使用 Sympy lambdify 函数实例化贝塞尔函数时出错
Error when instantiating bessel functions with Sympy lambdify function
我目前正在开发一个 python 程序，将 sympy 计算的符号表达式转换为包含所有数值的 numpy 数组。我用 sympy.lambdify 函数实例化符号表达式。
一些符号表达式包含贝塞尔函数，我将scipy.special.jv/jy等作为参数传递给lambdify函数。这是单个代码（来自程序的应用程序）：
m = Symbol('m')
expr= -1.06048319593874*(-(3.14159265358979*m*besselj(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*besselj(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I)))*bessely(27.9937791601866*m, 6.81776274795262*sqrt(-I))/(3.14159265358979*m*bessely(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*bessely(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I))) + besselj(27.9937791601866*m, 6.81776274795262*sqrt(-I)))*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m
nm = 5
vm = arange(nm) +1
bessel = {'besselj':jv,'besselk':kv,'besseli':iv,'bessely':yv}
libraries = [bessel, "numpy"]
result = lambdify(m, expr, modules=libraries)(vm)
In[1] : result
Out[1]:
array([ -7.51212638e-030 -3.22606326e-030j,
4.81143544e-046 +1.04405860e-046j,
1.97977798e-097 +3.02047228e-098j,
3.84986092e-124 +4.73598141e-125j,
1.12934434e-181 +1.21145535e-182j])
结果如我所料：一个 5 行的一维数组，每个整数值为符号 m。
问题是当我尝试在程序中实现它时。这是实现：
expr = list_symbolic_expr[index]
vcm = arange(Dim_col[0]) +1 #Dim_col = list of integer range values to instantiate
m = Symbol(str(Var_col[0])) #Var_col = list of symbolic integer parameters
print expr, m
smat = lambdify(m, expr, modules=libraries)(vcm)
这是使用 scipy.special 贝塞尔函数时的错误：
In [2]: run Get_System_Num
Out [2]:
-1.06048319593874*(-(3.14159265358979*m*besselj(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*besselj(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I)))*bessely(27.9937791601866*m, 6.81776274795262*sqrt(-I))/(3.14159265358979*m*bessely(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*bessely(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I))) + besselj(27.9937791601866*m, 6.81776274795262*sqrt(-I)))*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m m
File "Numeric\Get_System_Num.py", line 183, in get_Mat
smat = lambdify(m, expr, modules=libraries)(vcm)
File "<string>", line 1, in <lambda>
TypeError: ufunc 'jv' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''
这是使用 sympy.special 贝塞尔函数时的错误：
File "Numeric\Get_System_Num.py", line 183, in get_Mat
smat = lambdify(m, expr, modules=libraries)(vcm)
File "<string>", line 1, in <lambda>
File "C:\Users\Emile\Anaconda2\lib\site-packages\sympy\core\function.py", line 375, in __new__
result = super(Function, cls).__new__(cls, *args, **options)
File "C:\Users\Emile\Anaconda2\lib\site-packages\sympy\core\function.py", line 199, in __new__
evaluated = cls.eval(*args)
File "C:\Users\Emile\Anaconda2\lib\site-packages\sympy\functions\special\bessel.py", line 161, in eval
if nu.is_integer:
AttributeError: 'numpy.ndarray' object has no attribute 'is_integer'
似乎 lambdify 无法正确解释程序代码中 bessel 函数的输入值（无论是 scipy 还是 sympy bessel 函数），因为输入值似乎是一个数组对象，而这在单个代码中不是问题。但是我看不出它们之间的区别。
感谢您阅读本文，我希望有人可以帮助我解决这个问题。请告诉我是否需要更多信息来说明这个问题。
--编辑1--
我一直在寻找问题，甚至当我尝试对这个简单的表达式进行 lambdify 时：17469169.5935065*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m，将 "numpy" 作为模块参数，它给出了：'float' object has no attribute 'sin'
我也试过打包模块 'numexpr' 但没有成功，因为它停在另一个表达式：0.058**(46.6321243523316*k)，说：
File "C:\Users\Emile\Anaconda2\lib\site-packages\numexpr\necompiler.py", line 756, in evaluate
signature = [(name, getType(arg)) for (name, arg) in zip(names, arguments)]
