sqrt(-self.e_sq) self.i_inv = -self.i else: if self.Isq == '+': # I**2 = 1 self.i = self.e/sqrt(self.e_sq) self.i_inv = self.i else: # I**2 = -1 self.i = self.e/sqrt(-self.e_sq) self.i_inv = -self.i if self.debug: print('Exit Ga.__init__()') self._agrads = {} # cache of gradient operator with respect to vector a self.dslot = -1 # args slot for dervative, -1 for coordinates # mystery state used by the Mlt class self._mlt_a = [] # List of dummy vectors for Mlt calculations self._mlt_acoefs = [] # List of dummy vectors coefficients self._mlt_pdiffs = [] # List of lists dummy vector coefficients self._XOX = self.mv('XOX', 'vector') # cached vector for use in is_versor @_cached_property def coord_vec(self) -> Expr: """ Linear combination of coordinates and basis vectors. For example in orthogonal 3D :math:`x*e_x+y*e_y+z*e_z`. """ if self.coords is None: raise ValueError("Ga with no coords has no coord_vec") return sum([coord * base for coord, base in zip(self.coords, self.basis)]) def _reciprocal_of_basis_blade(self, blade: Symbol) -> Expr: r""" Compute the reciprocal :math:`e^I` of a basis blade :math:`e_I`. This is a blade :math:`e^I` such that :math:`\left<\e^Ie_J\right> = \delta_I^J` (:cite:`Hestenes`, eq 3.19). """ index = self.indexes_to_blades_dict.inverse[blade] r_blade = reduce(self.wedge, [ self.r_basis[i] for i in index[::-1] ], S.One) r_blade = r_blade.simplify() # normalize at the end if not self.is_ortho: # r_basis is already normalized if is_ortho is true r_blade /= (self.e_sq**len(index)) return r_blade @_cached_property def _reciprocal_blade_dict(self) -> lazy_dict: """ A dictionary mapping basis blades to their reciprocal blades. """ return lazy_dict({}, self._reciprocal_of_basis_blade)
[docs] def make_grad(self, a: Union[_mv.Mv, Sequence[Expr]], cmpflg: bool = False) -> mv.Dop: r""" Obtain a gradient operator with respect to the multivector a, :math:`\bm{\nabla}_a`.""" if not isinstance(a, mv.Mv): # This might be needed for Mlt, let's leave it till we're sure. # Convert to a multivector. a = sum((ai * ei for ai, ei in zip(a, self.mv_basis)), self.mv(S.Zero)) cache_key = (a, cmpflg) if cache_key in self._agrads: return self._agrads[cache_key] # make the grad and cache it grad_a = mv.Dop([ (self.mv(self._reciprocal_blade_dict[base]), dop.Pdop({coef: 1})) for coef, base in metric.linear_expand_terms(a.obj) ], ga=self, cmpflg=cmpflg) self._agrads[cache_key] = grad_a return grad_a
def __str__(self): return self.name def E(self) -> mv.Mv: # Unnoromalized pseudo-scalar return self.e def I(self) -> mv.Mv: # Noromalized pseudo-scalar return self.i @property def mv_I(self) -> _mv.Mv: # This exists for backwards compatibility. Note this is not `I()`! # galgebra 0.4.5 warnings.warn( "`ga.mv_I` is deprecated, use `ga.E()` instead, or perhaps `ga.I()`", DeprecationWarning, stacklevel=2) # default pseudoscalar return self.E() @property def mv_x(self) -> _mv.Mv: # This exists for backwards compatibility. # galgebra 0.4.5 warnings.warn( "`ga.mv_x` is deprecated, use `ga.mv(your_name, 'vector')` instead", DeprecationWarning, stacklevel=2) # testing vectors return mv.Mv('XxXx', 'vector', ga=self) def X(self): # galgebra 0.5.0 warnings.warn( "ga.X() is deprecated, use `ga.coord_vec` instead", DeprecationWarning, stacklevel=2) return self.coord_vec @property def Pdiffs(self) -> Dict[Symbol, _dop.Pdop]: # galgebra 0.4.5 warnings.warn( "ga.Pdiffs[x] is deprecated, use `Pdop(x)` instead", DeprecationWarning, stacklevel=2) return {x: dop.Pdop(x) for x in self.coords} @property def sPds(self) -> Dict[Symbol, _dop.Sdop]: # galgebra 0.4.5 warnings.warn( "ga.sPds[x] is deprecated, use `Sdop(x)` instead", DeprecationWarning, stacklevel=2) return {x: dop.Sdop(x) for x in self.coords} @property def Pdop_identity(self) -> _dop.Pdop: # galgebra 0.4.5 warnings.warn( "ga.Pdop_identity is deprecated, use `Pdop({})` instead", DeprecationWarning, stacklevel=2) return dop.Pdop({}) @property def blades_lst(self) -> List[Symbol]: # galgebra 0.5.0 warnings.warn( "ga.blades_lst is deprecated, use `ga.blades.flat[1:]` instead", DeprecationWarning, stacklevel=2) return list(self.blades.flat[1:]) @property def bases_lst(self) -> List[Symbol]: # galgebra 0.5.0 warnings.warn( "ga.bases_lst is deprecated, use `ga.bases.flat[1:]` instead", DeprecationWarning, stacklevel=2) return list(self.bases.flat[1:]) @property def indexes_lst(self) -> List[Tuple[int, ...]]: # galgebra 0.5.0 warnings.warn( "ga.blades_lst is deprecated, use `ga.indexes.flat[1:]` instead", DeprecationWarning, stacklevel=2) return list(self.indexes.flat[1:])
[docs] def mv(self, root=None, *args, **kwargs) -> Union[_mv.Mv, Tuple[_mv.Mv, ...]]: """ Instanciate and return a multivector for this, 'self', geometric algebra. """ if root is None: # Return ga basis and compute grad and rgrad return self.mv_basis # ensure that ga is not already in kwargs kwargs = dict(ga=self, **kwargs) if not isinstance(root, str): return mv.Mv(root, *args, **kwargs) if ' ' in root and ' ' not in args[0]: root_lst = root.split(' ') mv_lst = [] for root in root_lst: mv_lst.append(mv.Mv(root, *args, **kwargs)) return tuple(mv_lst) if ' ' in root and ' ' in args[0]: root_lst = root.split(' ') mvtype_lst = args[0].split(' ') if len(root_lst) != len(mvtype_lst): raise ValueError('In Ga.mv() for multiple multivectors and ' + 'multivector types incompatible args ' + str(root_lst) + ' and ' + str(mvtype_lst)) mv_lst = [] for root, mv_type in zip(root_lst, mvtype_lst): args_list = list(args) args_list[0] = mv_type args = tuple(args_list) mv_lst.append(mv.Mv(root, *args, **kwargs)) return tuple(mv_lst) return mv.Mv(root, *args, **kwargs)
[docs] def mvr(self, norm: bool = True) -> Tuple[_mv.Mv, ...]: r""" Returns tumple of reciprocal basis vectors. If norm=True or basis vectors are orthogonal the reciprocal basis is normalized in the sense that .. math:: {i}\cdot e^{j} = \delta_{i}^{j}. If the basis is not orthogonal and norm=False then .. math:: e_{i}\cdot e^{j} = I^{2}\delta_{i}^{j}. """ if norm and not self.is_ortho: return tuple([self.r_basis_mv[i] / self.e_sq for i in self.n_range]) else: return tuple(self.r_basis_mv)
