Source code for galgebra.mv

"""
Multivector and Linear Multivector Differential Operator
"""

import copy
import numbers
import operator
from functools import reduce
from typing import List, Any, Tuple, Union, TYPE_CHECKING

from sympy import (
    Symbol, Function, S, expand, Add,
    sin, cos, sinh, cosh, sqrt, trigsimp,
    simplify, diff, Expr, Abs, collect, SympifyError,
)
from sympy import exp as sympy_exp
from sympy import N as Nsympy
from sympy.printing.latex import LatexPrinter as _LatexPrinter
from sympy.printing.str import StrPrinter as _StrPrinter

from . import printer
from . import metric
from .printer import ZERO_STR
from ._utils import KwargParser as _KwargParser
from . import dop

if TYPE_CHECKING:
    from galgebra.ga import Ga

# This file does not and should not use these.
# Unfortunately, some of our examples do.
ONE = S.One
ZERO = S.Zero
HALF = S.Half


# Add custom settings to the builtin latex printer
_LatexPrinter._default_settings.update({
    'galgebra_mv_fmt': 1
})
_StrPrinter._default_settings.update({
    'galgebra_mv_fmt': 1
})


########################### Multivector Class ##########################


[docs] class Mv(printer.GaPrintable): """ Wrapper class for multivector objects (``self.obj``) so that it is easy to overload operators (``*``, ``^``, ``|``, ``<``, ``>``) for the various multivector products and for printing. Also provides a constructor to easily instantiate multivector objects. Additionally, the functionality of the multivector derivative have been added via the special vector ``grad`` so that one can take the geometric derivative of a multivector function ``A`` by applying ``grad`` from the left, ``grad*A``, or the right ``A*grad`` for both the left and right derivatives. The operator between the ``grad`` and the 'A' can be any of the multivector product operators. If ``f`` is a scalar function ``grad*f`` is the usual gradient of a function. If ``A`` is a vector function ``grad|f`` is the divergence of ``A`` and ``-I*(grad^A)`` is the curl of ``A`` (I is the pseudo scalar for the geometric algebra) Attributes ---------- obj : sympy.core.Expr The underlying sympy expression for this multivector """ ################### Multivector initialization ##################### # This is read by one code path in `galgebra.printer.Fmt`. Only one example # sets it. fmt = 1 dual_mode_lst = ['+I', 'I+', '+Iinv', 'Iinv+', '-I', 'I-', '-Iinv', 'Iinv-']
[docs] @staticmethod def setup(ga: 'Ga') -> Tuple['Mv', List['Mv'], 'Mv']: """ Set up constant multivectors required for multivector class for a given geometric algebra, `ga`. """ # copy basis in case the caller wanted to change it return ga.mv_I, list(ga.mv_basis), ga.mv_x
[docs] @staticmethod def Mul(A: 'Mv', B: 'Mv', op: str) -> 'Mv': """ Function for all types of geometric multiplications called by overloaded operators for ``*``, ``^``, ``|``, ``<``, and ``>``. """ if not isinstance(A, Mv): A = B.Ga.mv(A) if not isinstance(B, Mv): B = A.Ga.mv(B) if op == '*': return A * B elif op == '^': return A ^ B elif op == '|': return A | B elif op == '<': return A < B elif op == '>': return A > B else: raise ValueError('Operation ' + op + 'not allowed in Mv.Mul!')
def characterise_Mv(self) -> None: if self.char_Mv: return obj = expand(self.obj) if isinstance(obj, numbers.Number): self.i_grade = 0 self.is_blade_rep = True self.grades = [0] return if obj.is_commutative: self.i_grade = 0 self.is_blade_rep = True self.grades = [0] return if isinstance(obj, Add): args = obj.args else: if obj in self.Ga.blades.flat: self.is_blade_rep = True self.i_grade = self.Ga.blades_to_grades_dict[obj] self.grades = [self.i_grade] self.char_Mv = True self.blade_flg = True return else: args = [obj] grades = [] # print 'args =', args self.is_blade_rep = True for term in args: if term.is_commutative: if 0 not in grades: grades.append(0) else: c, nc = term.args_cnc(split_1=False) blade = nc[0] # print 'blade =', blade if blade in self.Ga.blades.flat: grade = self.Ga.blades_to_grades_dict[blade] if grade not in grades: grades.append(grade) else: self.char_Mv = True self.is_blade_rep = False self.i_grade = None return if len(grades) == 1: self.i_grade = grades[0] else: self.i_grade = None self.grades = grades self.char_Mv = True # helper methods called by __init__. Note that these names must not change, # as the part of the name after `_make_` is public API via the string # argument passed to __init__. # # The double underscores in argument names are to force the passing # positionally. When python 3.8 is the lowest supported version, we can # switch to using the / syntax from PEP570 @staticmethod def _make_grade(ga: 'Ga', __name_or_coeffs: Union[str, list, tuple], __grade: int, **kwargs) -> Expr: """ Make a pure grade multivector. """ def add_superscript(root, s): if not s: return root return '{}__{}'.format(root, s) grade = __grade kw = _KwargParser('_make_grade', kwargs) if isinstance(__name_or_coeffs, str): name = __name_or_coeffs f = kw.pop('f', False) kw.reject_remaining() if isinstance(f, bool): if f: # Is a multivector function of all coordinates return sum([Function(add_superscript(name, super_script), real=True)(*ga.coords) * base for super_script, base in zip(ga.blade_super_scripts[grade], ga.blades[grade])]) else: # Is a constant multivector function return sum([Symbol(add_superscript(name, super_script), real=True) * base for super_script, base in zip(ga.blade_super_scripts[grade], ga.blades[grade])]) else: # Is a multivector function of tuple f variables return sum([Function(add_superscript(name, super_script), real=True)(*f) * base for super_script, base in zip(ga.blade_super_scripts[grade], ga.blades[grade])]) elif isinstance(__name_or_coeffs, (list, tuple)): coeffs = __name_or_coeffs kw.reject_remaining() if len(coeffs) <= len(ga.blades[grade]): return sum([ coef * base for coef, base in zip(coeffs, ga.blades[grade][:len(coeffs)])]) else: raise ValueError("Too many coefficients") else: raise TypeError("Expected a string, list, or tuple") @staticmethod def _make_scalar(ga: 'Ga', __name_or_value: Union[str, Expr], **kwargs) -> Expr: """ Make a scalar multivector """ if isinstance(__name_or_value, str): name = __name_or_value return Mv._make_grade(ga, name, 0, **kwargs) else: value = __name_or_value return value @staticmethod def _make_vector(ga: 'Ga', __name_or_coeffs: Union[str, list, tuple], **kwargs) -> Expr: """ Make a vector multivector """ return Mv._make_grade(ga, __name_or_coeffs, 1, **kwargs) @staticmethod def _make_bivector(ga: 'Ga', __name_or_coeffs: Union[str, list, tuple], **kwargs) -> Expr: """ Make a bivector multivector """ return Mv._make_grade(ga, __name_or_coeffs, 2, **kwargs) @staticmethod def _make_pseudo(ga: 'Ga', __name_or_coeffs: Union[str, list, tuple], **kwargs) -> Expr: """ Make a pseudo scalar multivector """ return Mv._make_grade(ga, __name_or_coeffs, ga.n, **kwargs) @staticmethod def _make_mv(ga: 'Ga', __name: str, **kwargs) -> Expr: """ Make a general (2**n components) multivector """ if not isinstance(__name, str): raise TypeError("Must be a string") return reduce(operator.add, ( Mv._make_grade(ga, __name, grade, **kwargs) for grade in range(ga.n + 1) )) # ## GSG code starts ### @staticmethod def _make_even(ga: 'Ga', __name: str, **kwargs) -> Expr: """ Make a general even multivector """ if not isinstance(__name, str): raise TypeError("Must be a string") return reduce(operator.add, ( Mv._make_grade(ga, __name, grade, **kwargs) for grade in range(0, ga.n + 1, 2) )) # ## GSG code ends ### @staticmethod def _make_odd(ga: 'Ga', __name: str, **kwargs) -> Expr: """ Make a general odd multivector """ if not isinstance(__name, str): raise