File "C:\Users\Emile\Anaconda2\lib\site-packages\numexpr\necompiler.py", line 654, in getType
raise ValueError("unknown type %s" % a.dtype.name)
ValueError: unknown type object
所以我真的不知道这个问题的本质，我什至无法跟踪它，因为引发的异常来自 sympy 内部，我不知道如何在 sympy 函数中放置标志。
--编辑2--
这是 Get_System_Num class 的主体：首先是导入，然后我还从符号项目中导入数据，意思是：我要实例化的每个符号系数（列在Mat_Num);每个系数的实例化符号（在“list_Var”中列出）并取决于行和列；实际大矩阵中实例化矩阵的大小和位置（在list_Dim中列出）。由于这些列表在开始时是象征性的，所以我需要用我在 Pr_Num <=> 数字项目中输入的实际值替换符号，我在 sub_system_num 方法中进行。
from numpy import diag, zeros, concatenate, arange, meshgrid, array
from scipy.special import jv,kv,iv,yv
from sympy import srepr, Matrix, lambdify, Symbol
from Numeric.Sub_System_Num import Sub_System_Num
class Get_System_Num :
#Calling constructor
def __init__(self, Pr_Num) :
#Assigning inputs values to local values
self.Pr_Num = Pr_Num
self.Pr_Symb = Pr_Num.Pr_Symb
#Load data from Pr_Symb
self.Mat_Num_Index = self.Pr_Symb.Mat_Num_Index
#Subs symbols with numeric values
print "Substitute symbols with real values..."
self.Sub_System_Num = Sub_System_Num(self)
#Gathering the results
self.Mat_Num = self.Sub_System_Num.Mat_Num
self.Mat_Size = self.Sub_System_Num.Mat_Size
self.list_Index = self.Sub_System_Num.list_Index
self.list_Var_row = self.Sub_System_Num.list_Var.col(0)
self.list_Dim_row = self.Sub_System_Num.list_Dim.col(0)
self.list_Var_col = self.Sub_System_Num.list_Var.col(1)
self.list_Dim_col = self.Sub_System_Num.list_Dim.col(1)
self.count = 0
print "Compute numerical matrix..."
self.Mat = self.get_Mat()
这是填充数值矩阵的方法。我只提出了只有一个变量可以为行和列实例化的情况。这是 9 (3²) 种可能情况中的一种，因为每一行和每一列都有 0,1 或 2 个变量要实例化。关于 if srepr(coeff).__contains__(srepr(Var_row[0])) ：这会检查变量是否真的包含在 expr 中，如果没有，我会手动复制 coeff。由于表达式以前是用符号计算的，有时 sympy 可能会简化它们，这样变量就不再存在了。
def get_Mat(self) :
bessel = {'besselj':jv,'besselk':kv,'besseli':iv,'bessely':yv}
libraries = [bessel, 'numpy']
Mat = zeros((self.Mat_Size[0], self.Mat_Size[0]), dtype=complex)
for index in range(0, self.Mat_Num_Index.__len__()) :
Nb_row = self.list_Index[self.Mat_Num_Index[index][0], 0]
Nb_col = self.list_Index[self.Mat_Num_Index[index][1], 1]
Pos_row = self.list_Index[self.Mat_Num_Index[index][0], 2]
Pos_col = self.list_Index[self.Mat_Num_Index[index][1], 3]
Var_row = self.list_Var_row[self.Mat_Num_Index[index][0]]
Var_col = self.list_Var_col[self.Mat_Num_Index[index][1]]
Dim_row = self.list_Dim_row[self.Mat_Num_Index[index][0]]
Dim_col = self.list_Dim_col[self.Mat_Num_Index[index][1]]
coeff = self.Mat_Num[index]
#M(K or Z, M or Z)
if Var_row.__len__() == 1 :
if Var_col.__len__() == 1 :
if Var_row[0] == Var_col[0] : #M(K,M=K) or M(Z,Z)
vrk = arange(Dim_row[0]) +1
k = Symbol(str(Var_row[0]))
smat = lambdify(k, coeff, modules=libraries)(vrk)
if srepr(coeff).__contains__(srepr(Var_row[0])) :
smat = diag(smat)
print 'coeff n°', index, ' Case 1/1 M(K,M=K) or M(Z,Z) : i dependent '
print coeff, Var_col, Var_row
else :
smat = diag([smat]*Dim_row[0])
print 'coeff n°', index, ' Case 1/1 M(K,M=K) or M(Z,Z) : i non dependent'
print coeff, Var_col, Var_row
else :
if srepr(coeff).__contains__(srepr(Var_col[0]) and srepr(Var_row[0])) : #M(K,Z) or M(Z,M) or M(K, M)
print 'coeff n°', index, ' Case 1/1 M(K,Z) or M(Z,M) : i dependent '
print coeff, Var_col, Var_row, index
vrk = arange(Dim_row[0]) +1
vcm = arange(Dim_col[0]) +1
mcm, mrk = meshgrid(vcm, vrk)
k = Symbol(str(Var_row[0]))
i = Symbol(str(Var_col[0]))
smat = lambdify((k, i), coeff, modules=libraries)(mrk, mcm)