[docs] def bases_dict(self, prefix: str = None) -> Dict[str, Symbol]: ''' returns a dictionary mapping basis element names to their MultiVector instances, optionally for specific grades if you are lazy, you might do this to populate your namespace with the variables of a given layout. >>> locals().update(ga.bases()) ''' if prefix is None: prefix = 'e' bl = self.blades.flat[1:] # do not include the scalar, which is not named var_names = [prefix+''.join([k for k in str(b) if k.isdigit()]) for b in bl] return {key: val for key, val in zip(var_names, bl)}
def _build_grads(self) -> None: if not self.is_ortho: r_basis = [x / self.e_sq for x in self.r_basis_mv] else: r_basis = self.r_basis_mv if self.norm: r_basis = [x / e_norm for x, e_norm in zip(self.r_basis_mv, self.e_norm)] pdx = [dop.Pdop(x) for x in self.coords] self.grad = mv.Dop(r_basis, pdx, ga=self) self.rgrad = mv.Dop(r_basis, pdx, ga=self, cmpflg=True) def grads(self) -> Tuple[_mv.Dop, _mv.Dop]: if self.coords is None: raise ValueError("Ga must have been initialized with coords to compute grads") return self.grad, self.rgrad
[docs] def pdop(self, *args, **kwargs) -> _dop.Pdop: """ Shorthand to construct a :class:`~galgebra.dop.Pdop` """ # galgebra 0.4.5 warnings.warn( "`ga.pdop` is deprecated, use `Pdop()` directly.", DeprecationWarning, stacklevel=2) return dop.Pdop(*args, **kwargs)
[docs] def dop(self, *args, **kwargs) -> _mv.Dop: """ Shorthand to construct a :class:`~galgebra.mv.Dop` for this algebra """ return mv.Dop(*args, ga=self, **kwargs)
[docs] def sdop(self, *args, **kwargs) -> _dop.Sdop: """ Shorthand to construct a :class:`~galgebra.dop.Sdop` """ # galgebra 0.4.5 warnings.warn( "`ga.sdop` is deprecated, use `Sdop()` directly.", DeprecationWarning, stacklevel=2) return dop.Sdop(*args, **kwargs)
@property def lt_coords(self) -> List[Expr]: # galgebra 0.5.0 warnings.warn( "`ga.lt_coords` is deprecated, use the identical `ga.coords`.", DeprecationWarning, stacklevel=2) return self.coords @property def lt_x(self) -> Expr: # galgebra 0.5.0 warnings.warn( "`ga.lt_x` is deprecated, use the identical `ga.coord_vec`.", DeprecationWarning, stacklevel=2) return self.coord_vec
[docs] def lt(self, *args, **kwargs): """ Instanciate and return a linear transformation for this, 'self', geometric algebra. """ return lt.Lt(*args, ga=self, **kwargs)
[docs] def sm(self, *args, **kwargs) -> 'Sm': """ Instanciate and return a submanifold for this geometric algebra. See :class:`Sm` for instantiation inputs. """ return Sm(*args, ga=self, **kwargs)
def parametric(self, coords: List[Expr]) -> None: if not isinstance(coords, list): raise TypeError('In Ga.parametric coords = ' + str(coords) + ' is not a list.') if len(coords) != self.n: raise ValueError('In Ga.parametric number of parametric functions' + ' not equal to number of coordinates.') self.par_coords = {} for coord, par_coord in zip(self.coords, coords): self.par_coords[coord] = par_coord def basis_vectors(self) -> Tuple[Symbol, ...]: return tuple(self.basis) def _build_basis_base_symbol(self, base_index: Tuple[int, ...]) -> Symbol: """ Build a symbol used for the `base_rep` from the given tuple """ if not base_index: return S(1) return BasisBaseSymbol(*(self.basis[i] for i in base_index)) def _build_basis_blade_symbol(self, base_index: Tuple[int, ...]) -> Symbol: """ Build a symbol used for the `blade_rep` from the given tuple """ if self.wedge_print: return BasisBladeSymbol(*(self.basis[i] for i in base_index)) else: return BasisBladeNoWedgeSymbol(*(self.basis[i] for i in base_index)) def _print_basis_and_blade_debug(self) -> None: printer.oprint('indexes', self.indexes, 'list(indexes)', self.indexes.flat, 'blades', self.blades, 'list(blades)', self.blades.flat, 'indexes_to_blades_dict', self.indexes_to_blades_dict, 'blades_to_grades_dict', self.blades_to_grades_dict, 'blade_super_scripts', self.blade_super_scripts) if not self.is_ortho: printer.oprint('bases', self.bases, 'list(bases)', self.bases.flat, 'indexes_to_bases_dict', self.indexes_to_bases_dict, 'bases_to_grades_dict', self.bases_to_grades_dict) @_cached_property def indexes(self) -> GradedTuple[Tuple[int, ...]]: """ Index tuples of basis blades """ basis_indexes = tuple(self.n_range) return GradedTuple( tuple(combinations(basis_indexes, i)) for i in range(len(basis_indexes) + 1) ) @_cached_property def blades(self) -> GradedTuple[Symbol]: r""" Basis blades symbols by grade. The bases for the multivector (geometric) algebra are formed from all combinations of the bases of the vector space, including the empty combination which is the scalars. Each base is represented as a non-commutative symbol of the form .. math:: e_{i_{1}}\wedge e_{i_{2}}\wedge ...\wedge e_{i_{r}}. where :math:`0 < i_{1} < i_{2} < ... < i_{r}` and :math:`0 < r \le n` the dimension of the vector space and :math:`0 < i_{j} \le n`. The total number of all symbols of this form is :math:`2^{n}`. These are called the blade basis for the geometric algebra and any multivector can be represented by a linears combination of these blades. The number of basis vectors that are in the symbol for the blade is call the grade of the blade. Representing the multivector as a linear combination of blades gives a blade decomposition of the multivector. There is a linear mapping from :attr:`bases` to blades and blades to bases so that one can easily convert from one representation to another. """ return self.indexes._map( lambda index: self._build_basis_blade_symbol(index)) @_cached_property def indexes_to_blades_dict(self) -> OrderedBiMap[Tuple[int, ...], Symbol]: """ Bidirectional map from index tuples (:attr:`indices`) to basis blades (:attr:`blades`) """ return OrderedBiMap(list(zip(self.indexes.flat, self.blades.flat))) @_cached_property def blades_to_grades_dict(self) -> Dict[Symbol, int]: return { blade: igrade for igrade, grade in enumerate(self.blades) for blade in grade } @_cached_property def