Sau*_*tro 4 python matrix sympy automatic-differentiation differentiation
我们需要微分算子的两个矩阵[B]和[C]如：
B = sympy.Matrix([[ D(x), D(y) ],
[ D(y), D(x) ]])
C = sympy.Matrix([[ D(x), D(y) ]])
ans = B * sympy.Matrix([[x*y**2],
[x**2*y]])
print ans
[x**2 + y**2]
[ 4*x*y]
ans2 = ans * C
print ans2
[2*x, 2*y]
[4*y, 4*x]
这也可以用于计算矢量场的卷曲，例如：
culr = sympy.Matrix([[ D(x), D(y), D(z) ]])
field = sympy.Matrix([[ x**2*y, x*y*z, -x**2*y**2 ]])
要使用 Sympy 解决这个问题，必须创建以下 Python 类：
import sympy
class D( sympy.Derivative ):
def __init__( self, var ):
super( D, self ).__init__()
self.var = var
def __mul__(self, other):
return sympy.diff( other, self.var )
当微分运算符的矩阵在左边相乘时，这个类单独解决。这里diff只在要微分的函数已知时才执行。
为了解决当微分运算符矩阵在右边相乘时，__mul__核心类中的方法Expr必须按以下方式更改：
class Expr(Basic, EvalfMixin):
# ...
def __mul__(self, other):
import sympy
if other.__class__.__name__ == 'D':
return sympy.diff( self, other.var )
else:
return Mul(self, other)
#...
它工作得很好，但 Sympy 中应该有一个更好的本地解决方案来处理这个问题。有人知道它可能是什么吗？
此解决方案应用了其他答案和此处的提示。该D操作员可以定义为如下：
D(t)*2*t**3 = 6*t**2但2*t**3*D(t)什么也不做D必须具有is_commutative = FalseevaluateExpr()
D运算符并将mydiff()*应用到相应的右侧部分*：mydiff用于代替diff允许D创建更高的顺序，例如mydiff(D(t), t) = D(t,t)
该diff内部__mul__()的D保持，仅供参考，因为在当前的解决方案的evaluateExpr()实际执行分化的工作。创建了一个 python mudule 并保存为d.py.
import sympy
from sympy.core.decorators import call_highest_priority
from sympy import Expr, Matrix, Mul, Add, diff
from sympy.core.numbers import Zero
class D(Expr):
_op_priority = 11.
is_commutative = False
def __init__(self, *variables, **assumptions):
super(D, self).__init__()
self.evaluate = False
self.variables = variables
def __repr__(self):
return 'D%s' % str(self.variables)
def __str__(self):
return self.__repr__()
@call_highest_priority('__mul__')
def __rmul__(self, other):
return Mul(other, self)
@call_highest_priority('__rmul__')
def __mul__(self, other):
if isinstance(other, D):
variables = self.variables + other.variables
return D(*variables)
if isinstance(other, Matrix):
other_copy = other.copy()
for i, elem in enumerate(other):
other_copy[i] = self * elem
return other_copy
if self.evaluate:
return diff(other, *self.variables)
else:
return Mul(self, other)
def __pow__(self, other):
variables = self.variables
for i in range(other-1):
variables += self.variables
return D(*variables)
def mydiff(expr, *variables):
if isinstance(expr, D):
expr.variables += variables
return D(*expr.variables)
if isinstance(expr, Matrix):
expr_copy = expr.copy()
for i, elem in enumerate(expr):
expr_copy[i] = diff(elem, *variables)
return expr_copy
return diff(expr, *variables)
def evaluateMul(expr):
end = 0
if expr.args:
if isinstance(expr.args[-1], D):
if len(expr.args[:-1])==1:
cte = expr.args[0]
return Zero()
end = -1
for i in range(len(expr.args)-1+end, -1, -1):
arg = expr.args[i]
if isinstance(arg, Add):
arg = evaluateAdd(arg)
if isinstance(arg, Mul):
arg = evaluateMul(arg)
if isinstance(arg, D):
left = Mul(*expr.args[:i])
right = Mul(*expr.args[i+1:])
right = mydiff(right, *arg.variables)
ans = left * right
return evaluateMul(ans)
return expr
def evaluateAdd(expr):
newargs = []
for arg in expr.args:
if isinstance(arg, Mul):
arg = evaluateMul(arg)
if isinstance(arg, Add):
arg = evaluateAdd(arg)
if isinstance(arg, D):
arg = Zero()
newargs.append(arg)
return Add(*newargs)
#courtesy: /sf/answers/3380403491/
def disableNonCommutivity(expr):
replacements = {s: sympy.Dummy(s.name) for s in expr.free_symbols}
return expr.xreplace(replacements)
def evaluateExpr(expr):
if isinstance(expr, Matrix):
for i, elem in enumerate(expr):
elem = elem.expand()
expr[i] = evaluateExpr(elem)
return disableNonCommutivity(expr)
expr = expr.expand()
if isinstance(expr, Mul):
expr = evaluateMul(expr)
elif isinstance(expr, Add):
expr = evaluateAdd(expr)
elif isinstance(expr, D):
expr = Zero()
return disableNonCommutivity(expr)
示例 1：矢量场的卷曲。请注意，定义变量很重要，commutative=False因为它们的顺序Mul().args会影响结果，请参阅其他问题。
from d import D, evaluateExpr
from sympy import Matrix
sympy.var('x', commutative=False)
sympy.var('y', commutative=False)
sympy.var('z', commutative=False)
curl = Matrix( [[ D(x), D(y), D(z) ]] )
field = Matrix( [[ x**2*y, x*y*z, -x**2*y**2 ]] )
evaluateExpr( curl.cross( field ) )
# [-x*y - 2*x**2*y, 2*x*y**2, -x**2 + y*z]
示例 2：结构分析中使用的典型 Ritz 近似。
from d import D, evaluateExpr
from sympy import sin, cos, Matrix
sin.is_commutative = False
cos.is_commutative = False
g1 = []
g2 = []
g3 = []
sympy.var('x', commutative=False)
sympy.var('t', commutative=False)
sympy.var('r', commutative=False)
sympy.var('A', commutative=False)
m=5
n=5
for j in xrange(1,n+1):
for i in xrange(1,m+1):
g1 += [sin(i*x)*sin(j*t), 0, 0]
g2 += [ 0, cos(i*x)*sin(j*t), 0]
g3 += [ 0, 0, sin(i*x)*cos(j*t)]
g = Matrix( [g1, g2, g3] )
B = Matrix(\
[[ D(x), 0, 0],
[ 1/r*A, 0, 0],
[ 1/r*D(t), 0, 0],
[ 0, D(x), 0],
[ 0, 1/r*A, 1/r*D(t)],
[ 0, 1/r*D(t), D(x)-1/x],
[ 0, 0, 1],
[ 0, 1, 0]])
ans = evaluateExpr(B*g)
print_to_file()已经创建了一个函数来快速检查大表达式。
import sympy
import subprocess
def print_to_file( guy, append=False ):
flag = 'w'
if append: flag = 'a'
outfile = open(r'print.txt', flag)
outfile.write('\n')
outfile.write( sympy.pretty(guy, wrap_line=False) )
outfile.write('\n')
outfile.close()
subprocess.Popen( [r'notepad.exe', r'print.txt'] )
print_to_file( B*g )
print_to_file( ans, append=True )