elif not(srepr(coeff).__contains__(srepr(Var_row[0]))) : #M(Z,M)
print 'coeff n°', index, ' Case 1/1 M(Z,M) : i non dependent '
print coeff, Var_col, Var_row, index
vcm = arange(Dim_col[0]) +1
m = Symbol(str(Var_col[0]))
smat = lambdify(m, coeff, modules=libraries)(vcm)
smat = [smat]*Dim_row[0]
smat = concatenate(list(smat), axis=0)
elif not(srepr(coeff).__contains__(srepr(Var_col[0]))) : #M(K,Z)
print 'coeff n°', index, ' Case 1/1 M(K,Z) : i non dependent'
print coeff, Var_col, Var_row, index
vrk = arange(Dim_row[0]) +1
k = Symbol(str(Var_row[0]))
smat = lambdify(k, coeff, modules=libraries)(vrk)
smat = [smat]*Dim_col[0]
smat = concatenate(list(smat), axis=1)
self.count = self.count +1
Mat[Pos_row:Pos_row+Nb_row, Pos_col:Pos_col+Nb_col] = smat
return Mat
最后我用 lambdified 系数填充矩阵，在本例中是一个较小的矩阵大小 (Nb_row, Nb_col).
我会继续自己搜索，不要犹豫，询问更多详情！
--编辑3--
我发现如果我这样做 :
m = Symbol(str(Var_col[0]))
coeff = lambdify(m, coeff, modules=libraries)
for nm in range(0, Dim_col[0]) :
smat.append(coeff(nm+1))
它有效，但它更耗时（尽管比使用 subs 和 evalf 少得多）。当我使用数组对象（无论是 numpy 还是 sympy 数组类型）调用 lambdify 创建的 lambda 函数时出现错误。
我可以重现你的错误。基于这些输入：
m = Symbol('m')
expr = -1.06048319593874*(-(3.14159265358979*m*besselj(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*besselj(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I)))*bessely(27.9937791601866*m, 6.81776274795262*sqrt(-I))/(3.14159265358979*m*bessely(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*bessely(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I))) + besselj(27.9937791601866*m, 6.81776274795262*sqrt(-I)))*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m
nm = 2
bessel = {'besselj': jv,'besselk':kv,'besseli':iv,'bessely':yv}
libraries = [bessel, "numpy"]
它适用于 vm 中的整数：
vm = np.arange(nm, dtype=np.int) +1
result = lambdify(m, expr, modules=libraries)(vm)
但是复数会产生与您收到的相同的错误消息：
vm = np.arange(nm, dtype=np.complex) +1
result = lambdify(m, expr, modules=libraries)(vm)
错误信息：
TypeError Traceback (most recent call last)
<ipython-input-30-f0e31009275a> in <module>()
1 vm = np.arange(nm, dtype=np.complex) +1
----> 2 result = lambdify(m, expr, modules=libraries)(vm)
/Users/mike/anaconda/envs/py34/lib/python3.4/site-packages/numpy/__init__.py in <lambda>(_Dummy_27)
TypeError: ufunc 'jv' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''
因此，请检查您在 vcm 中获得的数据类型，即 arange(Dim_col[0]) +1 returns。
您只能取具有属性 real 的复数的实部。例如，vm.real 给出 vm 的实部。如果这是你想要的，你应该得到你的结果：
result = lambdify(m, expr, modules=libraries).(vm.real)
我目前正在开发一个 python 程序，将 sympy 计算的符号表达式转换为包含所有数值的 numpy 数组。我用 sympy.lambdify 函数实例化符号表达式。
一些符号表达式包含贝塞尔函数，我将scipy.special.jv/jy等作为参数传递给lambdify函数。这是单个代码（来自程序的应用程序）：
m = Symbol('m')
expr= -1.06048319593874*(-(3.14159265358979*m*besselj(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*besselj(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I)))*bessely(27.9937791601866*m, 6.81776274795262*sqrt(-I))/(3.14159265358979*m*bessely(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*bessely(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I))) + besselj(27.9937791601866*m, 6.81776274795262*sqrt(-I)))*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m
nm = 5
vm = arange(nm) +1
bessel = {'besselj':jv,'besselk':kv,'besseli':iv,'bessely':yv}
libraries = [bessel, "numpy"]
result = lambdify(m, expr, modules=libraries)(vm)
In[1] : result
Out[1]:
array([ -7.51212638e-030 -3.22606326e-030j,