bases(self) -> GradedTuple[Symbol]: r""" Bases (non-commutative sympy symbols) by grade. If the basis vectors are not orthogonal a second set of symbols is required in addition to the :attr:`blades`, given by: .. math:: e_{i_{1}}e_{i_{2}}...e_{i_{r}} where :math:`0 < i_{1} < i_{2} < ... < i_{r}` and :math:`0 < r \le n` the dimension of the vector space and :math:`0 < i_{j} \le n`. The total number of all symbols of this form is :math:`2^{n}`. Any multivector can be represented as a linear combination of these bases. For the case of an orthogonal set of basis vectors the bases and blades are identical, and so this attribute raises :exc:`ValueError`. """ if self.is_ortho: raise ValueError("There is no need for bases in orthogonal algebras") return self.indexes._map( lambda index: self._build_basis_base_symbol(index)) @_cached_property def indexes_to_bases_dict(self) -> OrderedBiMap[Tuple[int, ...], Symbol]: """ Bidirectional map from index tuples (:attr:`indices`) to basis bases (:attr:`bases`) """ return OrderedBiMap(list(zip(self.indexes.flat, self.bases.flat))) @_cached_property def bases_to_grades_dict(self) -> Dict[Symbol, int]: return { blade: igrade for igrade, grade in enumerate(self.bases) for blade in grade } @_cached_property def basis_super_scripts(self) -> List[str]: if self.coords is None: base0 = str(self.basis[0]) if '_' in base0: sub_index = base0.index('_') return [str(base)[sub_index + 1:] for base in self.basis] else: return [str(i + 1) for i in self.n_range] else: return [str(coord) for coord in self.coords] @_cached_property def blade_super_scripts(self) -> GradedTuple[str]: return self.indexes._map(lambda base_index: ''.join( self.basis_super_scripts[i] for i in base_index )) @_cached_property def mv_blades(self) -> GradedTuple[_mv.Mv]: """ :class:`mv.Mv` instances corresponding to :attr:`blades`. """ return self.blades._map(lambda blade: mv.Mv(blade, ga=self)) @_cached_property def mv_basis(self) -> Tuple[_mv.Mv, ...]: """ :class:`mv.Mv` instances corresponding to :attr:`basis`. """ return tuple(mv.Mv(obj, ga=self) for obj in self.basis) @property def indexes_to_bases(self): # galgebra 0.5.0 warnings.warn( "`ga.indexes_to_bases` is deprecated, use `ga.indexes_to_bases_dict.items()`", DeprecationWarning, stacklevel=2) return self.indexes_to_bases_dict.items() @property def indexes_to_blades(self): # galgebra 0.5.0 warnings.warn( "`ga.indexes_to_blades` is deprecated, use `ga.indexes_to_blades_dict.items()`", DeprecationWarning, stacklevel=2) return self.indexes_to_blades_dict.items() @property def bases_to_indexes(self): # galgebra 0.5.0 warnings.warn( "`ga.bases_to_indexes` is deprecated, use `ga.indexes_to_bases_dict.inverse.items()`", DeprecationWarning, stacklevel=2) return self.indexes_to_bases_dict.inverse.items() @property def blades_to_indexes(self): # galgebra 0.5.0 warnings.warn( "`ga.blades_to_indexes` is deprecated, use `ga.indexes_to_blades_dict.inverse.items()`", DeprecationWarning, stacklevel=2) return self.indexes_to_blades_dict.inverse.items() @property def bases_to_indexes_dict(self) -> OrderedBiMap: # galgebra 0.5.0 warnings.warn( "`ga.bases_to_indexes_dict` is deprecated, use `ga.indexes_to_bases_dict.inverse.`", DeprecationWarning, stacklevel=2) return self.indexes_to_bases_dict.inverse @property def blades_to_indexes_dict(self) -> OrderedBiMap: # galgebra 0.5.0 warnings.warn( "`ga.blades_to_indexes_dict` is deprecated, use `ga.indexes_to_blades_dict.inverse`", DeprecationWarning, stacklevel=2) return self.indexes_to_blades_dict.inverse @property def mul_table_dict(self): # galgebra 0.5.0 warnings.warn( "`ga.mul_table_dict` is deprecated, use `ga.mul.table_dict`", DeprecationWarning, stacklevel=2) return self.mul.table_dict @property def wedge_table_dict(self): # galgebra 0.5.0 warnings.warn( "`ga.wedge_table_dict` is deprecated, use `ga.wedge.table_dict`", DeprecationWarning, stacklevel=2) return self.wedge.table_dict @property def dot_table_dict(self): # galgebra 0.5.0 warnings.warn( "`ga.dot_table_dict` is deprecated, use `ga.hestenes_dot.table_dict`", DeprecationWarning, stacklevel=2) return self.hestenes_dot.table_dict @property def left_contract_table_dict(self): # galgebra 0.5.0 warnings.warn( "`ga.left_contract_table_dict` is deprecated, use `ga.left_contract.table_dict`", DeprecationWarning, stacklevel=2) return self.left_contract.table_dict @property def right_contract_table_dict(self): # galgebra 0.5.0 warnings.warn( "`ga.right_contract_table_dict` is deprecated, use `ga.right_contract.table_dict`", DeprecationWarning, stacklevel=2) return self.right_contract.table_dict def _build_connection(self): # Partial derivatives of multivector bases multiplied (*,^,|,<,>) # on left and right (True and False) by reciprocal basis vectors. self.connect = {('*', True): [], ('^', True): [], ('|', True): [], ('<', True): [], ('>', True): [], ('*', False): [], ('^', False): [], ('|', False): [], ('<', False): [], ('>', False): []} # Partial derivatives of multivector bases self._dbases = {} ######## Functions for Calculation products of blades/bases ######## # ******************* Geometric Product (*) ********************** # def geometric_product_basis_blades(self, blade12: Tuple[Symbol, Symbol]) -> Expr: # galgebra 0.5.0 warnings.warn( "`ga.geometric_product_basis_blades` is deprecated, use `ga.mul.of_basis_blades`", DeprecationWarning, stacklevel=2) return self.mul.of_basis_blades(*blade12)