TypeError("Must be a string") return reduce(operator.add, ( Mv._make_grade(ga, __name, grade, **kwargs) for grade in range(1, ga.n + 1, 2) ), S.Zero) # base case needed in case n == 0 # aliases _make_grade2 = _make_bivector _make_spinor = _make_even # alias for compatibility with old code ### GSSG: removed alias `_make_even = _make_spinor_` def __init__(self, *args, ga: 'Ga', recp=None, coords=None, **kwargs): """ __init__(self, *args, ga, recp=None, **kwargs) Note this constructor is overloaded, based on the type and number of positional arguments: .. class:: Mv(*, ga, recp=None) :noindex: Create a zero multivector .. class:: Mv(expr, /, *, ga, recp=None) :noindex: Create a multivector from an existing vector or sympy expression .. class:: Mv(coeffs, grade, /, ga, recp=None) :noindex: Create a multivector constant with a given grade .. class:: Mv(name, category, /, *cat_args, ga, recp=None, f=False) :noindex: Create a multivector constant with a given category .. class:: Mv(name, grade, /, ga, recp=None, f=False) :noindex: Create a multivector variable or function of a given grade .. class:: Mv(coeffs, category, /, *cat_args, ga, recp=None) :noindex: Create a multivector variable or function of a given category ``*`` and ``/`` in the signatures above are python 3.8 syntax, and respectively indicate the boundaries between positional-only, normal, and keyword-only arguments. Parameters ---------- ga : ~galgebra.ga.Ga Geometric algebra to be used with multivectors recp : object, optional Normalization for reciprocal vector. Unused. name : str Name of this multivector, if it is a variable or function coeffs : sequence Sequence of coefficients for the given category. This is only meaningful category : str One of: * ``"grade"`` - this takes an additional argument, the grade to create, in ``cat_args`` * ``"scalar"`` * ``"vector"`` * ``"bivector"`` / ``"grade2"`` * ``"pseudo"`` * ``"mv"`` * ``"even"`` / ``"spinor"`` * ``"odd"`` f : bool, tuple True if function of coordinates, or a tuple of those coordinates. Only valid if a name is passed coords : This argument is always accepted but ignored. It is incorrectly described internally as the coordinates to be used with multivector functions. """ ### GSG: removed mention of "spinor" under `category : str` in above docstring kw = _KwargParser('__init__', kwargs) self.Ga = ga self.recp = recp # not used self.char_Mv = False self.i_grade = None # if pure grade mv, grade value self.grades = None # list of grades in mv self.is_blade_rep = True # flag for blade representation self.blade_flg = None # if is_blade is called flag is set self.versor_flg = None # if is_versor is called flag is set self.coords = self.Ga.coords self.title = None if len(args) == 0: # default constructor 0 self.obj = S.Zero self.i_grade = 0 kw.reject_remaining() elif len(args) == 1 and not isinstance(args[0], str): # copy constructor x = args[0] if isinstance(x, Mv): self.obj = x.obj self.is_blade_rep = x.is_blade_rep self.i_grade = x.i_grade self.characterise_Mv() else: if isinstance(x, Expr): # copy constructor for obj expression self.obj = x else: # copy constructor for scalar obj expression self.obj = S(x) self.is_blade_rep = True self.characterise_Mv() kw.reject_remaining() else: if isinstance(args[1], str): make_args = list(args) mode = make_args.pop(1) make_func = getattr(Mv, '_make_{}'.format(mode), None) if make_func is None: raise ValueError('{!r} is not an allowed multivector type.'.format(mode)) self.obj = make_func(self.Ga, *make_args, **kwargs) elif isinstance(args[1], int): # args[1] = r (integer) Construct grade r multivector if args[1] == 0: # _make_scalar interprets its coefficient argument differently make_args = list(args) make_args.pop(1) self.obj = Mv._make_scalar(self.Ga, *make_args, **kwargs) else: self.obj = Mv._make_grade(self.Ga, *args, **kwargs) else: raise TypeError("Expected string or int") if isinstance(args[0], str): self.title = args[0] self.characterise_Mv() def _sympy_(self): """ Hook used by sympy.sympify """ raise SympifyError(self, TypeError( "Cannot safely convert an `Mv` instance to a sympy object. " "Use `mv.obj` to obtain the internal sympy object, but note that " "this does not overload the geometric operators, and will not " "track the associated `Ga` instance." )) ################# Multivector member functions ##################### def reflect_in_blade(self, blade: 'Mv') -> 'Mv': # Reflect mv in blade # See Mv class functions documentation if blade.is_blade(): self.characterise_Mv() blade.characterise_Mv() blade_inv = blade.rev() / blade.qform() # ### GSG replaced .norm2() by .qform() grade_dict = self.Ga.grade_decomposition(self) blade_grade = blade.i_grade reflect = Mv(0, 'scalar', ga=self.Ga) for grade in list(grade_dict.keys()): if (grade * (blade_grade + 1)) % 2 == 0: reflect += blade * grade_dict[grade] * blade_inv else: reflect -= blade * grade_dict[grade] * blade_inv return reflect else: raise ValueError(str(blade) + 'is not a blade in reflect_in_blade(self, blade)') def project_in_blade(self, blade: 'Mv') -> 'Mv': # See Mv class functions documentation if blade.is_blade(): blade.characterise_Mv() blade_inv = blade.rev() / blade.qform() # ### GSG replaced .norm2() by .qform() return (self < blade) * blade_inv # < is left contraction else: raise ValueError(str(blade) + 'is not a blade in project_in_blade(self, blade)') def rotate_multivector(self, itheta: 'Mv', hint: str = '-'): Rm = (-itheta/S(2)).exp(hint) Rp = (itheta/S(2)).exp(hint) return Rm * self * Rp
[docs] def base_rep(self) -> 'Mv': """ Express as a linear combination of geometric products """ if not self.is_blade_rep: return self b = copy.copy(self) b.obj = self.Ga.blade_to_base_rep(self.obj) b.is_blade_rep = False return b
[docs] def blade_rep(self) -> 'Mv': """ Express as a linear combination of blades """ if self.is_blade_rep: return self b = copy.copy(self) b.obj = self.Ga.base_to_blade_rep(self.obj) b.is_blade_rep = True return b
def __hash__(self) -> int: if self.is_scalar(): # ensure we match equality return hash(self.obj) else: return hash((self.Ga, self.obj)) def __eq__(self, A): if isinstance(A, Mv): diff = (self - A).expand().simplify() # diff = (self - A).expand() return diff.obj == S.Zero else: return self.is_scalar() and self.obj == A """ def __eq__(self, A): if not isinstance(A, Mv): if not self.is_scalar(): return False if expand(self.obj) == expand(A): return True else: return False if self.is_blade_rep != A.is_blade_rep: self = self.blade_rep() A = A.blade_rep() coefs, bases = metric.linear_expand(self.obj) Acoefs, Abases = metric.linear_expand(A.obj) if len(bases) != len(Abases): return False if set(bases) != set(Abases): return False for base in bases: index = bases.index(base) indexA = Abases.index(base) if expand(coefs[index]) != expand(Acoefs[index]): return False return True """ def __neg__(self): return Mv(-self.obj, ga=self.Ga) def _arithmetic_op(self, A, op, name: str): """ Common implementation for + and - """ if isinstance(A, dop._BaseDop): return NotImplemented if not isinstance(A, Mv): return Mv(op(self.obj, A), ga=self.Ga) if self.Ga != A.Ga: raise ValueError( 'In {} operation Mv arguments are not from same geometric ' 'algebra'.format(name)) if self.is_blade_rep == A.is_blade_rep: return Mv(op(self.obj, A.obj), ga=self.Ga) else: if self.is_blade_rep: A = A.blade_rep() else: self = self.blade_rep() return Mv(op(self.obj, A.obj), ga=self.Ga) def __add__(self, A): return self._arithmetic_op(A, lambda a, b: a + b, '+') def __radd__(self, A): return self._arithmetic_op(A, lambda a, b: b + a, '+') def __sub__(self, A): return self._arithmetic_op(A, lambda a, b: a - b, '-') def __rsub__(self, A): return self._arithmetic_op(A, lambda a, b: b - a, '-') def __mul__(self, A): if isinstance(A, dop._BaseDop): return NotImplemented if not isinstance(A, Mv): return Mv(expand(A * self.obj), ga=self.Ga) if self.Ga != A.Ga: raise ValueError('In * operation Mv arguments are not from same geometric algebra') if self.is_scalar(): return Mv(self.obj * A, ga=self.Ga) if self.is_blade_rep and A.is_blade_rep: self = self.base_rep() A = A.base_rep() selfxA = Mv(self.Ga.mul(self.obj, A.obj), ga=self.Ga) selfxA.is_blade_rep = False return selfxA.blade_rep() elif self.is_blade_rep: self = self.base_rep() selfxA = Mv(self.Ga.mul(self.obj, A.obj), ga=self.Ga) selfxA.is_blade_rep = False return selfxA.blade_rep() elif A.is_blade_rep: A = A.base_rep() selfxA = Mv(self.Ga.mul(self.obj, A.obj), ga=self.Ga) selfxA.is_blade_rep = False return selfxA.blade_rep() else: return Mv(self.Ga.mul(self.obj, A.obj), ga=self.Ga) def __rmul__(self, A): if isinstance(A, dop._BaseDop): return NotImplemented return Mv(expand(A * self.obj), ga=self.Ga) def __truediv__(self, A): if isinstance(A, Mv): return self * A.inv() else: return self * (S.One/A) def __str__(self): return printer.GaPrinter()._print(self) def __getitem__(self, key: int) -> 'Mv': ''' get a specified grade of a multivector ''' return self.grade(key) def _sympystr(self, print_obj: printer.GaPrinter) -> str: # note: this just replaces `self` for the rest of this function obj = expand(self.obj) obj = metric.Simp.apply(obj) self = Mv(obj, ga=self.Ga) if self.i_grade == 0: return print_obj._print(self.obj) if self.is_blade_rep or self.Ga.is_ortho: base_keys = self.Ga.blades.flat grade_keys = self.Ga.blades_to_grades_dict else: base_keys = self.Ga.bases.flat grade_keys = self.Ga.bases_to_grades_dict if isinstance(self.obj, Add): # collect coefficients of bases if self.obj.is_commutative: return self.obj args = self.obj.args terms = {} # dictionary with base indexes as keys grade0 = S.Zero for arg in args: c, nc = arg.args_cnc() c = reduce(operator.mul, c, S.One) if len(nc) > 0: base = nc[0] if base in base_keys: index = base_keys.index(base) if index in terms: c_tmp, base, g_keys = terms[index] terms[index] = (c_tmp + c, base, g_keys) else: terms[index] = (c, base, grade_keys[base]) else: grade0 += c if grade0 != S.Zero: terms[-1] = (grade0, S.One, -1) terms = list(terms.items()) sorted_terms = sorted(terms, key=operator.itemgetter(0)) # sort via base indexes s = print_obj._print(sorted_terms[0][1][0] * sorted_terms[0][1][1]) if print_obj._settings['galgebra_mv_fmt'] == 3: s = ' ' + s + '\n' if print_obj._settings['galgebra_mv_fmt'] == 2: s = ' ' + s old_grade = sorted_terms[0][1][2] for (key, (c, base, grade)) in sorted_terms[1:]: term = print_obj._print(c * base) if print_obj._settings['galgebra_mv_fmt'] == 2 and old_grade != grade: # one grade per line old_grade = grade s += '\n' if term[0] == '-': term = ' - ' + term[1:] else: term = ' + ' + term if print_obj._settings['galgebra_mv_fmt'] == 3: # one base per line s += term + '\n' else: # one multivector per line s += term if s[-1] == '\n': s = s[:-1] return s else: return print_obj._print(self.obj) def _latex(self, print_obj: _LatexPrinter) -> str: if self.obj == S.Zero: return ZERO_STR first_line = True def append_plus(c_str): nonlocal first_line if first_line: first_line = False return c_str else: c_str = c_str.strip() if c_str[0] == '-': return ' ' + c_str else: return ' + ' + c_str # str representation of multivector # note: this just replaces `self` for the rest of this function obj = expand(self.obj) obj = metric.Simp.apply(obj) self = Mv(obj, ga=self.Ga) if self.obj == S.Zero: return ZERO_STR if self.is_blade_rep or self.Ga.is_ortho: base_keys = self.Ga.blades.flat grade_keys = self.Ga.blades_to_grades_dict else: base_keys = self.Ga.bases.flat grade_keys = self.Ga.bases_to_grades_dict if isinstance(self.obj, Add): args = self.obj.args else: args = [self.obj] terms = {} # dictionary with base indexes as keys grade0 = S.Zero for arg in args: c, nc = arg.args_cnc(split_1=False) c = reduce(operator.mul, c, S.One) if len(nc) > 0: base = nc[0] if base in base_keys: index = base_keys.index(base) if index in terms: c_tmp, base, g_keys = terms[index] terms[index] = (c_tmp + c, base, g_keys) else: terms[index] = (c, base, grade_keys[base]) else: grade0 += c if grade0 != S.Zero: terms[-1] = (grade0, S.One, 0) terms = list(terms.items()) sorted_terms = sorted(terms, key=operator.itemgetter(0)) # sort via base indexes if len(sorted_terms) == 1 and sorted_terms[0][1][2] == 0: # scalar return print_obj._print(printer.coef_simplify(sorted_terms[0][1][0])) lines = [] old_grade = -1 s = '' for (index, (coef, base, grade)) in sorted_terms: coef = printer.coef_simplify(coef) # coef = simplify(coef) l_coef = print_obj._print(coef) if l_coef == '1' and base != S.One: l_coef = '' if l_coef == '-1' and base != S.One: l_coef = '-' if base == S.One: l_base = '' else: l_base = print_obj._print(base) if isinstance(coef, Add): cb_str = '\\left ( ' + l_coef + '\\right ) ' + l_base else: cb_str = l_coef + ' ' + l_base if print_obj._settings['galgebra_mv_fmt'] == 3: # One base per line lines.append(append_plus(cb_str)) elif print_obj._settings['galgebra_mv_fmt'] == 2: # One grade per line if grade != old_grade: old_grade = grade if not first_line: lines.append(s) s = append_plus(cb_str) else: s += append_plus(cb_str) else: # One multivector per line s += append_plus(cb_str) if print_obj._settings['galgebra_mv_fmt'] == 2: lines.append(s) if print_obj._settings['galgebra_mv_fmt'] >= 2: if len(lines) == 1: return lines[0] s = ' \\begin{aligned}[t] ' for line in lines: s += ' & ' + line + ' \\\\ ' s = s[:-3] + ' \\end{aligned} ' return s def __xor__(self, A): # wedge (^) product if isinstance(A, dop._BaseDop): return NotImplemented if not isinstance(A, Mv): return Mv(A * self.obj, ga=self.Ga) if self.Ga != A.Ga: raise ValueError('In ^ operation Mv arguments are not from same geometric algebra') if self.is_scalar(): return self * A self = self.blade_rep() A = A.blade_rep() return Mv(self.Ga.wedge(self.obj, A.obj), ga=self.Ga) def __rxor__(self, A): # wedge (^) product if isinstance(A, dop._BaseDop): return NotImplemented assert not isinstance(A, Mv) return Mv(A * self.obj, ga=self.Ga) def __or__(self, A): # dot (|) product if isinstance(A, dop._BaseDop): return NotImplemented if not isinstance(A, Mv): return Mv(ga=self.Ga) if self.Ga != A.Ga: raise ValueError('In | operation Mv arguments are not from same geometric algebra') self = self.blade_rep() A = A.blade_rep() return Mv(self.Ga.hestenes_dot(self.obj, A.obj), ga=self.Ga) def __ror__(self, A): # dot (|) product if isinstance(A, dop._BaseDop): return NotImplemented assert not isinstance(A, Mv) return Mv(ga=self.Ga) def __pow__(self, n): # Integer power operator if not isinstance(n, int): raise ValueError('!!!!Multivector power can only be to integer power!!!!') if n < 0: return (self**(-n)).inv() result = Mv(S.One, 'scalar', ga=self.Ga) for x in range(n): result *= self return result def __lshift__(self, A): # anti-comutator (<<) return S.Half * (self * A + A * self) def __rshift__(self, A): # comutator (>>) return S.Half * (self * A - A * self) def __rlshift__(self, A): # anti-comutator (<<) return S.Half * (A * self + self * A) def __rrshift__(self, A): # comutator (>>) return S.Half * (A * self - self * A) def __lt__(self, A): # left contraction (<) if isinstance(A, Dop): # Cannot return `NotImplemented` here, as that would call `A > self` return A.Mul(self, A, op='<') elif isinstance(A, dop._BaseDop): raise TypeError( "'<' not supported between instances of 'Mv' and {!r}" .format(type(A).__name__) ) if not isinstance(A, Mv): # sympy scalar return Mv(A * self.scalar(), ga=self.Ga) if self.Ga != A.Ga: raise ValueError('In < operation Mv arguments are not from same geometric algebra') self = self.blade_rep() A = A.blade_rep() return Mv(self.Ga.left_contract(self.obj, A.obj), ga=self.Ga) def __gt__(self, A): # right contraction (>) if isinstance(A, Dop): # Cannot return `NotImplemented` here, as that would call `A < self` return A.Mul(self, A, op='>') elif isinstance(A, dop._BaseDop): raise TypeError( "'>' not supported between instances of 'Mv' and {!r}" .format(type(A).__name__) ) if not isinstance(A, Mv): # sympy scalar return Mv(A * self.obj, ga=self.Ga) if self.Ga != A.Ga: raise ValueError('In > operation Mv arguments are not from same geometric algebra') self = self.blade_rep() A = A.blade_rep() return Mv(self.Ga.right_contract(self.obj, A.obj), ga=self.Ga)
