mpmath和scipy之间的关系:类型错误 [英] Relation between mpmath and scipy: Type Error

问题描述

Scipy具有许多特殊功能，特别是Bessel函数jn(始终由大写字母J_n(x)表示)和球形Bessel函数spherical_jn(由小写字母j_n(x)表示).另一方面，mpmath具有quadosc，这是一种用于集成快速振荡功能(如jn和spherical_jn)的特殊方法. 我得到的问题似乎是 quadosc 来自mpmath的不支持，例如，来自scipy的jn作为使该积分不可或缺的输入.我的意思是，如果我使用从numpy导入的quad，则不会获得TypeError的任何错误，但是在x很大时不很适合评估J_n(x)或j_n(x)的积分.

(***)在网站

from mpmath import findroot, quadosc, inf, j0

j0zero = lambda n: findroot(j0, pi*(n-0.25)) # ***
I = quadosc(j0, [0, inf], zeros=j0zero)
print(I)
I = 1.0 # OK, this is the correct answer.

但是，如果我使用从numpy导入的J_n(x):

from scipy.special import jn

f = lambda x: jn(0,x)
j0zero = lambda n: findroot(f, pi*(n-0.25))
II = quadosc(f, [0, inf], zeros=j0zero)
print(II)

然后我遇到以下错误(添加了回溯)

TypeError Traceback (most recent call last)
~/anaconda3/lib/python3.7/site-packages/mpmath/calculus/optimization.py in findroot(ctx, f, x0, solver, tol, verbose, verify, **kwargs)
    927  try:
--> 928   fx = f(*x0)
    929    multidimensional = isinstance(fx, (list, tuple, ctx.matrix))
<ipython-input-449-aeebd9a1e908> in <lambda>(x)
      2 
----> 3 f = lambda x: jn(0,x)
      4 j0zero = lambda n: findroot(f, pi*(n-0.25))

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''

During handling of the above exception, another exception occurred:
TypeError  Traceback (most recent call last)
<ipython-input-449-aeebd9a1e908> in <module>
      3 f = lambda x: jn(0,x)
      4 j0zero = lambda n: findroot(f, pi*(n-0.25))
----> 5 II = quadosc(f, [0, inf], zeros=j0zero)
      6 print(II)

~/anaconda3/lib/python3.7/site-packages/mpmath/calculus/quadrature.py in quadosc(ctx, f, interval, omega, period, zeros)
     998   # raise ValueError("zeros do not appear to be correctly indexed")
     999   n = 1
 -> 1000  s = ctx.quadgl(f, [a, zeros(n)])
    1001  def term(k):
    1002  return ctx.quadgl(f, [zeros(k), zeros(k+1)]

~/anaconda3/lib/python3.7/site-packages/mpmath/calculus/optimization.py in findroot(ctx, f, x0, solver, tol, verbose, verify, **kwargs)
    929     multidimensional = isinstance(fx, (list, tuple, ctx.matrix))
    930     except TypeError:
--> 931     fx = f(x0[0])
    932     multidimensional = False
    933     if 'multidimensional' in kwargs:

另一方面，如果我使用quad，那么我得到了

from scipy.integrate import quad

f = lambda x: jn(0,x)
III = quad(f,0,inf)[0]
print(III)
III = -21.154674722694516 # What is an incorrect answer.

那么我该如何在mpmath的quadosc中使用来自scipy的函数jn?如何解决此错误?感谢您的所有帮助.

解决方案

In [31]: one,two = mpmath.mpmathify(1), mpmath.mpmathify(2)                     

In [32]: one,two                                                                
Out[32]: (mpf('1.0'), mpf('2.0'))

In [33]: one+two                                                                
Out[33]: mpf('3.0')

In [34]: jn(1,2)                                                                
Out[34]: 0.5767248077568736

In [35]: jn(one,two)                                                            
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-35-ec48c25f686b> in <module>
----> 1 jn(one,two)

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''

mpmath数字不能在jn/jv中使用.

以同样的方式，可以在mpmath函数中使用mpmath数字，但不能使用等效的numpy:

In [41]: mpmath.sin(one)                                                        
Out[41]: mpf('0.8414709848078965')

In [42]: np.sin(1)                                                              
Out[42]: 0.8414709848078965

In [43]: np.sin(one)                                                            
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
AttributeError: 'mpf' object has no attribute 'sin'

The above exception was the direct cause of the following exception:

TypeError                                 Traceback (most recent call last)
<ipython-input-43-38fc918b311a> in <module>
----> 1 np.sin(one)

TypeError: loop of ufunc does not support argument 0 of type mpf which has no callable sin method

您可以使用以下方式转换mpmath正常的python编号:

In [44]: float(one)                                                             
Out[44]: 1.0

In [45]: jn(float(one),float(two))                                              
Out[45]: 0.5767248077568736

np.float64(one)也可以，但是jn不喜欢np.float128(one).显然，jn已针对C double进行了编译，但没有针对精度更高的浮点数进行编译.

mpmath是否谈论将其与numpy一起使用?我见过mpmath与sympy一起使用，但没有与numpy一起使用.