[docs] def reduce_basis(self, blst): r""" Repetitively applies :meth:`reduce_basis_loop` to `blst` product representation until normal form is realized for non-orthogonal basis If the basis vectors are represented by the non- commutative symbols :math:`e_1,...,e_n` then a grade :math:`r` base is the geometric product :math:`e_{i_1}e_{i_2}\cdots e_{i_r}` where :math:`i_1<i_2<\ldots<i_r` (normal form). Then in galgebra this base is represented by a single indexed non-commutative symbol with indexes :math:`[i_1,i_2,\ldots,i_r]`. The total number of these bases in an n-dimensional vector space is :math:`2^n`. :meth:`reduce_basis` takes the geometric products of basis vectors that are not in normal form (out of order) and reduces them to a sum of bases that are in normal form (in order). It does this by recursively applying the geometric algebra formula .. math:: e_ie_j = 2(e_i \cdot e_j) - e_je_i where the scalar product :math:`e_i \cdot e_j` is obtained from the metric tensor of the vector space. This also allows one to calculate the geometric product of any two bases and grade of the geometric algebra, and form the multiplication table. """ blst = list(blst) if blst == []: # blst represents scalar blst_coef = [1] blst_expand = [[]] return blst_coef, blst_expand blst_expand = [blst] blst_coef = [1] blst_flg = [False] # reduce untill all blst revise flgs are True while not reduce(operator.and_, blst_flg): for i in range(len(blst_flg)): if not blst_flg[i]: # keep revising if revise flg is False tmp = Ga.reduce_basis_loop(self.g, blst_expand[i]) if isinstance(tmp, bool): blst_flg[i] = tmp # revision of blst_expand[i] complete elif len(tmp) == 3: # blst_expand[i] contracted blst_coef[i] = tmp[0] * blst_coef[i] blst_expand[i] = tmp[1] blst_flg[i] = tmp[2] else: # blst_expand[i] revised blst_coef[i] = -blst_coef[i] # if revision force one more pass in case revision # causes repeated index previous to revised pair of # indexes blst_flg[i] = False blst_expand[i] = tmp[3] blst_coef.append(-blst_coef[i] * tmp[0]) blst_expand.append(tmp[1]) blst_flg.append(tmp[2]) new_blst_coef = [] new_blst_expand = [] for coef, xpand in zip(blst_coef, blst_expand): if xpand in new_blst_expand: i = new_blst_expand.index(xpand) new_blst_coef[i] += coef else: new_blst_expand.append(xpand) new_blst_coef.append(coef) return new_blst_coef, new_blst_expand
[docs] @staticmethod def reduce_basis_loop(g, blst): r""" blst is a list of integers :math:`[i_{1},\ldots,i_{r}]` representing the geometric product of r basis vectors :math:`a_{{i_1}}\cdots a_{{i_r}}`. :meth:`reduce_basis_loop` searches along the list :math:`[i_{1},\ldots,i_{r}]` untill it finds :math:`i_{j} = i_{j+1}` and in this case contracts the list, or if :math:`i_{j} > i_{j+1}` it revises the list (:math:`\sim i_{j}` means remove :math:`i_{j}` from the list) * Case 1: If :math:`i_{j} = i_{j+1}`, return :math:`a_{i_{j}}^2` and :math:`[i_{1},\ldots,\sim i_{j},\sim i_{j+1},\ldots,i_{r}]` * Case 2: If :math:`i_{j} > i_{j+1}`, return :math:`a_{i_{j}}.a_{i_{j+1}}`, :math:`[i_{1},\ldots,\sim i_{j},\sim i_{j+1},\ldots,i_{r}]`, and :math:`[i_{1},\ldots,i_{j+1},i_{j},\ldots,i_{r}]` This is an implementation of the formula .. math:: e_i e_j = 2(e_i \cdot e_j) - e_j e_i Where :math:`e_i` and :math:`e_j` are basis vectors. """ nblst = len(blst) # number of basis vectors if nblst <= 1: return True # a scalar or vector is already reduced for jstep in range(1, nblst): istep = jstep - 1 if blst[istep] == blst[jstep]: # basis vectorindex is repeated i = blst[istep] # save basis vector index if len(blst) > 2: blst = blst[:istep] + blst[jstep + 1:] # contract blst else: blst = [] if len(blst) <= 1 or jstep == nblst - 1: blst_flg = True # revision of blst is complete else: blst_flg = False # more revision needed return g[i, i], blst, blst_flg if blst[istep] > blst[jstep]: # blst not in normal order blst1 = blst[:istep] + blst[jstep + 1:] # contract blst a1 = 2 * g[blst[jstep], blst[istep]] # coef of contraction blst = blst[:istep] + [blst[jstep]] + [blst[istep]] + blst[jstep + 1:] # revise blst if len(blst1) <= 1: blst1_flg = True # revision of blst is complete else: blst1_flg = False # more revision needed return a1, blst1, blst1_flg, blst return True # revision complete, blst in normal order
# ****************** Outer/wedge (^) product ********************* #
[docs] @staticmethod def blade_reduce(lst: List[int]) -> Tuple[int, Optional[List[int]]]: """ Reduce wedge product of basis vectors to normal order. `lst` is a list of indicies of basis vectors. blade_reduce sorts the list and determines if the overall number of exchanges in the list is odd or even, returning sign changes (``sgn``) and sorted list. If any two indicies in list are equal (wedge product is zero) ``sgn = 0`` and ``lst = None`` are returned. """ sgn = S(1) for i in range(1, len(lst)): save = lst[i] j = i while j > 0 and lst[j - 1] > save: sgn = -sgn lst[j] = lst[j - 1] j -= 1 lst[j] = save if lst[j] == lst[j - 1]: return S(0), None return sgn, lst
def wedge_product_basis_blades(self, blade12: Tuple[Symbol, Symbol]) -> Expr: # galgebra 0.5.0 warnings.warn( "`ga.wedge_product_basis_blades` is deprecated, use `ga.wedge.of_basis_blades`", DeprecationWarning, stacklevel=2) return self.wedge.of_basis_blades(*blade12) # ***** Dot (|) product, reft (<) and right (>) contractions ***** # def _dot_product_method(self, mode: str) -> _SingleGradeProductFunction: if mode == '|': return self.hestenes_dot elif mode == '<': return self.left_contract elif mode == '>': return self.right_contract else: raise ValueError('mode={!r} not allowed'.format(mode)) def dot_product_basis_blades(self, blade12: Tuple[Symbol, Symbol], mode: str) -> Expr: # galgebra 0.5.0 warnings.warn( "`ga.dot_product_basis_blades` is deprecated, use `ga.<which-dot>.of_basis_blades` " "where `<which-dot>` is one of `hestenes_dot`, `left_contract`, and `right_contract`", DeprecationWarning, stacklevel=2) return self._dot_product_method(mode)._of_basis_blades_ortho(*blade12) def non_orthogonal_dot_product_basis_blades(self, blade12: Tuple[Symbol, Symbol], mode: str) -> Expr: # galgebra 0.5.0 warnings.warn( "`ga.non_orthogonal_dot_product_basis_blades` is deprecated, use `ga.