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467 lines
14 KiB
467 lines
14 KiB
from sympy.assumptions.ask import ask, Q
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from sympy.assumptions.refine import handlers_dict
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from sympy.core import Basic, sympify, S
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from sympy.core.mul import mul, Mul
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from sympy.core.numbers import Number, Integer
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from sympy.core.symbol import Dummy
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from sympy.functions import adjoint
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from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
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do_one, new)
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from sympy.matrices.common import ShapeError, NonInvertibleMatrixError
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from sympy.matrices.matrices import MatrixBase
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from .inverse import Inverse
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from .matexpr import MatrixExpr
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from .matpow import MatPow
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from .transpose import transpose
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from .permutation import PermutationMatrix
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from .special import ZeroMatrix, Identity, GenericIdentity, OneMatrix
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# XXX: MatMul should perhaps not subclass directly from Mul
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class MatMul(MatrixExpr, Mul):
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"""
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A product of matrix expressions
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Examples
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========
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>>> from sympy import MatMul, MatrixSymbol
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>>> A = MatrixSymbol('A', 5, 4)
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>>> B = MatrixSymbol('B', 4, 3)
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>>> C = MatrixSymbol('C', 3, 6)
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>>> MatMul(A, B, C)
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A*B*C
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"""
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is_MatMul = True
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identity = GenericIdentity()
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def __new__(cls, *args, evaluate=False, check=True, _sympify=True):
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if not args:
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return cls.identity
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# This must be removed aggressively in the constructor to avoid
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# TypeErrors from GenericIdentity().shape
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args = list(filter(lambda i: cls.identity != i, args))
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if _sympify:
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args = list(map(sympify, args))
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obj = Basic.__new__(cls, *args)
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factor, matrices = obj.as_coeff_matrices()
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if check:
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validate(*matrices)
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if not matrices:
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# Should it be
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#
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# return Basic.__neq__(cls, factor, GenericIdentity()) ?
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return factor
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if evaluate:
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return canonicalize(obj)
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return obj
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@property
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def shape(self):
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matrices = [arg for arg in self.args if arg.is_Matrix]
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return (matrices[0].rows, matrices[-1].cols)
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def could_extract_minus_sign(self):
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return self.args[0].could_extract_minus_sign()
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def _entry(self, i, j, expand=True, **kwargs):
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# Avoid cyclic imports
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from sympy.concrete.summations import Sum
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from sympy.matrices.immutable import ImmutableMatrix
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coeff, matrices = self.as_coeff_matrices()
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if len(matrices) == 1: # situation like 2*X, matmul is just X
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return coeff * matrices[0][i, j]
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indices = [None]*(len(matrices) + 1)
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ind_ranges = [None]*(len(matrices) - 1)
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indices[0] = i
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indices[-1] = j
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def f():
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counter = 1
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while True:
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yield Dummy("i_%i" % counter)
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counter += 1
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dummy_generator = kwargs.get("dummy_generator", f())
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for i in range(1, len(matrices)):
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indices[i] = next(dummy_generator)
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for i, arg in enumerate(matrices[:-1]):
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ind_ranges[i] = arg.shape[1] - 1
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matrices = [arg._entry(indices[i], indices[i+1], dummy_generator=dummy_generator) for i, arg in enumerate(matrices)]
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expr_in_sum = Mul.fromiter(matrices)
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if any(v.has(ImmutableMatrix) for v in matrices):
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expand = True
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result = coeff*Sum(
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expr_in_sum,
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*zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
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)
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# Don't waste time in result.doit() if the sum bounds are symbolic
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if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
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expand = False
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return result.doit() if expand else result
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def as_coeff_matrices(self):
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scalars = [x for x in self.args if not x.is_Matrix]
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matrices = [x for x in self.args if x.is_Matrix]
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coeff = Mul(*scalars)
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if coeff.is_commutative is False:
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raise NotImplementedError("noncommutative scalars in MatMul are not supported.")
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return coeff, matrices
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def as_coeff_mmul(self):
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coeff, matrices = self.as_coeff_matrices()
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return coeff, MatMul(*matrices)
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def _eval_transpose(self):
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"""Transposition of matrix multiplication.
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Notes
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=====
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The following rules are applied.