[docs] def collect(self, deep=False) -> 'Mv': """ group coeffients of blades of multivector so there is only one coefficient per grade """ """ # dead code self.obj = expand(self.obj) if self.is_blade_rep or Mv.Ga.is_ortho: c = self.Ga.blades.flat else: c = self.Ga.bases.flat self.obj = self.obj.collect(c) return self """ obj_dict = {} for coef, base in metric.linear_expand_terms(self.obj): if base in list(obj_dict.keys()): obj_dict[base] += coef else: obj_dict[base] = coef obj = S.Zero for base in list(obj_dict.keys()): if deep: obj += collect(obj_dict[base])*base else: obj += obj_dict[base]*base return Mv(obj, ga=self.Ga)
def is_scalar(self) -> bool: grades = self.Ga.grades(self.obj) return grades == [0] def is_vector(self) -> bool: grades = self.Ga.grades(self.obj) return grades == [1]
[docs] def is_blade(self) -> bool: """ True is self is blade, otherwise False sets self.blade_flg and returns value """ if self.blade_flg is not None: return self.blade_flg else: if self.is_versor(): if self.i_grade is not None: self.blade_flg = True else: self.blade_flg = False else: self.blade_flg = False return self.blade_flg
def is_base(self) -> bool: coefs, _bases = metric.linear_expand(self.obj) return coefs == [S.One]
[docs] def is_versor(self) -> bool: """ Test for versor (geometric product of vectors) This follows Leo Dorst's test for a versor. Leo Dorst, 'Geometric Algebra for Computer Science,' p.533 Sets self.versor_flg and returns value """ if self.versor_flg is not None: return self.versor_flg self.characterise_Mv() self.versor_flg = False self_rev = self.rev() # see if self*self.rev() is a scalar test = self*self_rev if not test.is_scalar(): return self.versor_flg # see if self*x*self.rev() returns a vector for x an arbitrary vector test = self * self.Ga._XOX * self.rev() self.versor_flg = test.is_vector() return self.versor_flg
r''' ### GSG start code ### def is_versor(self) -> bool: """ Presumes `self` is an invertible multivector. Returns True if `self` is a versor (geometric product of invertible vectors). This test follows results in lecture notes by Lundholm and Svensson. """ x = self.Ga.mv('', 1) # generic 1-vector return (self.g_invol() * x * self.inv()).is_vector() ### GSG end code ### ''' def is_zero(self) -> bool: return self.obj == 0
[docs] def scalar(self) -> Expr: """ return scalar part of multivector as sympy expression """ return self.Ga.scalar_part(self.obj)
[docs] def get_grade(self, r: int) -> 'Mv': """ return r-th grade of multivector as a multivector """ return Mv(self.Ga.get_grade(self.obj, r), ga=self.Ga)
def components(self) -> List['Mv']: cb = metric.linear_expand_terms(self.obj) cb = sorted(cb, key=lambda x: self.Ga.blades.flat.index(x[1])) return [self.Ga.mv(coef * base) for coef, base in cb]
[docs] def get_coefs(self, grade: int) -> List[Expr]: """ Like ``blade_coefs(self.Ga.mv_blades[grade])``, but requires all components to be of that grade. Raises ------ ValueError: If the multivector is not of the given grade. """ blade_lst = self.Ga.blades[grade] coef_lst = [S.Zero] * len(blade_lst) for coef, blade in metric.linear_expand_terms(self.obj): if coef == S.Zero: continue # TODO: why does expansion return this? try: base_i = blade_lst.index(blade) except ValueError: raise ValueError( "MultiVector has a {} component which is not grade {}" .format(blade, grade) ) from None coef_lst[base_i] += coef return coef_lst
[docs] def blade_coefs(self, blade_lst: List['Mv'] = None) -> List[Expr]: """ For a multivector, A, and a list of basis blades, blade_lst return a list (sympy expressions) of the coefficients of each basis blade in blade_lst """ if blade_lst is None: blade_lst = self.Ga.mv_blades.flat else: for blade in blade_lst: if not blade.is_base() or not blade.is_blade(): raise ValueError("%s expression isn't a basis blade" % blade) blade_lst = [x.obj for x in blade_lst] coefs, bases = metric.linear_expand(self.obj) coef_lst = [] for blade in blade_lst: if blade in bases: coef_lst.append(coefs[bases.index(blade)]) else: coef_lst.append(S.Zero) return coef_lst
[docs] def proj(self, bases_lst: List['Mv']) -> 'Mv': """ Project multivector onto a given list of bases. That is find the part of multivector with the same bases as in the bases_lst. """ bases_lst = [x.obj for x in bases_lst] obj = 0 for coef, base in metric.linear_expand_terms(self.obj): if base in bases_lst: obj += coef * base return Mv(obj, ga=self.Ga)
def dual(self) -> 'Mv': mode = self.Ga.dual_mode_value sign = S.One if '-' in mode: sign = -sign if 'Iinv' in mode: I = self.Ga.i_inv else: I = self.Ga.i if mode[0] == '+' or mode[0] == '-': return sign * I * self else: return sign * self * I # ## GSG code starts ###
[docs] def undual(self) -> 'Mv': """ Inverse method to multivector method `.dual()`, so both `A.dual().undual()` and `A.undual().dual` return `A`. """ return self.Ga.I()**2 * self.dual()
# ## GSG code ends ###
[docs] def even(self) -> 'Mv': """ return even parts of multivector """ return Mv(self.Ga.even_odd(self.obj, True), ga=self.Ga)
[docs] def odd(self) -> 'Mv': """ return odd parts of multivector """ return Mv(self.Ga.even_odd(self.obj, False), ga=self.Ga)
# ## GSG code starts ###
[docs] def g_invol(self) -> 'Mv': """ - Returns grade involute of multivector `self`; negates `self`'s odd grade part but preserves its even grade part. - Grade involution is its own inverse operation. """ return self.even() - self.odd()
# ## GSG code ends ### def rev(self) -> 'Mv': self = self.blade_rep() return Mv(self.Ga.reverse(self.obj), ga=self.Ga) __invert__ = rev # allow `~x` to call x.rev() # ## GSG code starts ###
[docs] def ccon(self) -> 'Mv': """ - Returns Clifford conjugate of multivector `self`, i.e. returns the reverse of self's grade involute. - Clifford conjugation is its own inverse operation. """ return self.g_invol().rev()
# ## GSG code ends ### # ## GSG code starts ###
[docs] def sp(self, B, switch='') -> Expr: # scalar product """ - Returns scalar product of multivectors self and B. - Object returned is a real expression, not a 0-vector. - switch can be either '' (the empty string) or 'rev'. The latter causes left factor self to be reversed before its product with B is taken. """ if not isinstance(B, Mv): raise ValueError("Right factor of sp must be a multivector") if switch not in ['', 'rev']: raise ValueError("switch must be '' or 'rev'.") if switch == '': return (self * B).scalar() if switch == 'rev': return (self.rev() * B).scalar()