<which-dot>.of_basis_blades` " "where `<which-dot>` is one of `hestenes_dot`, `left_contract`, and `right_contract`", DeprecationWarning, stacklevel=2) return self._dot_product_method(mode)._of_basis_blades_non_ortho(*blade12) ############# Non-Orthogonal Tables and Dictionaries ############### @property def basic_mul_table_dict(self) -> OrderedDict[Mul, Expr]: # galgebra 0.5.0 warnings.warn( "`ga.basic_mul_table_dict` is deprecated, use `ga.mul.table_dict`", DeprecationWarning, stacklevel=2) return self.basic_mul.table_dict @property def basic_mul_table(self): # galgebra 0.5.0 warnings.warn( "`ga.basic_mul_table` is deprecated, use `ga.basic_mul.table_dict.items()`", DeprecationWarning, stacklevel=2) return list(self.basic_mul.table_dict.items()) @property def basic_mul_keys(self): # galgebra 0.5.0 warnings.warn( "`ga.basic_mul_keys` is deprecated, use `ga.basic_mul.table_dict.keys()`", DeprecationWarning, stacklevel=2) return list(self.basic_mul.table_dict.keys()) @property def basic_mul_values(self): # galgebra 0.5.0 warnings.warn( "`ga.basic_mul_values` is deprecated, use `ga.basic_mul.table_dict.values()`", DeprecationWarning, stacklevel=2) return list(self.basic_mul.table_dict.values()) def non_orthogonal_bases_products(self, base12: Tuple[Symbol, Symbol]) -> Expr: # galgebra 0.5.0 warnings.warn( "`ga.non_orthogonal_bases_products` is deprecated, use `ga.basic_mul.of_basis_bases`", DeprecationWarning, stacklevel=2) return self.basic_mul.of_basis_bases(*base12) @_cached_property def blade_expansion_dict(self) -> OrderedDict[Symbol, Expr]: """ dictionary expanding blade basis in terms of base basis """ blade_expansion_dict = OrderedDict() for blade, index in zip(self.blades.flat, self.indexes.flat): grade = len(index) if grade <= 1: blade_expansion_dict[blade] = blade else: a = self.indexes_to_blades_dict[index[:1]] A = self.indexes_to_blades_dict[index[1:]] Aexpand = blade_expansion_dict[A] # Formula for outer (^) product of a vector and grade-r multivector # a^A_{r} = (a*A + (-1)^{r}*A*a)/2 # The folowing evaluation takes the most time for setup it is the due to # the substitution required for the multiplications a_W_A = half * (self.basic_mul(a, Aexpand) - ((-1) ** grade) * self.basic_mul(Aexpand, a)) blade_expansion_dict[blade] = expand(a_W_A) if self.debug: print('blade_expansion_dict =', blade_expansion_dict) return blade_expansion_dict @_cached_property def base_expansion_dict(self) -> OrderedDict[Symbol, Expr]: """ dictionary expanding base basis in terms of blade basis """ base_expansion_dict = OrderedDict() for base, blade, index in zip(self.bases.flat, self.blades.flat, self.indexes.flat): grade = len(index) if grade <= 1: base_expansion_dict[base] = base else: # back substitution of tridiagonal system tmp = self.blade_expansion_dict[blade] tmp = tmp.subs(base, -blade) tmp = -tmp.subs(base_expansion_dict) base_expansion_dict[base] = expand(tmp) if self.debug: print('base_expansion_dict =', self.base_expansion_dict) return base_expansion_dict @property def base_expansion(self): # galgebra 0.5.0 warnings.warn( "`ga.base_expansion` is deprecated, use `ga.base_expansion_dict.items()`", DeprecationWarning, stacklevel=2) return list(self.base_expansion_dict.items()) @property def blade_expansion(self): # galgebra 0.5.0 warnings.warn( "`ga.blade_expansion` is deprecated, use `ga.blade_expansion_dict.items()`", DeprecationWarning, stacklevel=2) return list(self.blade_expansion_dict.items()) def base_to_blade_rep(self, A): if self.is_ortho: return A else: # return expand(A).subs(self.base_expansion_dict) return nc_subs(expand(A), self.base_expansion_dict.items()) def blade_to_base_rep(self, A): if self.is_ortho: return A else: # return expand(A).subs(self.blade_expansion_dict) return nc_subs(expand(A), self.blade_expansion_dict.items()) ###### Products (*,^,|,<,>) for multivector representations ######## @_cached_property def basic_mul(self) -> BaseProductFunction: r""" The geometic product of objects in base form, :math:`A B` """ if self.is_ortho: raise ValueError("No need for this operation for orthogonal algebras") return _BaseGeometricProductFunction(self) def Mul(self, A: Expr, B: Expr, mode: str = '*') -> Expr: # Unifies all products into one function if mode == '*': return self.mul(A, B) elif mode == '^': return self.wedge(A, B) elif mode == '|': return self.hestenes_dot(A, B) elif mode == '<': return self.left_contract(A, B) elif mode == '>': return self.right_contract(A, B) else: raise ValueError('Unknown multiplication operator {!r}', mode) @_cached_property def mul(self) -> BladeProductFunction: r""" The geometric product, :math:`A B` """ return _GeometricProductFunction(self) @_cached_property def wedge(self) -> BladeProductFunction: r""" The wedge product, :math:`A \wedge B` """ return _WedgeProductFunction(self) @_cached_property def hestenes_dot(self) -> BladeProductFunction: r""" The hestenes dot product, :math:`A \bullet B` """ return _HestenesDotFunction(self) @_cached_property def scalar_product(self) -> BladeProductFunction: r""" The scalar product, :math:`A * B` """ return _ScalarProductFunction(self) @_cached_property def left_contract(self) -> BladeProductFunction: r""" The left contraction, :math:`A \rfloor B` """ return _LeftContractFunction(self) @_cached_property def right_contract(self) -> BladeProductFunction: r""" The right contraction, :math:`A \lfloor B` """ return _RightContractFunction(self)
[docs] def dot(self, A: Expr, B: Expr) -> Expr: r""" Inner product ``|``, ``<``, or ``>``. The :attr:`dot_mode` attribute determines which of these is used. """ # forbid something silly like setting dot_mode to the wedge or geometric # product if self.dot_mode in '^*': raise ValueError('"' + str(self.dot_mode) + '" not a legal mode in dot') return self.Mul(A, B, mode=self.dot_mode)
######################## Helper Functions ##########################
[docs] def grade_decomposition(self, A: _MaybeMv) -> Dict[int, _MaybeMv]: """ Returns dictionary with grades as keys of grades of A. For example if A is a rotor the dictionary keys would be 0 and 2. For a vector the single key would be 1. Note A can be input as a multivector or an multivector object (sympy expression). If A is a multivector the dictionary entries are multivectors. If A is a sympy expression (in this case a linear combination of non-commutative symbols) the dictionary entries are sympy expressions. """ if isinstance(A, mv.Mv): A.blade_rep() A.characterise_Mv() Aobj = expand(A.obj) else: Aobj = A coefs, blades = metric.linear_expand(Aobj) grade_dict = {} for coef, blade in zip(coefs, blades): if blade == one: if 0 in list(grade_dict.keys()): grade_dict[0] += coef else: grade_dict[0] = coef else: grade = self.blades_to_grades_dict[blade] if grade in grade_dict: grade_dict[grade] += coef * blade else: grade_dict[grade] = coef * blade if isinstance(A, mv.Mv): for grade in list(grade_dict.keys()): grade_dict[grade] = self.mv(grade_dict[grade]) return grade_dict