4.81143544e-046 +1.04405860e-046j,
1.97977798e-097 +3.02047228e-098j,
3.84986092e-124 +4.73598141e-125j,
1.12934434e-181 +1.21145535e-182j])
结果如我所料：一个 5 行的一维数组，每个整数值为符号 m。
问题是当我尝试在程序中实现它时。这是实现：
expr = list_symbolic_expr[index]
vcm = arange(Dim_col[0]) +1 #Dim_col = list of integer range values to instantiate
m = Symbol(str(Var_col[0])) #Var_col = list of symbolic integer parameters
print expr, m
smat = lambdify(m, expr, modules=libraries)(vcm)
这是使用 scipy.special 贝塞尔函数时的错误：
In [2]: run Get_System_Num
Out [2]:
-1.06048319593874*(-(3.14159265358979*m*besselj(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*besselj(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I)))*bessely(27.9937791601866*m, 6.81776274795262*sqrt(-I))/(3.14159265358979*m*bessely(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*bessely(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I))) + besselj(27.9937791601866*m, 6.81776274795262*sqrt(-I)))*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m m
File "Numeric\Get_System_Num.py", line 183, in get_Mat
smat = lambdify(m, expr, modules=libraries)(vcm)
File "<string>", line 1, in <lambda>
TypeError: ufunc 'jv' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''
这是使用 sympy.special 贝塞尔函数时的错误：
File "Numeric\Get_System_Num.py", line 183, in get_Mat
smat = lambdify(m, expr, modules=libraries)(vcm)
File "<string>", line 1, in <lambda>
File "C:\Users\Emile\Anaconda2\lib\site-packages\sympy\core\function.py", line 375, in __new__
result = super(Function, cls).__new__(cls, *args, **options)
File "C:\Users\Emile\Anaconda2\lib\site-packages\sympy\core\function.py", line 199, in __new__
evaluated = cls.eval(*args)
File "C:\Users\Emile\Anaconda2\lib\site-packages\sympy\functions\special\bessel.py", line 161, in eval
if nu.is_integer:
AttributeError: 'numpy.ndarray' object has no attribute 'is_integer'
似乎 lambdify 无法正确解释程序代码中 bessel 函数的输入值（无论是 scipy 还是 sympy bessel 函数），因为输入值似乎是一个数组对象，而这在单个代码中不是问题。但是我看不出它们之间的区别。
感谢您阅读本文，我希望有人可以帮助我解决这个问题。请告诉我是否需要更多信息来说明这个问题。
--编辑1--
我一直在寻找问题，甚至当我尝试对这个简单的表达式进行 lambdify 时：17469169.5935065*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m，将 "numpy" 作为模块参数，它给出了：'float' object has no attribute 'sin'
我也试过打包模块 'numexpr' 但没有成功，因为它停在另一个表达式：0.058**(46.6321243523316*k)，说：
File "C:\Users\Emile\Anaconda2\lib\site-packages\numexpr\necompiler.py", line 756, in evaluate
signature = [(name, getType(arg)) for (name, arg) in zip(names, arguments)]
File "C:\Users\Emile\Anaconda2\lib\site-packages\numexpr\necompiler.py", line 654, in getType
raise ValueError("unknown type %s" % a.dtype.name)
ValueError: unknown type object
所以我真的不知道这个问题的本质，我什至无法跟踪它，因为引发的异常来自 sympy 内部，我不知道如何在 sympy 函数中放置标志。
--编辑2--
这是 Get_System_Num class 的主体：首先是导入，然后我还从符号项目中导入数据，意思是：我要实例化的每个符号系数（列在Mat_Num);每个系数的实例化符号（在“list_Var”中列出）并取决于行和列；实际大矩阵中实例化矩阵的大小和位置（在list_Dim中列出）。由于这些列表在开始时是象征性的，所以我需要用我在 Pr_Num <=> 数字项目中输入的实际值替换符号，我在 sub_system_num 方法中进行。
from numpy import diag, zeros, concatenate, arange, meshgrid, array
from scipy.special import jv,kv,iv,yv
from sympy import srepr, Matrix, lambdify, Symbol
from Numeric.Sub_System_Num import Sub_System_Num
class Get_System_Num :
#Calling constructor
def __init__(self, Pr_Num) :
#Assigning inputs values to local values
self.Pr_Num = Pr_Num
self.Pr_Symb = Pr_Num.Pr_Symb
#Load data from Pr_Symb
self.Mat_Num_Index = self.Pr_Symb.Mat_Num_Index
#Subs symbols with numeric values
print "Substitute symbols with real values..."
