167 lines
5.2 KiB
Python
167 lines
5.2 KiB
Python
from sympy.core.basic import Basic
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from sympy.core.expr import Expr, ExprBuilder
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from sympy.core.singleton import S
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from sympy.core.sorting import default_sort_key
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from sympy.core.symbol import uniquely_named_symbol
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from sympy.core.sympify import sympify
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from sympy.matrices.matrices import MatrixBase
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from sympy.matrices.common import NonSquareMatrixError
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class Trace(Expr):
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"""Matrix Trace
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Represents the trace of a matrix expression.
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Examples
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========
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>>> from sympy import MatrixSymbol, Trace, eye
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>>> A = MatrixSymbol('A', 3, 3)
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>>> Trace(A)
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Trace(A)
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>>> Trace(eye(3))
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Trace(Matrix([
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[1, 0, 0],
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[0, 1, 0],
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[0, 0, 1]]))
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>>> Trace(eye(3)).simplify()
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3
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"""
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is_Trace = True
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is_commutative = True
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def __new__(cls, mat):
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mat = sympify(mat)
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if not mat.is_Matrix:
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raise TypeError("input to Trace, %s, is not a matrix" % str(mat))
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if mat.is_square is False:
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raise NonSquareMatrixError("Trace of a non-square matrix")
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return Basic.__new__(cls, mat)
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def _eval_transpose(self):
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return self
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def _eval_derivative(self, v):
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from sympy.concrete.summations import Sum
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from .matexpr import MatrixElement
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if isinstance(v, MatrixElement):
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return self.rewrite(Sum).diff(v)
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expr = self.doit()
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if isinstance(expr, Trace):
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# Avoid looping infinitely:
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raise NotImplementedError
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return expr._eval_derivative(v)
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def _eval_derivative_matrix_lines(self, x):
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from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction
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r = self.args[0]._eval_derivative_matrix_lines(x)
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for lr in r:
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if lr.higher == 1:
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lr.higher = ExprBuilder(
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ArrayContraction,
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[
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ExprBuilder(
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ArrayTensorProduct,
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[
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lr._lines[0],
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lr._lines[1],
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]
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),
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(1, 3),
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],
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validator=ArrayContraction._validate
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)
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else:
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# This is not a matrix line:
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lr.higher = ExprBuilder(
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ArrayContraction,
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[
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ExprBuilder(
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ArrayTensorProduct,
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[
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lr._lines[0],
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lr._lines[1],
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lr.higher,
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]
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),
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(1, 3), (0, 2)
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]
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)
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lr._lines = [S.One, S.One]
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lr._first_pointer_parent = lr._lines
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lr._second_pointer_parent = lr._lines
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lr._first_pointer_index = 0
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lr._second_pointer_index = 1
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return r
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@property
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def arg(self):
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return self.args[0]
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def doit(self, **hints):
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if hints.get('deep', True):
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arg = self.arg.doit(**hints)
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try:
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return arg._eval_trace()
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except (AttributeError, NotImplementedError):
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return Trace(arg)
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else:
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# _eval_trace would go too deep here
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if isinstance(self.arg, MatrixBase):
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return trace(self.arg)
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else:
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return Trace(self.arg)
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def as_explicit(self):
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return Trace(self.arg.as_explicit()).doit()
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def _normalize(self):
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# Normalization of trace of matrix products. Use transposition and
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# cyclic properties of traces to make sure the arguments of the matrix
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# product are sorted and the first argument is not a transposition.
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from sympy.matrices.expressions.matmul import MatMul
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from sympy.matrices.expressions.transpose import Transpose
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trace_arg = self.arg
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if isinstance(trace_arg, MatMul):
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def get_arg_key(x):
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a = trace_arg.args[x]
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if isinstance(a, Transpose):
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a = a.arg
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return default_sort_key(a)
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indmin = min(range(len(trace_arg.args)), key=get_arg_key)
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if isinstance(trace_arg.args[indmin], Transpose):
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trace_arg = Transpose(trace_arg).doit()
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indmin = min(range(len(trace_arg.args)), key=lambda x: default_sort_key(trace_arg.args[x]))
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trace_arg = MatMul.fromiter(trace_arg.args[indmin:] + trace_arg.args[:indmin])
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return Trace(trace_arg)
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return self
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def _eval_rewrite_as_Sum(self, expr, **kwargs):
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from sympy.concrete.summations import Sum
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i = uniquely_named_symbol('i', expr)
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s = Sum(self.arg[i, i], (i, 0, self.arg.rows - 1))
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return s.doit()
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def trace(expr):
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"""Trace of a Matrix. Sum of the diagonal elements.
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Examples
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========
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>>> from sympy import trace, Symbol, MatrixSymbol, eye
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>>> n = Symbol('n')
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>>> X = MatrixSymbol('X', n, n) # A square matrix
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>>> trace(2*X)
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2*Trace(X)
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>>> trace(eye(3))
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3
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"""
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return Trace(expr).doit()
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