# ## GSG code ends ### def diff(self, coord) -> 'Mv': if self.Ga.coords is None: obj = diff(self.obj, coord) elif coord not in self.Ga.coords: if self.Ga.par_coords is None: obj = diff(self.obj, coord) elif coord not in self.Ga.par_coords: obj = diff(self.obj, coord) else: obj = diff(self.obj, coord) for x_coord in self.Ga.coords: f = self.Ga.par_coords[x_coord] if f != S.Zero: tmp1 = self.Ga.pDiff(self.obj, x_coord) tmp2 = diff(f, coord) obj += tmp1 * tmp2 else: obj = self.Ga.pDiff(self.obj, coord) return Mv(obj, ga=self.Ga) def pdiff(self, var) -> 'Mv': return Mv(self.Ga.pDiff(self.obj, var), ga=self.Ga)
[docs] def Grad(self, coords, mode: str = '*', left: bool = True) -> 'Mv': """ Returns various derivatives (``*``, ``^``, ``|``, ``<``, ``>``) of multivector functions with respect to arbitrary coordinates, 'coords'. This would be used where you have a multivector function of both the basis coordinate set and and auxiliary coordinate set. Consider for example a linear transformation in which the matrix coefficients depend upon the manifold coordinates, but the vector being transformed does not and you wish to take the divergence of the linear transformation with respect to the linear argument. """ return Mv(self.Ga.Diff(self, mode, left, coords=coords), ga=self.Ga)
[docs] def exp(self, hint: str = '-') -> 'Mv': # Calculate exponential of multivector """ Only works if square of multivector is a scalar. If square is a number we can determine if square is > or < zero and hence if one should use trig or hyperbolic functions in expansion. If square is not a number use 'hint' to determine which type of functions to use in expansion """ self = self.blade_rep() self_sq = self * self if self_sq.is_scalar(): sq = simplify(self_sq.obj) # sympy expression for self**2 if sq == S.Zero: # sympy expression for self**2 = 0 return self + S.One coefs, bases = metric.linear_expand(self.obj) if len(coefs) == 1: # Exponential of scalar * base base = bases[0] base_Mv = self.Ga.mv(base) base_sq = (base_Mv*base_Mv).scalar() if hint == '-': # base^2 < 0 base_n = sqrt(-base_sq) return self.Ga.mv(cos(base_n*coefs[0]) + sin(base_n*coefs[0])*(bases[0]/base_n)) else: # base^2 > 0 base_n = sqrt(base_sq) return self.Ga.mv(cosh(base_n*coefs[0]) + sinh(base_n*coefs[0])*(bases[0]/base_n)) if sq.is_number: # Square is number, can test for sign if sq > S.Zero: norm = sqrt(sq) value = self.obj / norm tmp = Mv(cosh(norm) + sinh(norm) * value, ga=self.Ga) tmp.is_blade_rep = True return tmp else: norm = sqrt(-sq) value = self.obj / norm tmp = Mv(cos(norm) + sin(norm) * value, ga=self.Ga) tmp.is_blade_rep = True return tmp else: if hint == '+': norm = simplify(sqrt(sq)) value = self.obj / norm tmp = Mv(cosh(norm) + sinh(norm) * value, ga=self.Ga) tmp.is_blade_rep = True return tmp else: norm = simplify(sqrt(-sq)) value = self.obj / norm tmp = Mv(cos(norm) + sin(norm) * value, ga=self.Ga) tmp.is_blade_rep = True return tmp else: raise ValueError('"' + str(self) + '**2" is not a scalar in exp.')
def set_coef(self, igrade: int, ibase: int, value: Expr) -> None: if self.blade_rep: base = self.Ga.blades[igrade][ibase] else: base = self.Ga.bases[igrade][ibase] coefs, bases = metric.linear_expand(self.obj) bases_lst = list(bases) # python 2.5 if base in bases: self.obj += (value - coefs[bases_lst.index(base)]) * base else: self.obj += value * base
[docs] def Fmt(self, fmt: int = 1, title: str = None) -> printer.GaPrintable: """ Set format for printing of multivectors * `fmt=1` - One multivector per line * `fmt=2` - One grade per line * `fmt=3` - one base per line Usage for multivector ``A`` example is:: A.Fmt('2', 'A') output is:: 'A = '+str(A) with one grade per line. Works for both standard printing and for latex. """ if fmt is not None: obj = printer._WithSettings(self, dict(galgebra_mv_fmt=fmt)) else: obj = self return printer._FmtResult(obj, title)
def _repr_latex_(self) -> str: # overloaded to include the inferred title if self.title is not None: return printer._FmtResult(self, self.title)._repr_latex_() return super()._repr_latex_() # ## GSG code starts ###
[docs] def qform(self) -> Expr: """ - Returns the quadratic form of multivector self. - Return value is a real SymPy expression, NOT a GAlgebra 0-vector. - Expression necessarily represents a nonnegative number only when self's geometric algebra has a Euclidean metric. """ return simplify((self.rev()*self).scalar())
# ## GSG code ends ### # ## GSG code starts ###
[docs] def norm2(self, hint: str = '0') -> Expr: """ - Returns the normsquared of multivector self, defined as the absolute value of the quadratic form at self. - Return value is a real SymPy expression, NOT a GAlgebra 0-vector. Whether numeric or symbolic, A.norm2() always represents a nonnegative number. - String values '+', '-', or '0' of hint respectively determine whether the quadratic form, the absolute value of which is the norm squared, should be regarded as nonnegative, nonpositive, or of unknown sign, except when that quantity's sign can be determined by other considerations, such as the metric being Euclidean. """ quadform = self.qform() # the quadratic form at `self` # Case2: metric is positive definite if self.Ga.g.is_positive_definite: return quadform return metric.abs_with_hint(quadform, hint)
# ## GSG code ends ### # ## GSG code starts ###
[docs] def norm(self, hint='0') -> Expr: """ - Returns the norm of multivector self, defined as the square root of self's norm squared. - Whether numeric or symbolic, returned value is a real SymPy expression that necessarily represents a nonnegative number. Returned value is NOT a GAlgebra 0-vector. - String values '+', '-', or '0' of hint respectively determine whether the quadratic form from which the norm ultimately derives should be regarded as nonnegative, nonpositive, or of unknown sign, except when the quadratic form's sign can be determined by other considerations, such as the metric being Euclidean. """ return simplify(metric.square_root_of_expr(self.norm2(hint), hint='+'))
# ## GSG code ends ### __abs__ = norm # allow `abs(x)` to call z.norm() # ## GSG code starts ###
[docs] def mag2(self) -> Expr: """ - Returns the magnitude squared of multivector self, defined as the sum of the absolute values of the norm squareds of self's grade parts. - Returned value is a real SymPy expression, NOT a GAlgebra 0-vector. Expression necesssarily represents a nonnegative number. - The magnitude squared differs from the norm squared of `self` when the metric is non-Euclidean. """ total = 0 for k in range(self.Ga.n + 1): total += Abs(self.grade(k).norm2()) return total
# ## GSG code ends ### # ## GSG code starts ###
[docs] def mag(self) -> Expr: """ - Returns the magnitude of multivector self, defined as the square root of the magnitude squared. - The magnitude necessarily agrees with the norm only when the metric is Euclidean. Otherwise the magnitude is greater than or equal to the norm. """ return simplify(sqrt(self.mag2()))
# ## GSG code ends ### def inv(self) -> 'Mv': if self.is_scalar(): # self is a scalar return self.Ga.mv(S.One/self.obj) self_sq = self * self if self_sq.is_scalar(): # self*self is a scalar """ if self_sq.scalar() == S.Zero: raise ValueError('!!!!In multivector inverse, A*A is zero!!!!') """ return (S.One/self_sq.obj)*self self_rev = self.rev() self_self_rev = self * self_rev if self_self_rev.is_scalar(): # self*self.rev() is a scalar """ if self_self_rev.scalar() == S.Zero: raise ValueError('!!!!In multivector inverse A*A.rev() is zero!!!!') """ return (S.One/self_self_rev.obj) * self_rev raise TypeError('In inv() for self =' + str(self) + 'self, or self*self or self*self.rev() is not a scalar') def func(self, fct) -> 'Mv': # Apply function, fct, to each coefficient of multivector s = S.Zero for coef, base in metric.linear_expand_terms(self.obj): s += fct(coef) * base fct_self = Mv(s, ga=self.Ga) fct_self.characterise_Mv() return fct_self def trigsimp(self) -> 'Mv': return self.func(trigsimp)
[docs] def simplify(self, modes=simplify) -> 'Mv': """ Simplify a multivector by scalar (sympy) simplifications. `modes` is an operation or sequence of operations to apply to the the coefficients of a multivector expansion. """ if not isinstance(modes, (list, tuple)): modes = [modes] obj = S.Zero for coef, base in metric.linear_expand_terms(self.obj): for mode in modes: coef = mode(coef) obj += coef * base return Mv(obj, ga=self.Ga)