[docs] def split_multivector(self, A: _MaybeMv) -> Tuple[Union[Expr, int], Union[Expr, int]]: """ Split multivector :math:`A` into commutative part :math:`a` and non-commutative part :math:`A'` so that :math:`A = a+A'` """ if isinstance(A, mv.Mv): return self.split_multivector(A.obj) else: A = expand(A) if isinstance(A, Add): a = sum([x for x in A.args if x.is_commutative]) Ap = sum([x for x in A.args if not x.is_commutative]) return (a, Ap) elif isinstance(A, Symbol): if A.is_commutative: return (A, 0) else: return (0, A) else: if A.is_commutative: return (A, 0) else: return (0, A)
[docs] def remove_scalar_part(self, A: _MaybeMv) -> Union[Expr, int]: """ Return non-commutative part (sympy object) of ``A.obj``. """ if isinstance(A, mv.Mv): return self.remove_scalar_part(A.obj) else: if isinstance(A, Add): A = expand(A) return sum([x for x in A.args if not x.is_commutative]) elif isinstance(A, Symbol): if A.is_commutative: return 0 else: return A else: if A.is_commutative: return 0 else: return A
def scalar_part(self, A: _MaybeMv) -> Union[Expr, int]: if isinstance(A, mv.Mv): return self.scalar_part(A.obj) else: A = expand(A) if isinstance(A, Add): return sum([x for x in A.args if x.is_commutative]) elif isinstance(A, Symbol): if A.is_commutative: return A else: return 0 else: if A.is_commutative: return A else: return 0 """ else: if A.is_commutative: return A else: return zero """ def grades(self, A: Expr) -> List[int]: # Return list of grades present in A A = self.base_to_blade_rep(A) A = expand(A) blades = set() if isinstance(A, Add): args = A.args else: args = [A] for term in args: blade = term.args_cnc()[1] l_blade = len(blade) if l_blade > 0: blades.add(blade[0]) else: blades.add(one) return sorted({ self.blades_to_grades_dict[blade] for blade in blades }) def reverse(self, A: Expr) -> Expr: # Calculates reverse of A (see documentation) A = expand(A) blades = {} if isinstance(A, Add): args = A.args else: if A.is_commutative: return A else: args = [A] for term in args: if term.is_commutative: if 0 in blades: blades[0] += term else: blades[0] = term else: _c, nc = term.args_cnc() blade = nc[0] grade = self.blades_to_grades_dict[blade] if grade in blades: blades[grade] += term else: blades[grade] = term s = zero for grade in blades: if (grade * (grade - 1)) / 2 % 2 == 0: s += blades[grade] else: s -= blades[grade] return s def get_grade(self, A: Expr, r: int) -> Expr: # Return grade r of A, <A>_{r} coefs, bases = metric.linear_expand(A) return sum(( coef * base for coef, base in zip(coefs, bases) if self.blades_to_grades_dict[base] == r ), S(0)) def even_odd(self, A: Expr, even: bool = True) -> Expr: # Return even or odd part of A A = expand(A) if A.is_commutative and even: return A if isinstance(A, Add): args = A.args else: args = [A] s = zero for term in args: if term.is_commutative: if even: s += term else: c, nc = term.args_cnc(split_1=False) blade = nc[0] grade = self.blades_to_grades_dict[blade] if even and grade % 2 == 0: s += Mul._from_args(c) * blade elif not even and grade % 2 == 1: s += Mul._from_args(c) * blade return s @_cached_property def e_sq(self) -> Expr: r""" If ``self.gsym = True`` then :math:`E_{n}^2` is not evaluated, but is represented as :math:`E_{n}^2 = (-1)^{n*(n-1)/2}\operatorname{det}(g)` where :math:`\operatorname{det}(g)` the determinant of the metric tensor can be general scalar function of the coordinates. """ if self.gsym is not None: # Define square of pseudo-scalar in terms of metric tensor # determinant n = self.n return (-1) ** (n*(n - 1)//2) * self.detg else: return simplify(expand((self.e*self.e).scalar())) ##################### Multivector derivatives ###################### @_cached_property def r_basis(self) -> List[Expr]: r""" Reciprocal basis vectors :math:`e^{j}` as linear combination of basis vector symbols. These satisfy .. math:: e^{j}\cdot e_{k} = \delta_{k}^{j} where :math:`\delta_{k}^{j}` is the kronecker delta. We use the formula from Doran and Lasenby 4.94: .. math:: e^{j} = (-1)^{j-1}e_{1} \wedge ...e_{j-1} \wedge e_{j+1} \wedge ... \wedge e_{n}*E_{n}^{-1} where :math:`E_{n} = e_{1}\wedge ...\wedge e_{n}`. For non-orthogonal basis :math:`e^{j}` is not normalized and must be divided by :math:`E_{n}^2` (``self.e_sq``) in any relevant calculations. """ if self.debug: print('Enter r_basis.\n') if self.is_ortho: r_basis = [self.basis[i] / self.g[i, i] for i in self.n_range] else: duals = list(self.blades[self.n - 1]) # After reverse, the j-th of them is exactly e_{1}^...e_{j-1}^e_{j+1}^...^e_{n} duals.reverse() sgn = 1 r_basis = [] for dual in duals: dual_base_rep = self.blade_to_base_rep(dual) # {E_n}^{-1} = \frac{E_n}{{E_n}^{2}} # r_basis_j = sgn * duals[j] * E_n so it's not normalized, missing a factor of {E_n}^{-2} """ print('blades list =', self.blades.flat) print('debug =', expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e.obj)))) print('collect arg =', expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e.obj)))) """ r_basis_j = metric.collect(expand(self.base_to_blade_rep(self.mul(sgn * dual_base_rep, self.e.obj))), self.blades.flat) r_basis.append(r_basis_j) # sgn = (-1)**{j-1} sgn = -sgn if self.debug: printer.oprint('E', self.e, 'E**2', self.e_sq, 'unnormalized reciprocal basis =\n', r_basis) print('reciprocal basis test =') for ei in self.basis: for ej in r_basis: ei_dot_ej = self.hestenes_dot(ei, ej) if ei_dot_ej == zero: print('e_{i}|e_{j} = ' + str(ei_dot_ej)) else: print('e_{i}|e_{j} = ' + str(expand(ei_dot_ej / self.e_sq))) return