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Transposition for matrix multiplied with another matrix:
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`\\left(A B\\right)^{T} = B^{T} A^{T}`
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Transposition for matrix multiplied with scalar:
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`\\left(c A\\right)^{T} = c A^{T}`
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Transpose
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"""
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coeff, matrices = self.as_coeff_matrices()
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return MatMul(
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coeff, *[transpose(arg) for arg in matrices[::-1]]).doit()
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def _eval_adjoint(self):
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return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()
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def _eval_trace(self):
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factor, mmul = self.as_coeff_mmul()
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if factor != 1:
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from .trace import trace
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return factor * trace(mmul.doit())
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else:
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raise NotImplementedError("Can't simplify any further")
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def _eval_determinant(self):
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from sympy.matrices.expressions.determinant import Determinant
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factor, matrices = self.as_coeff_matrices()
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square_matrices = only_squares(*matrices)
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return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))
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def _eval_inverse(self):
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try:
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return MatMul(*[
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arg.inverse() if isinstance(arg, MatrixExpr) else arg**-1
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for arg in self.args[::-1]]).doit()
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except ShapeError:
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return Inverse(self)
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def doit(self, **kwargs):
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deep = kwargs.get('deep', True)
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if deep:
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args = [arg.doit(**kwargs) for arg in self.args]
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else:
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args = self.args
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# treat scalar*MatrixSymbol or scalar*MatPow separately
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expr = canonicalize(MatMul(*args))
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return expr
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# Needed for partial compatibility with Mul
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def args_cnc(self, **kwargs):
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coeff_c = [x for x in self.args if x.is_commutative]
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coeff_nc = [x for x in self.args if not x.is_commutative]
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return [coeff_c, coeff_nc]
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def _eval_derivative_matrix_lines(self, x):
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from .transpose import Transpose
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with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
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lines = []
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for ind in with_x_ind:
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left_args = self.args[:ind]
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right_args = self.args[ind+1:]
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if right_args:
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right_mat = MatMul.fromiter(right_args)
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else:
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right_mat = Identity(self.shape[1])
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if left_args:
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left_rev = MatMul.fromiter([Transpose(i).doit() if i.is_Matrix else i for i in reversed(left_args)])
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else:
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left_rev = Identity(self.shape[0])
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d = self.args[ind]._eval_derivative_matrix_lines(x)
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for i in d:
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i.append_first(left_rev)
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i.append_second(right_mat)
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lines.append(i)
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return lines
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mul.register_handlerclass((Mul, MatMul), MatMul)
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def validate(*matrices):
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""" Checks for valid shapes for args of MatMul """
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for i in range(len(matrices)-1):
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A, B = matrices[i:i+2]
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if A.cols != B.rows:
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raise ShapeError("Matrices %s and %s are not aligned"%(A, B))
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# Rules
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def newmul(*args):
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if args[0] == 1:
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args = args[1:]
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return new(MatMul, *args)
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def any_zeros(mul):
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if any(arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
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for arg in mul.args):
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matrices = [arg for arg in mul.args if arg.is_Matrix]
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return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
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return mul
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def merge_explicit(matmul):
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""" Merge explicit MatrixBase arguments
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>>> from sympy import MatrixSymbol, Matrix, MatMul, pprint
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>>> from sympy.matrices.expressions.matmul import merge_explicit
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>>> A = MatrixSymbol('A', 2, 2)
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>>> B = Matrix([[1, 1], [1, 1]])
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>>> C = Matrix([[1, 2], [3, 4]])
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>>> X = MatMul(A, B, C)
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>>> pprint(X)
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[1 1] [1 2]
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A*[ ]*[ ]
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[1 1] [3 4]
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>>> pprint(merge_explicit(X))
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[4 6]
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A*[ ]
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[4 6]
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>>> X = MatMul(B, A, C)
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>>> pprint(X)
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[1 1] [1 2]
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[ ]*A*[ ]
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[1 1] [3 4]
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>>> pprint(merge_explicit(X))
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[1 1] [1 2]
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[ ]*A*[ ]
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[1 1] [3 4]
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"""
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if not any(isinstance(arg, MatrixBase) for arg in matmul.args):
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return matmul
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newargs = []
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last = matmul.args[0]
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for arg in matmul.args[1:]:
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if isinstance(arg, (MatrixBase, Number)) and isinstance(last, (MatrixBase, Number)):
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last = last * arg
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else:
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newargs.append(last)
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last = arg
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newargs.append(last)
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return MatMul(*newargs)
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def remove_ids(mul):
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""" Remove Identities from a MatMul
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This is a modified version of sympy.strategies.rm_id.
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This is necesssary because MatMul may contain both MatrixExprs and Exprs
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as args.
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See Also
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========
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sympy.strategies.rm_id
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"""
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# Separate Exprs from MatrixExprs in args
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factor, mmul = mul.as_coeff_mmul()
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# Apply standard rm_id for MatMuls
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result = rm_id(lambda x: x.is_Identity is True)(mmul)
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if result != mmul:
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return newmul(factor, *result.args) # Recombine and return
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else:
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return mul
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def factor_in_front(mul):
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factor, matrices = mul.as_coeff_matrices()
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if factor != 1:
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return newmul(factor, *matrices)
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return mul
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def combine_powers(mul):
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r"""Combine consecutive powers with the same base into one, e.g.