[docs] def subs(self, *args, **kwargs) -> 'Mv': """ Perform a substitution on each coefficient separately """ obj = sum(( coef.subs(*args, **kwargs) * base for coef, base in metric.linear_expand_terms(self.obj) ), S.Zero) return Mv(obj, ga=self.Ga)
def expand(self) -> 'Mv': obj = sum(( expand(coef) * base for coef, base in metric.linear_expand_terms(self.obj) ), S.Zero) return Mv(obj, ga=self.Ga) def list(self) -> List[Expr]: return self.blade_coefs(self.Ga.mv_blades[1]) def grade(self, r=0) -> 'Mv': return self.get_grade(r)
[docs] def pure_grade(self) -> int: """ For pure grade return grade. If not pure grade return negative of maximum grade """ self.characterise_Mv() if self.i_grade is not None: return self.i_grade return -self.grades[-1]
def _eval_derivative_n_times(self, x, n) -> 'Mv': for i in range(n): self = self.Ga.pDiff(self, x) return self
[docs] def compare(A: Mv, B: Mv) -> Union[Expr, int]: """ Determine if ``B = c*A`` where c is a scalar. If true return c otherwise return 0. """ if isinstance(A, Mv) and isinstance(B, Mv): Acoefs, Abases = metric.linear_expand(A.obj) Bcoefs, Bbases = metric.linear_expand(B.obj) if len(Acoefs) != len(Bcoefs): return 0 if Abases != Bbases: return 0 if Bcoefs[0] != 0 and Abases[0] == Bbases[0]: c = simplify(Acoefs[0]/Bcoefs[0]) print('c =', c) else: return 0 for acoef, abase, bcoef, bbase in zip(Acoefs[1:], Abases[1:], Bcoefs[1:], Bbases[1:]): print(acoef, '\n', abase, '\n', bcoef, '\n', bbase) if bcoef != 0 and abase == bbase: print('c-a/b =', simplify(c-(acoef/bcoef))) if simplify(acoef/bcoef) != c: return 0 else: pass else: return 0 return c else: raise TypeError('In compare both arguments are not multivectors\n')
################# Multivector Differential Operator Class ##############
[docs] class Dop(dop._BaseDop): r""" Differential operator class for multivectors. The operators are of the form .. math:: D = D^{i_{1}...i_{n}}\partial_{i_{1}...i_{n}} where the :math:`D^{i_{1}...i_{n}}` are multivector functions of the coordinates :math:`x_{1},...,x_{n}` and :math:`\partial_{i_{1}...i_{n}}` are partial derivative operators .. math:: \partial_{i_{1}...i_{n}} = \frac{\partial^{i_{1}+...+i_{n}}}{\partial{x_{1}^{i_{1}}}...\partial{x_{n}^{i_{n}}}}. If :math:`*` is any multivector multiplicative operation then the operator D operates on the multivector function :math:`F` by the following definitions .. math:: D*F = D^{i_{1}...i_{n}}*\partial_{i_{1}...i_{n}}F returns a multivector and .. math:: F*D = F*D^{i_{1}...i_{n}}\partial_{i_{1}...i_{n}} returns a differential operator. If the :attr:`cmpflg` in the operator is set to ``True`` the operation returns .. math:: F*D = (\partial_{i_{1}...i_{n}}F)*D^{i_{1}...i_{n}} a multivector function. For example the representation of the grad operator in 3d would be: .. math:: D^{i_{1}...i_{n}} &= [e_x,e_y,e_z] \\ \partial_{i_{1}...i_{n}} &= [(1,0,0),(0,1,0),(0,0,1)]. See LaTeX documentation for definitions of operator algebraic operations ``+``, ``-``, ``*``, ``^``, ``|``, ``<``, and ``>``. Attributes ---------- ga : ~galgebra.ga.Ga Associated geometric algebra cmpflg : bool Complement flag terms : list of tuples """ def __init_from_coef_and_pdop(self, coefs: List[Any], pdiffs: List['dop.Pdop']): if len(coefs) != len(pdiffs): raise ValueError('In Dop.__init__ coefficent list and Pdop list must be same length.') self.terms = tuple(zip(coefs, pdiffs)) def __init_from_terms(self, terms: Union[ List[Tuple[Mv, dop.Pdop]], List[Tuple[dop.Sdop, Mv]], ]): if len(terms) == 0: self.terms = () elif all( isinstance(coef, Mv) and isinstance(pdiff, dop.Pdop) for coef, pdiff in terms ): # Mv expansion [(Mv, Pdop)] self.terms = tuple(terms) elif all( isinstance(sdop, dop.Sdop) and isinstance(coef, Mv) for sdop, coef in terms ): # Sdop expansion [(Sdop, Mv)] self.terms = dop._consolidate_terms( (coef * mv, pdiff) for (sdop, mv) in terms for (coef, pdiff) in sdop.terms ) else: raise TypeError( 'In Dop.__init__ terms are neither (Mv, Pdop) pairs or ' '(Sdop, Mv) pairs, got {}'.format(terms)) def __init__(self, *args, ga: 'Ga', cmpflg: bool = False, debug: bool = False) -> None: """ Parameters ---------- ga : Associated geometric algebra cmpflg : bool Complement flag for Dop debug : bool True to print out debugging information """ if ga is None: raise ValueError('ga argument to Dop() must not be None') self.cmpflg = cmpflg self.Ga = ga if len(args) == 2: self.__init_from_coef_and_pdop(*args) elif len(args) == 1: self.__init_from_terms(*args) else: # count include self, as python usually does raise TypeError( "Dop() takes from 1 to 2 positional arguments but {} were " "given".format(len(args)))
[docs] def simplify(self, modes=simplify) -> 'Dop': """ Simplify each multivector coefficient of a partial derivative """ return Dop( [(coef.simplify(modes=modes), pd) for coef, pd in self.terms], ga=self.Ga, cmpflg=self.cmpflg )
[docs] def consolidate_coefs(self) -> 'Dop': """ Remove zero coefs and consolidate coefs with repeated pdiffs. """ return Dop(dop._consolidate_terms(self.terms), ga=self.Ga, cmpflg=self.cmpflg)
@staticmethod def Add(dop1, dop2): if isinstance(dop1, Dop) and isinstance(dop2, Dop): if dop1.Ga != dop2.Ga: raise ValueError('In Dop.Add Dop arguments are not from same geometric algebra') if dop1.cmpflg != dop2.cmpflg: raise ValueError('In Dop.Add complement flags have different values: %s vs. %s' % (dop1.cmpflg, dop2.cmpflg)) return Dop(dop._merge_terms(dop1.terms, dop2.terms), cmpflg=dop1.cmpflg, ga=dop1.Ga) else: # convert values to multiplicative operators if isinstance(dop1, Dop): if not isinstance(dop2, Mv): dop2 = dop1.Ga.mv(dop2) dop2 = Dop([(dop2, dop.Pdop({}))], cmpflg=dop1.cmpflg, ga=dop1.Ga) elif isinstance(dop2, Dop): if not isinstance(dop1, Mv): dop1 = dop2.Ga.mv(dop1) dop1 = Dop([(dop1, dop.Pdop({}))], cmpflg=dop2.cmpflg, ga=dop2.Ga) else: raise TypeError("Neither argument is a Dop instance") return Dop.Add(dop1, dop2) def __add__(self, dop): return Dop.Add(self, dop) def __radd__(self, dop): return Dop.Add(dop, self) def __neg__(self): return Dop( [(-coef, pdiff) for coef, pdiff in self.terms], ga=self.Ga, cmpflg=self.cmpflg ) def __sub__(self, dop): return Dop.Add(self, -dop) def __rsub__(self, dop): return Dop.Add(dop, -self) @staticmethod def Mul(dopl, dopr, op='*'): # General multiplication of Dop's # cmpflg is True if the Dop operates on the left argument and # False if the Dop operates on the right argument if isinstance(dopl, Dop) and isinstance(dopr, Dop): if dopl.Ga != dopr.Ga: raise ValueError('In Dop.Mul Dop arguments are not from same geometric algebra') ga = dopl.Ga if dopl.cmpflg != dopr.cmpflg: raise ValueError('In Dop.Mul Dop arguments do not have same cmplfg') if not dopl.cmpflg: # dopl and dopr operate on right argument product = sum(( Dop.Mul(coef, pdiff(dopr), op=op) for coef, pdiff in dopl.terms ), Dop([], ga=ga, cmpflg=False)) else: # dopl and dopr operate on left argument product = sum(( Dop.Mul(pdiff(dopl), coef, op=op) for coef, pdiff in dopr.terms ), Dop([], ga=ga, cmpflg=True)) else: if not isinstance(dopl, Dop): # dopl is a scalar or Mv and dopr is Dop if isinstance(dopl, Mv) and dopl.Ga != dopr.Ga: raise ValueError('In Dop.Mul Dop arguments are not from same geometric algebra') else: dopl = dopr.Ga.mv(dopl) ga = dopl.Ga if not dopr.cmpflg: # dopr operates on right argument product = Dop([ (Mv.Mul(dopl, coef, op=op), pdiff) for coef, pdiff in dopr.terms ], ga=ga) else: return sum([ Mv.Mul(pdiff(dopl), coef, op=op) for coef, pdiff in dopr.terms ], Mv(0, ga=ga)) else: # dopr is a scalar or a multivector if isinstance(dopr, Mv) and dopl.Ga != dopr.Ga: raise ValueError('In Dop.Mul Dop arguments are not from same geometric algebra') ga = dopl.Ga if not dopl.cmpflg: # dopl operates on right argument return sum([ Mv.Mul(coef, pdiff(dopr), op=op) for coef, pdiff in dopl.terms ], Mv(0, ga=ga)) else: product = Dop([ (Mv.Mul(coef, dopr, op=op), pdiff) for coef, pdiff in dopl.terms ], ga=ga, cmpflg=True) # returns Dop complement return product.consolidate_coefs() def TSimplify(self): return Dop([ (metric.Simp.apply(coef), pdiff) for coef, pdiff in self.terms ], ga=self.Ga) def __truediv__(self, dopr): if isinstance(dopr, (Dop, Mv)): raise TypeError('In Dop.