self.Sub_System_Num = Sub_System_Num(self)
#Gathering the results
self.Mat_Num = self.Sub_System_Num.Mat_Num
self.Mat_Size = self.Sub_System_Num.Mat_Size
self.list_Index = self.Sub_System_Num.list_Index
self.list_Var_row = self.Sub_System_Num.list_Var.col(0)
self.list_Dim_row = self.Sub_System_Num.list_Dim.col(0)
self.list_Var_col = self.Sub_System_Num.list_Var.col(1)
self.list_Dim_col = self.Sub_System_Num.list_Dim.col(1)
self.count = 0
print "Compute numerical matrix..."
self.Mat = self.get_Mat()
这是填充数值矩阵的方法。我只提出了只有一个变量可以为行和列实例化的情况。这是 9 (3²) 种可能情况中的一种，因为每一行和每一列都有 0,1 或 2 个变量要实例化。关于 if srepr(coeff).__contains__(srepr(Var_row[0])) ：这会检查变量是否真的包含在 expr 中，如果没有，我会手动复制 coeff。由于表达式以前是用符号计算的，有时 sympy 可能会简化它们，这样变量就不再存在了。
def get_Mat(self) :
bessel = {'besselj':jv,'besselk':kv,'besseli':iv,'bessely':yv}
libraries = [bessel, 'numpy']
Mat = zeros((self.Mat_Size[0], self.Mat_Size[0]), dtype=complex)
for index in range(0, self.Mat_Num_Index.__len__()) :
Nb_row = self.list_Index[self.Mat_Num_Index[index][0], 0]
Nb_col = self.list_Index[self.Mat_Num_Index[index][1], 1]
Pos_row = self.list_Index[self.Mat_Num_Index[index][0], 2]
Pos_col = self.list_Index[self.Mat_Num_Index[index][1], 3]
Var_row = self.list_Var_row[self.Mat_Num_Index[index][0]]
Var_col = self.list_Var_col[self.Mat_Num_Index[index][1]]
Dim_row = self.list_Dim_row[self.Mat_Num_Index[index][0]]
Dim_col = self.list_Dim_col[self.Mat_Num_Index[index][1]]
coeff = self.Mat_Num[index]
#M(K or Z, M or Z)
if Var_row.__len__() == 1 :
if Var_col.__len__() == 1 :
if Var_row[0] == Var_col[0] : #M(K,M=K) or M(Z,Z)
vrk = arange(Dim_row[0]) +1
k = Symbol(str(Var_row[0]))
smat = lambdify(k, coeff, modules=libraries)(vrk)
if srepr(coeff).__contains__(srepr(Var_row[0])) :
smat = diag(smat)
print 'coeff n°', index, ' Case 1/1 M(K,M=K) or M(Z,Z) : i dependent '
print coeff, Var_col, Var_row
else :
smat = diag([smat]*Dim_row[0])
print 'coeff n°', index, ' Case 1/1 M(K,M=K) or M(Z,Z) : i non dependent'
print coeff, Var_col, Var_row
else :
if srepr(coeff).__contains__(srepr(Var_col[0]) and srepr(Var_row[0])) : #M(K,Z) or M(Z,M) or M(K, M)
print 'coeff n°', index, ' Case 1/1 M(K,Z) or M(Z,M) : i dependent '
print coeff, Var_col, Var_row, index
vrk = arange(Dim_row[0]) +1
vcm = arange(Dim_col[0]) +1
mcm, mrk = meshgrid(vcm, vrk)
k = Symbol(str(Var_row[0]))
i = Symbol(str(Var_col[0]))
smat = lambdify((k, i), coeff, modules=libraries)(mrk, mcm)
elif not(srepr(coeff).__contains__(srepr(Var_row[0]))) : #M(Z,M)
