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