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$$A \times A^2 \Rightarrow A^3$$
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This also cancels out the possible matrix inverses using the
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knowledgebase of :class:`~.Inverse`, e.g.,
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$$ Y \times X \times X^{-1} \Rightarrow Y $$
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"""
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factor, args = mul.as_coeff_matrices()
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new_args = [args[0]]
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for B in args[1:]:
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A = new_args[-1]
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if A.is_square == False or B.is_square == False:
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new_args.append(B)
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continue
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if isinstance(A, MatPow):
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A_base, A_exp = A.args
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else:
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A_base, A_exp = A, S.One
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if isinstance(B, MatPow):
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B_base, B_exp = B.args
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else:
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B_base, B_exp = B, S.One
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if A_base == B_base:
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new_exp = A_exp + B_exp
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new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
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continue
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elif not isinstance(B_base, MatrixBase):
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try:
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B_base_inv = B_base.inverse()
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except NonInvertibleMatrixError:
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B_base_inv = None
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if B_base_inv is not None and A_base == B_base_inv:
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new_exp = A_exp - B_exp
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new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
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continue
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new_args.append(B)
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return newmul(factor, *new_args)
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def combine_permutations(mul):
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"""Refine products of permutation matrices as the products of cycles.
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"""
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args = mul.args
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l = len(args)
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if l < 2:
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return mul
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result = [args[0]]
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for i in range(1, l):
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A = result[-1]
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B = args[i]
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if isinstance(A, PermutationMatrix) and \
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isinstance(B, PermutationMatrix):
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cycle_1 = A.args[0]
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cycle_2 = B.args[0]
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result[-1] = PermutationMatrix(cycle_1 * cycle_2)
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else:
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result.append(B)
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return MatMul(*result)
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def combine_one_matrices(mul):
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"""
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Combine products of OneMatrix
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e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4)
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"""
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factor, args = mul.as_coeff_matrices()
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new_args = [args[0]]
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for B in args[1:]:
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A = new_args[-1]
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if not isinstance(A, OneMatrix) or not isinstance(B, OneMatrix):
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new_args.append(B)
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continue
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new_args.pop()
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new_args.append(OneMatrix(A.shape[0], B.shape[1]))
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factor *= A.shape[1]
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return newmul(factor, *new_args)
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def distribute_monom(mul):
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"""
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Simplify MatMul expressions but distributing
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rational term to MatMul.
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e.g. 2*(A+B) -> 2*A + 2*B
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"""
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args = mul.args
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if len(args) == 2:
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from .matadd import MatAdd
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if args[0].is_MatAdd and args[1].is_Rational:
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return MatAdd(*[MatMul(mat, args[1]).doit() for mat in args[0].args])
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if args[1].is_MatAdd and args[0].is_Rational:
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return MatAdd(*[MatMul(args[0], mat).doit() for mat in args[1].args])
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return mul
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rules = (
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distribute_monom, any_zeros, remove_ids, combine_one_matrices, combine_powers, unpack, rm_id(lambda x: x == 1),
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merge_explicit, factor_in_front, flatten, combine_permutations)
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canonicalize = exhaust(typed({MatMul: do_one(*rules)}))
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def only_squares(*matrices):
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"""factor matrices only if they are square"""
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if matrices[0].rows != matrices[-1].cols:
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raise RuntimeError("Invalid matrices being multiplied")
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out = []
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start = 0
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for i, M in enumerate(matrices):
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if M.cols == matrices[start].rows:
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out.append(MatMul(*matrices[start:i+1]).doit())
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start = i+1
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return out
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def refine_MatMul(expr, assumptions):
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"""
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>>> from sympy import MatrixSymbol, Q, assuming, refine
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>>> X = MatrixSymbol('X', 2, 2)
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>>> expr = X * X.T
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>>> print(expr)
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X*X.T
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>>> with assuming(Q.orthogonal(X)):
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... print(refine(expr))
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I
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"""
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newargs = []
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exprargs = []
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for args in expr.args:
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if args.is_Matrix:
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exprargs.append(args)
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else:
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newargs.append(args)
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last = exprargs[0]
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for arg in exprargs[1:]:
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if arg == last.T and ask(Q.orthogonal(arg), assumptions):
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last = Identity(arg.shape[0])
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elif arg == last.conjugate() and ask(Q.unitary(arg), assumptions):
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last = Identity(arg.shape[0])
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else:
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newargs.append(last)
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last = arg
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newargs.append(last)
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return MatMul(*newargs)
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handlers_dict['MatMul'] = refine_MatMul
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