__truediv__ dopr must be a sympy scalar.') return Dop([ (coef / dopr, pdiff) for coef, pdiff in self.terms ], ga=self.Ga, cmpflg=self.cmpflg) def __mul__(self, dopr): # * geometric product return Dop.Mul(self, dopr, op='*') def __rmul__(self, dopl): # * geometric product return Dop.Mul(dopl, self, op='*') def __xor__(self, dopr): # ^ outer product return Dop.Mul(self, dopr, op='^') def __rxor__(self, dopl): # ^ outer product return Dop.Mul(dopl, self, op='^') def __or__(self, dopr): # | inner product return Dop.Mul(self, dopr, op='|') def __ror__(self, dopl): # | inner product return Dop.Mul(dopl, self, op='|') def __lt__(self, dopr): # < left contraction return Dop.Mul(self, dopr, op='<') def __gt__(self, dopr): # > right contraction return Dop.Mul(self, dopr, op='>') def __eq__(self, other): if isinstance(other, Dop): if self.Ga != other.Ga: return NotImplemented diff = self - other return len(diff.terms) == 0 else: return NotImplemented def is_scalar(self) -> bool: return all( not isinstance(coef, Mv) or coef.is_scalar() for coef, pdiff in self.terms ) def components(self) -> Tuple['Dop', ...]: return tuple( Dop([(sdop, Mv(base, ga=self.Ga))], ga=self.Ga) for sdop, base in self.Dop_mv_expand() ) def Dop_mv_expand(self, modes=None) -> List[Tuple[Expr, Expr]]: coefs = [] bases = [] self.consolidate_coefs() for coef, pdiff in self.terms: if isinstance(coef, Mv) and not coef.is_scalar(): for mv_coef, mv_base in metric.linear_expand_terms(coef.obj): if mv_base in bases: index = bases.index(mv_base) coefs[index] += dop.Sdop([(mv_coef, pdiff)]) else: bases.append(mv_base) coefs.append(dop.Sdop([(mv_coef, pdiff)])) else: if isinstance(coef, Mv): mv_coef = coef.obj else: mv_coef = coef if S.One in bases: index = bases.index(S.One) coefs[index] += dop.Sdop([(mv_coef, pdiff)]) else: bases.append(S.One) coefs.append(dop.Sdop([(mv_coef, pdiff)])) if modes is not None: for i in range(len(coefs)): coefs[i] = coefs[i].simplify(modes) terms = list(zip(coefs, bases)) return sorted(terms, key=lambda x: self.Ga.blades.flat.index(x[1])) def _sympystr(self, print_obj: _StrPrinter) -> str: if len(self.terms) == 0: return ZERO_STR mv_terms = self.Dop_mv_expand(modes=simplify) s = '' for sdop, base in mv_terms: str_base = print_obj._print(base) str_sdop = print_obj._print(sdop) if base == S.One: s += str_sdop else: if len(sdop.terms) > 1: if self.cmpflg: s += '(' + str_sdop + ')*' + str_base else: s += str_base + '*(' + str_sdop + ')' else: if str_sdop[0] == '-' and not isinstance(sdop.terms[0][0], Add): if self.cmpflg: s += str_sdop + '*' + str_base else: s += '-' + str_base + '*' + str_sdop[1:] else: if self.cmpflg: s += str_sdop + '*' + str_base else: s += str_base + '*' + str_sdop s += ' + ' s = s.replace('+ -', '-') return s[:-3] def _latex(self, print_obj: _LatexPrinter) -> str: if len(self.terms) == 0: return ZERO_STR self.consolidate_coefs() mv_terms = self.Dop_mv_expand(modes=simplify) s = '' for sdop, base in mv_terms: str_base = print_obj._print(base) str_sdop = print_obj._print(sdop) if base == S.One: s += str_sdop else: if str_sdop == '1': s += str_base if str_sdop == '-1': s += '-' + str_base if str_sdop[1:] != '1': s += ' ' + str_sdop[1:] else: if len(sdop.terms) > 1: if self.cmpflg: s += r'\left ( ' + str_sdop + r'\right ) ' + str_base else: s += str_base + ' ' + r'\left ( ' + str_sdop + r'\right ) ' else: if str_sdop[0] == '-' and not isinstance(sdop.terms[0][0], Add): if self.cmpflg: s += str_sdop + str_base else: s += '-' + str_base + ' ' + str_sdop[1:] else: if self.cmpflg: s += str_sdop + ' ' + str_base else: s += str_base + ' ' + str_sdop s += ' + ' s = s.replace('+ -', '-') return s[:-3] def Fmt(self, fmt: int = 1, title: str = None) -> printer.GaPrintable: if fmt is not None: obj = printer._WithSettings(self, dict(galgebra_mv_fmt=fmt)) else: obj = self return printer._FmtResult(obj, title) def _eval_derivative_n_times(self, x, n): return Dop(dop._eval_derivative_n_times_terms(self.terms, x, n), cmpflg=self.cmpflg, ga=self.Ga)
################################# Alan Macdonald's additions #########################
[docs] def Nga(x, prec=5): """ Like :func:`sympy.N`, but also works on multivectors For multivectors with coefficients that contain floating point numbers, this rounds all these numbers to a precision of ``prec`` and returns the rounded multivector. """ if isinstance(x, Mv): return Mv(Nsympy(x.obj, prec), ga=x.Ga) else: return Nsympy(x, prec)
def printeigen(M): # Print eigenvalues, multiplicities, eigenvectors of M. evects = M.eigenvects() for i in range(len(evects)): # i iterates over eigenvalues print(('Eigenvalue =', evects[i][0], ' Multiplicity =', evects[i][1], ' Eigenvectors:')) for j in range(len(evects[i][2])): # j iterates over eigenvectors of a given eigenvalue result = '[' for k in range(len(evects[i][2][j])): # k iterates over coordinates of an eigenvector result += str(trigsimp(evects[i][2][j][k]).evalf(3)) if k != len(evects[i][2][j]) - 1: result += ', ' result += '] ' print(result) def printGS(M, norm=False): # Print Gram-Schmidt output. from sympy import GramSchmidt global N N = GramSchmidt(M, norm) result = '[ ' for i in range(len(N)): result += '[' for j in range(len(N[0])): result += str(trigsimp(N[i][j]).evalf(3)) if j != len(N[0]) - 1: result += ', ' result += '] ' if j != len(N[0]) - 1: result += ' ' result += ']' print(result) def printrref(matrix, vars="xyzuvwrs"): # Print rref of matrix with variables. rrefmatrix = matrix.rref()[0] rows, cols = rrefmatrix.shape if len(vars) < cols - 1: print('Not enough variables.') return for i in range(rows): result = '' for j in range(cols - 1): result += str(rrefmatrix[i, j]) + vars[j] if j != cols - 2: result += ' + ' result += ' = ' + str(rrefmatrix[i, cols - 1]) print(result) def com(A, B): raise ImportError( """mv.com is removed, please use galgebra.ga.Ga.com(A, B) instead.""") def correlation(u, v, dec=3): # Compute the correlation coefficient of vectors u and v. rows, cols = u.shape uave = 0 vave = 0 for i in range(rows): uave += u[i] vave += v[i] uave = uave / rows vave = vave / rows ulocal = u[:, :] # Matrix copy vlocal = v[:, :] for i in range(rows): ulocal[i] -= uave vlocal[i] -= vave return ulocal.dot(vlocal) / (ulocal.norm() * vlocal.norm()). evalf(dec)
[docs] def cross(v1: Mv, v2: Mv) -> Mv: r""" If ``v1`` and ``v2`` are 3-dimensional Euclidean vectors, compute the vector cross product :math:`v_{1}\times v_{2} = -I{\lp {v_{1}{\wedge}v_{2}} \rp }`. """ if v1.is_vector() and v2.is_vector() and v1.Ga == v2.Ga and v1.Ga.n == 3: return -v1.Ga.I() * (v1 ^ v2) else: raise ValueError(str(v1) + ' and ' + str(v2) + ' not compatible for cross product.')
