# 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  # pragma: no cover

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
`