r_basis def _update_de_from_rbasis(self): # Replace reciprocal basis vectors with expansion in terms of # basis vectors in derivatives of basis vectors. de = self.de if de is not None: for x_i in self.n_range: for jb in self.n_range: if not self.is_ortho: de[x_i][jb] = metric.Simp.apply(de[x_i][jb].subs(self.r_basis_dict) / self.e_sq) else: de[x_i][jb] = metric.Simp.apply(de[x_i][jb].subs(self.r_basis_dict)) @_cached_property def g_inv(self) -> Matrix: """ inverse of metric tensor, g^{ij} """ g_inv = eye(self.n) for i in self.n_range: rx_i = self.r_symbols[i] for j in self.n_range: rx_j = self.r_symbols[j] if j >= i: g_inv[i, j] = self.hestenes_dot(self.r_basis_dict[rx_i], self.r_basis_dict[rx_j]) if not self.is_ortho: g_inv[i, j] /= self.e_sq**2 else: g_inv[i, j] = g_inv[j, i] return simplify(g_inv) @_cached_property def r_basis_dict(self) -> Dict[Symbol, Expr]: """ Dictionary to represent reciprocal basis vectors as expansions in terms of basis vectors. ``{reciprocal basis symbol: linear combination of basis symbols, ...}`` """ return { r_symbol: r_base for r_symbol, r_base in zip(self.r_symbols, self.r_basis) } @_cached_property def r_basis_mv(self) -> List[_mv.Mv]: """ List of reciprocal basis vectors in terms of basis multivectors. """ return [mv.Mv(r_base, ga=self) for r_base in self.r_basis]
[docs] def er_blade(self, er, blade, mode='*', left=True): r""" Product (``*``, ``^``, ``|``, ``<``, ``>``) of reciprocal basis vector 'er' and basis blade 'blade' needed for application of derivatives to multivectors. left is 'True' means 'er' is multiplying 'blade' on the left, 'False' is for 'er' multiplying 'blade' on the right. Symbolically for left geometric product: .. math:: e^{j}*(e_{i_{1}}\wedge ...\wedge e_{i_{r}}) """ if mode == '*': base = self.blade_to_base_rep(blade) if left: return self.base_to_blade_rep(self.mul(er, base)) else: return self.base_to_blade_rep(self.mul(base, er)) elif mode == '^': if left: return self.wedge(er, blade) else: return self.wedge(blade, er) else: if left: return self.Mul(er, blade, mode=mode) else: return self.Mul(blade, er, mode=mode)
[docs] def blade_derivation(self, blade: Symbol, ib: Union[int, Symbol]) -> Expr: """ Calculate derivatives of basis blade 'blade' using derivative of basis vectors calculated by metric. 'ib' is the index of the coordinate the derivation is with respect to or the coordinate symbol. These are requried for the calculation of the geometric derivatives in curvilinear coordinates or for more general manifolds. 'blade_derivation' caches the results in a dictionary, ``self._dbases``, so that the derivation for a given blade and coordinate is never calculated more that once. Note that the return value is not a multivector, but linear combination of basis blade symbols. """ if isinstance(ib, int): coord = self.coords[ib] else: coord = ib ib = self.coords.index(coord) key = (coord, blade) if key in self._dbases: return self._dbases[key] index = self.indexes_to_blades_dict.inverse[blade] grade = len(index) # differentiate each basis vector separately and sum db = S.Zero for i in range(grade): db += reduce(self.wedge, [ self.indexes_to_blades_dict[index[:i]], self.de[ib][index[i]], self.indexes_to_blades_dict[index[i + 1:]] ]) self._dbases[key] = db return db
[docs] def pDiff(self, A: _mv.Mv, coord: Union[List, Symbol]) -> _mv.Mv: """ Compute partial derivative of multivector function 'A' with respect to coordinate 'coord'. """ if isinstance(coord, list): # Perform multiple partial differentiation where coord is # a list of differentiation orders for each coordinate and # the coordinate is determinded by the list index. If the # element in the list is zero no differentiation is to be # performed for that coordinate index. dA = copy.copy(A) # Make copy of A for i in self.n_range: x = self.coords[i] xn = coord[i] if xn > 0: # Differentiate with respect to coordinate x for _j in range(xn): # xn > 1 multiple differentiation dA = self.pDiff(dA, x) return dA # Simple partial differentiation, once with respect to a single # variable, but including case of non-constant basis vectors dA = self.mv(expand(diff(A.obj, coord))) if self.connect_flg and self.dslot == -1 and not A.is_scalar(): # Basis blades are function of coordinates B = self.remove_scalar_part(A) if B != zero: if isinstance(B, Add): args = B.args else: args = [B] for term in args: if not term.is_commutative: c, nc = term.args_cnc(split_1=False) x = self.blade_derivation(nc[0], coord) if x != zero: dA += reduce(operator.mul, c, x) return dA
[docs] def grad_sqr(self, A, grad_sqr_mode, mode, left): r""" Calculate :math:`(grad *_{1} grad) *_{2} A` or :math:`A *_{2} (grad *_{1} grad)` where ``grad_sqr_mode`` = :math:`*_{1}` = ``*``, ``^``, or ``|`` and ``mode`` = :math:`*_{2}` = ``*``, ``^``, or ``|``. """ Sop, Bop = Ga.DopFop[(grad_sqr_mode, mode)] print('(Sop, Bop) =', Sop, Bop) print('grad_sqr:A =', A) s = zero if Sop is False and Bop is False: return s dA_i = [] for coord_i in self.coords: dA_i.append(self.pDiff(A, coord_i)) print('dA_i =', dA_i) if Sop: for i in self.n_range: coord_i = self.coords[i] if self.connect_flg: s += self.grad_sq_scl_connect[coord_i] * dA_i[i] for j in self.n_range: d2A_j = self.pDiff(dA_i[i], self.coords[j]) s += self.g_inv[i, j] * d2A_j if Bop and self.connect_flg: for i in self.n_range: coord_i = self.coords[i] print('mode =', mode) print('dA_i[i] =', dA_i[i]) if left: if mode == '|': s += self.dot(self.grad_sq_mv_connect[coord_i], dA_i[i]) if mode == '^': s += self.wedge(self.grad_sq_mv_connect[coord_i], dA_i[i]) if mode == '*': s += self.mul(self.grad_sq_mv_connect[coord_i], dA_i[i]) else: if mode == '|': s += self.dot(dA_i[i], self.grad_sq_mv_connect[coord_i]) if mode == '^': s += self.wedge(dA_i[i], self.grad_sq_mv_connect[coord_i]) if mode == '*': s += self.mul(dA_i[i], self.grad_sq_mv_connect[coord_i]) return s