print 'coeff n°', index, ' Case 1/1 M(Z,M) : i non dependent '
print coeff, Var_col, Var_row, index
vcm = arange(Dim_col[0]) +1
m = Symbol(str(Var_col[0]))
smat = lambdify(m, coeff, modules=libraries)(vcm)
smat = [smat]*Dim_row[0]
smat = concatenate(list(smat), axis=0)
elif not(srepr(coeff).__contains__(srepr(Var_col[0]))) : #M(K,Z)
print 'coeff n°', index, ' Case 1/1 M(K,Z) : i non dependent'
print coeff, Var_col, Var_row, index
vrk = arange(Dim_row[0]) +1
k = Symbol(str(Var_row[0]))
smat = lambdify(k, coeff, modules=libraries)(vrk)
smat = [smat]*Dim_col[0]
smat = concatenate(list(smat), axis=1)
self.count = self.count +1
Mat[Pos_row:Pos_row+Nb_row, Pos_col:Pos_col+Nb_col] = smat
return Mat
最后我用 lambdified 系数填充矩阵，在本例中是一个较小的矩阵大小 (Nb_row, Nb_col).
我会继续自己搜索，不要犹豫，询问更多详情！
--编辑3--
我发现如果我这样做 :
m = Symbol(str(Var_col[0]))
coeff = lambdify(m, coeff, modules=libraries)
for nm in range(0, Dim_col[0]) :
smat.append(coeff(nm+1))
它有效，但它更耗时（尽管比使用 subs 和 evalf 少得多）。当我使用数组对象（无论是 numpy 还是 sympy 数组类型）调用 lambdify 创建的 lambda 函数时出现错误。
我可以重现你的错误。基于这些输入：
m = Symbol('m')
expr = -1.06048319593874*(-(3.14159265358979*m*besselj(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*besselj(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I)))*bessely(27.9937791601866*m, 6.81776274795262*sqrt(-I))/(3.14159265358979*m*bessely(27.9937791601866*m, 4.46681007624482*sqrt(-I)) - 0.501286290793831*sqrt(-I)*bessely(27.9937791601866*m + 1, 4.46681007624482*sqrt(-I))) + besselj(27.9937791601866*m, 6.81776274795262*sqrt(-I)))*sin(0.942966379693359*m)*cos(1.5707963267949*m)/m
nm = 2
bessel = {'besselj': jv,'besselk':kv,'besseli':iv,'bessely':yv}
libraries = [bessel, "numpy"]
它适用于 vm 中的整数：
vm = np.arange(nm, dtype=np.int) +1
result = lambdify(m, expr, modules=libraries)(vm)
但是复数会产生与您收到的相同的错误消息：
vm = np.arange(nm, dtype=np.complex) +1
result = lambdify(m, expr, modules=libraries)(vm)
错误信息：
TypeError Traceback (most recent call last)
<ipython-input-30-f0e31009275a> in <module>()
1 vm = np.arange(nm, dtype=np.complex) +1
----> 2 result = lambdify(m, expr, modules=libraries)(vm)
/Users/mike/anaconda/envs/py34/lib/python3.4/site-packages/numpy/__init__.py in <lambda>(_Dummy_27)
TypeError: ufunc 'jv' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''
因此，请检查您在 vcm 中获得的数据类型，即 arange(Dim_col[0]) +1 returns。
您只能取具有属性 real 的复数的实部。例如，vm.real 给出 vm 的实部。如果这是你想要的，你应该得到你的结果：
result = lambdify(m, expr, modules=libraries).(vm.real)