[docs] def dual(A: Mv) -> Mv: """ Equivalent to :meth:`Mv.dual` """ if isinstance(A, Mv): return A.dual() else: raise ValueError('A not a multivector in dual(A)')
# ## GSG code starts ###
[docs] def undual(A: Mv) -> Mv: """ Equivalent to :meth: `Mv.undual`. Inverse function to multivector function `dual`, so both `undual(dual(A))` and `dual(undual(A))` return `A`. """ if isinstance(A, Mv): return A.undual() else: raise ValueError('A not a multivector in undual(A).')
# ## GSG code ends ###
[docs] def even(A: Mv) -> Mv: """ Equivalent to :meth:`Mv.even` """ if not isinstance(A, Mv): raise ValueError('A = ' + str(A) + ' not a multivector in even(A).') return A.even()
[docs] def odd(A: Mv) -> Mv: """ Equivalent to :meth:`Mv.odd` """ if not isinstance(A, Mv): raise ValueError('A = ' + str(A) + ' not a multivector in even(A).') return A.odd()
# ## GSG code starts ###
[docs] def g_invol(A: Mv) -> Mv: """ Equivalent to :meth: `Mv.g_invol`. - Returns grade involute of multivector `A`; negates `A`'s odd grade part but preserves its even grade part. - Grade involution is its own inverse operation. """ if not isinstance(A, Mv): raise ValueError('A not a multivector in g_invol(A)') return A.g_invol()
# ## GSG code ends ###
[docs] def exp(A: Union[Mv, Expr], hint: str = '-') -> Union[Mv, Expr]: """ If ``A`` is a multivector then ``A.exp(hint)`` is returned. If ``A`` is a *sympy* expression the *sympy* expression :math:`e^{A}` is returned (see :func:`sympy.exp`). """ if isinstance(A, Mv): return A.exp(hint) else: return sympy_exp(A)
[docs] def grade(A: Mv, r: int = 0) -> Mv: """ Equivalent to :meth:`Mv.grade` """ if isinstance(A, Mv): return A.grade(r) else: raise ValueError('A not a multivector in grade(A, r)')
[docs] def inv(A: Mv) -> Mv: """ Equivalent to :meth:`Mv.inv` """ if not isinstance(A, Mv): raise ValueError('A = ' + str(A) + ' not a multivector in inv(A).') return A.inv()
# ## GSG code starts ###
[docs] def qform(A: Mv) -> Expr: """ - Equivalent to :meth:`Mv.qform`. - qform(A) returns the quadratic form at multivector A. - Returned value is a real SymPy expression, NOT a GAlgebra 0-vector. - Expression necessarily represents a nonnegative number only when A's geometric algebra has a Euclidean metric. """ if not isinstance(A, Mv): raise TypeError('A not a multivector in qform(A)') return A.qform()
# ## GSG code ends ### # ## GSG code starts ###
[docs] def norm2(A: Mv, hint: str = '0') -> Expr: """ - Equivalent to :meth:`Mv.norm2` - Returns the normsquared of multivector self, defined as the absolute value of the quadratic form at self. - norm2(A() returns a real SymPy expression, NOT a GAlgebra 0-vector. Whether numeric or symbolic, norm2(A) always represents a nonnegative number. - String values '+', '-', or '0' of hint respectively determine whether the quadratic form, the absolute value of which is the norm squared, should be regarded as nonnegative, nonpositive, or of unknown sign, except when that quantity's sign can be determined by other considerations, such as the metric being Euclidean. """ if not isinstance(A, Mv): raise TypeError('A not a multivector in norm2(A)') return A.norm2(hint)
# ## GSG code ends ### # ## GSG code starts ###
[docs] def norm(A: Mv, hint: str = '0') -> Expr: """ - Equivalent to :meth:`Mv.norm` - Whether numeric nor symbolic, returned value is a real SymPy expression that necessarily represents a nonnegative number. Returned value is NOT a GAlgebra 0-vector. - String values '+', '-', or '0' of hint respectively determine whether a symbolic self.norm2() expression should be regarded as nonnegative, nonpositive, or of unknown sign. """ if not isinstance(A, Mv): raise TypeError('A not a multivector in norm(A)') return A.norm(hint=hint)
# ## GSG code ends ### # ## GSG code starts ###
[docs] def mag2(A: Mv) -> Expr: """ - Equivalent to :meth:`Mv.mag2` - Returns the magnitude squared of multivector self, defined as the sum of the absolute values of the norm squareds of self's grade parts. - Returned value is a real SymPy expression, NOT a GAlgebra 0-vector. Expression necesssarily represents a nonnegative number. - The magnitude squared differs from the normsquared of `self` when the metric is non-Euclidean. """ if not isinstance(A, Mv): raise TypeError('A not a multivector in mag2(A)') return A.mag2()
# ## GSG code ends ### # ## GSG code starts ###
[docs] def mag(A: Mv) -> Expr: """ - Equivalent to :meth:`Mv.mag` - Returns the magnitude of multivector self, defined as the square root of the magnitude squared. - The magnitude necessarily agrees with the norm only when the metric is Euclidean. Otherwise the magnitude is greater than or equal to the norm. """ if not isinstance(A, Mv): raise TypeError('A not a multivector in mag(A)') return A.mag()
# ## GSG code ends ###
[docs] def proj(B: Mv, A: Mv) -> Mv: """ Equivalent to :meth:`Mv.project_in_blade` """ if isinstance(A, Mv): return A.project_in_blade(B) else: raise ValueError('A not a multivector in proj(B, A)')
[docs] def rot(itheta: Mv, A: Mv, hint: str = '-') -> Mv: """ Equivalent to ``A.rotate_multivector(itheta, hint)`` where ``itheta`` is the bi-vector blade defining the rotation. For the use of ``hint`` see the method :meth:`Mv.rotate_multivector`. """ if isinstance(A, Mv): return A.rotate_multivector(itheta, hint) else: raise ValueError('A not a multivector in rotate(A, itheta)')
[docs] def refl(B: Mv, A: Mv) -> Mv: r""" Reflect multivector :math:`A` in blade :math:`B`. Returns :math:`\sum_{r}(-1)^{s(r+1)}B{\left < {A} \right >}_{r}B^{-1}`. if :math:`B` has grade :math:`s`. Equivalent to :meth:`Mv.reflect_in_blade` """ if isinstance(A, Mv): return A.reflect_in_blade(B) else: raise ValueError('A not a multivector in reflect(B, A)')
[docs] def rev(A: Mv) -> Mv: """ Equivalent to :meth:`Mv.rev` """ if isinstance(A, Mv): return A.rev() else: raise ValueError('A not a multivector in rev(A)')
# ## GSG code starts ###
[docs] def ccon(A: Mv) -> Mv: """ - Equivalent to :meth: `Mv.ccon`. - Returns Clifford conjugate of multivector `self`, i.e. returns the reverse of self's grade involute. - Clifford conjugation is its own inverse operation. """ if not isinstance(A, Mv): raise ValueError('A not a multivector in ccon(A)') return A.ccon()
# ## GSG code ends ###
[docs] def scalar(A: Mv) -> Expr: """ Equivalent to :meth:`Mv.scalar` """ if not isinstance(A, Mv): raise ValueError('A = ' + str(A) + ' not a multivector in scalar(A).') return A.scalar()
# ## GSG code starts ###
[docs] def sp(A: Mv, B: Mv, switch='') -> Expr: """ - Equivalent to :meth: `Mv.sp`. - Returns scalar product of multivectors A and B. - Returns a real SymPy expression, not a GAlgebra 0-vector. - switch can be either '' (the empty string) or 'rev'. The latter causes left factor A to be reversed before its product with B is taken. """ if not isinstance(A, Mv): raise ValueError("Left factor of sp must be a multivector") return A.sp(B, switch)
# ## GSG code ends ###