[docs] def connection(self, rbase, key_base, mode, left): """ Compute required multivector connections of the form (Einstein summation convention) :math:`e^{j}*(D_{j}e_{i_{1}...i_{r}})` and :math:`(D_{j}e_{i_{1}...i_{r}})*e^{j}` where :math:`*` could be ``*``, ``^``, ``|``, ``<``, or ``>`` depending upon the mode, and :math:`e^{j}` are reciprocal basis vectors. """ mode_key = (mode, left) keys = [i for i, j in self.connect[mode_key]] if left: key = rbase * key_base else: key = key_base * rbase if key not in keys: keys.append(key) C = zero for ib in self.n_range: x = self.blade_derivation(key_base, ib) if self.norm: x /= self.e_norm[ib] C += self.er_blade(self.r_basis[ib], x, mode, left) # Update connection dictionaries self.connect[mode_key].append((key, C)) return C
[docs] def ReciprocalFrame(self, basis: Sequence[_mv.Mv], mode: str = 'norm') -> Tuple[_mv.Mv, ...]: r""" Compute the reciprocal frame :math:`v^i` of a set of vectors :math:`v_i`. Parameters ---------- basis : The sequence of vectors :math:`v_i` defining the input frame. mode : * ``"norm"`` -- indicates that the reciprocal vectors should be normalized such that their product with the input vectors is 1, :math:`v^i \cdot v_j = \delta_{ij}`. * ``"append"`` -- indicates that instead of normalizing, the normalization coefficient :math:`E^2` should be appended to the returned tuple. One can divide by this coefficient to normalize the vectors. The returned vectors are such that :math:`v^i \cdot v_j = E^2\delta_{ij}`. .. deprecated:: 0.5.0 Arbitrary strings are interpreted as ``"append"``, but in future will be an error """ dim = len(basis) indexes = tuple(range(dim)) index = [()] for i in indexes[-2:]: index.append(tuple(combinations(indexes, i + 1))) MFbasis = [] for igrade in index[-2:]: grade = [] for iblade in igrade: blade = self.mv(S(1), 'scalar') for ibasis in iblade: blade ^= basis[ibasis] blade = blade.trigsimp() grade.append(blade) MFbasis.append(grade) E = MFbasis[-1][0] E_sq = trigsimp((E * E).scalar()) duals = copy.copy(MFbasis[-2]) duals.reverse() sgn = S(1) rbasis = [] for dual in duals: recpv = (sgn * dual * E).trigsimp() rbasis.append(recpv) sgn = -sgn if mode == 'norm': for i in range(dim): rbasis[i] = rbasis[i] / E_sq else: if mode != 'append': # galgebra 0.5.0 warnings.warn( "Mode {!r} not understood, falling back to {!r} but this " "is deprecated".format(mode, 'append'), DeprecationWarning, stacklevel=2) rbasis.append(E_sq) return tuple(rbasis)
def Mlt(self, *args, **kwargs): return lt.Mlt(args[0], self, *args[1:], **kwargs)
[docs]class Sm(Ga): """ Submanifold is a geometric algebra defined on a submanifold of a base geometric algebra defined on a manifold. The submanifold is defined by a mapping from the coordinates of the base manifold to the coordinates of the submanifold. The inputs required to define the submanifold are: Notes ----- The 'Ga' member function 'sm' can be used to instantiate the submanifold via (o3d is the base manifold):: coords = u, v = symbols('u, v', real=True) sm_example = o3d.sm([sin(u)*cos(v), sin(u)*sin(v), cos(u)], coords) eu, ev = sm_example.mv() sm_grad = sm_example.grad """ # __u is to emulate a Python 3.8 positional-only argument, with a clearer # spelling than `*args`. def __init__(self, __u, __coords, *, ga, norm=False, name=None, root='e', debug=False): """ Parameters ---------- u : The coordinate map defining the submanifold which is a list of functions of coordinates of the base manifold in terms of the coordinates of the submanifold. for example if the manifold is a unit sphere then - ``u = [sin(u)*cos(v), sin(u)*sin(v), cos(u)]``. Alternatively, a parametric vector function of the basis vectors of the base manifold. The coefficients of the bases are functions of the coordinates (``coords``). In this case we would call the submanifold a "vector" manifold and additional characteristics of the manifold can be calculated since we have given an explicit embedding of the manifold in the base manifold. coords : The coordinate list for the submanifold, for example ``[u, v]``. debug : True for debug output root : str Root symbol for basis vectors name : str Name of submanifold norm : bool Normalize basis if True ga : Base Geometric Algebra """ # print '!!!Enter Sm!!!' u = __u coords = __coords if ga is None: raise ValueError('Base geometric algebra must be specified for submanifold.') g_base = ga.g_raw n_base = ga.n n_sub = len(coords) # Construct names of basis vectors """ basis_str = '' for x in coords: basis_str += root + '_' + str(x) + ' ' basis_str = basis_str[:-1] """ # print 'u =', u if isinstance(u, mv.Mv): # Define vector manifold self.ebasis = [] for coord in coords: # Partial derivation of vector function to get basis vectors self.ebasis.append(u.diff(coord)) # print 'sm ebasis =', self.ebasis self.g = [] for b1 in self.ebasis: # Metric tensor from dot products of basis vectors tmp = [] for b2 in self.ebasis: tmp.append(b1 | b2) self.g.append(tmp) else: if len(u) != n_base: raise ValueError('In submanifold dimension of base manifold' + ' not equal to dimension of mapping.') dxdu = [] for x_i in u: tmp = [] for u_j in coords: tmp.append(diff(x_i, u_j)) dxdu.append(tmp) # print 'dxdu =', dxdu sub_pairs = list(zip(ga.coords, u)) # Construct metric tensor form coordinate maps g = eye(n_sub) # Zero n_sub x n_sub sympy matrix n_range = list(range(n_sub)) for i in n_range: for j in n_range: s = zero for k in ga.n_range: for l in ga.n_range: s += dxdu[k][i] * dxdu[l][j] * g_base[k, l].subs(sub_pairs) g[i, j] = trigsimp(s) Ga.__init__(self, root, g=g, coords=coords, norm=norm, debug=debug) if isinstance(u, mv.Mv): # Construct additional functions for vector manifold # self.r_basis_mv under construction pass self.ga = ga self.u = u if debug: print('Exit Sm.__init__()') def vpds(self) -> Tuple[_mv.Dop, _mv.Dop]: if not self.is_ortho: r_basis = [x / self.e_sq for x in self.r_basis_mv] else: r_basis = self.r_basis_mv if self.norm: r_basis = [x / e_norm for x, e_norm in zip(self.r_basis_mv, self.e_norm)] pdx = [self.Pdiffs[x] for x in self.coords] self.vpd = mv.Dop(r_basis, pdx, ga=self) self.rvpd = mv.Dop(r_basis, pdx, ga=self, cmpflg=True) return self.vpd, self.rvpd