Scipy have a lot of special functions, in particular Bessel functions jn (always denoted by uppercase letter J_n(x)) and spherical Bessel functions spherical_jn (denoted by lowercase letter j_n(x)). On the other hand mpmath have quadosc, a special method for integrate rapidly oscillating functions, like jn and spherical_jn. The problem I had obtained is that seems that quadosc from mpmath not support, e.g, jn from scipy as an input to make this integral. I means, if I use quad imported from numpy, so nothing error of TypeError is obtained, but quad is not very appropriate to evaluate integrals of J_n(x) or j_n(x) when x is very large.

(***) In the site SymPy find by "Oscillatory quadrature (quadosc)", this example come from there.

from mpmath import findroot, quadosc, inf, j0

j0zero = lambda n: findroot(j0, pi*(n-0.25)) # ***
I = quadosc(j0, [0, inf], zeros=j0zero)
print(I)
I = 1.0 # OK, this is the correct answer.

But if I use J_n(x) imported from numpy:

from scipy.special import jn

f = lambda x: jn(0,x)
j0zero = lambda n: findroot(f, pi*(n-0.25))
II = quadosc(f, [0, inf], zeros=j0zero)
print(II)

then I got the following error (Edited: added the Traceback)

TypeError Traceback (most recent call last)
~/anaconda3/lib/python3.7/site-packages/mpmath/calculus/optimization.py in findroot(ctx, f, x0, solver, tol, verbose, verify, **kwargs)
    927  try:
--> 928   fx = f(*x0)
    929    multidimensional = isinstance(fx, (list, tuple, ctx.matrix))
<ipython-input-449-aeebd9a1e908> in <lambda>(x)
      2 
----> 3 f = lambda x: jn(0,x)
      4 j0zero = lambda n: findroot(f, pi*(n-0.25))

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''

During handling of the above exception, another exception occurred:
TypeError  Traceback (most recent call last)
<ipython-input-449-aeebd9a1e908> in <module>
      3 f = lambda x: jn(0,x)
      4 j0zero = lambda n: findroot(f, pi*(n-0.25))
----> 5 II = quadosc(f, [0, inf], zeros=j0zero)
      6 print(II)

~/anaconda3/lib/python3.7/site-packages/mpmath/calculus/quadrature.py in quadosc(ctx, f, interval, omega, period, zeros)
     998   # raise ValueError("zeros do not appear to be correctly indexed")
     999   n = 1
 -> 1000  s = ctx.quadgl(f, [a, zeros(n)])
    1001  def term(k):
    1002  return ctx.quadgl(f, [zeros(k), zeros(k+1)]

~/anaconda3/lib/python3.7/site-packages/mpmath/calculus/optimization.py in findroot(ctx, f, x0, solver, tol, verbose, verify, **kwargs)
    929     multidimensional = isinstance(fx, (list, tuple, ctx.matrix))
    930     except TypeError:
--> 931     fx = f(x0[0])
    932     multidimensional = False
    933     if 'multidimensional' in kwargs:

On the other hand, if I use quad then I got

from scipy.integrate import quad

f = lambda x: jn(0,x)
III = quad(f,0,inf)[0]
print(III)
III = -21.154674722694516 # What is an incorrect answer.

So how can I use a function jn that come from scipy inside a quadosc of mpmath? How can I fix this error? Thanks for all help.

解决方案

In [31]: one,two = mpmath.mpmathify(1), mpmath.mpmathify(2)                     

In [32]: one,two                                                                
Out[32]: (mpf('1.0'), mpf('2.0'))

In [33]: one+two                                                                
Out[33]: mpf('3.0')

In [34]: jn(1,2)                                                                
Out[34]: 0.5767248077568736

In [35]: jn(one,two)                                                            
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-35-ec48c25f686b> in <module>
----> 1 jn(one,two)

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''

mpmath numbers cannot be use in jn/jv.

In the same way, a mpmath number can be used in a mpmath function, but not the equivalent numpy:

In [41]: mpmath.sin(one)                                                        
Out[41]: mpf('0.8414709848078965')

In [42]: np.sin(1)                                                              
Out[42]: 0.8414709848078965

In [43]: np.sin(one)                                                            
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
AttributeError: 'mpf' object has no attribute 'sin'

The above exception was the direct cause of the following exception:

TypeError                                 Traceback (most recent call last)
<ipython-input-43-38fc918b311a> in <module>
----> 1 np.sin(one)

TypeError: loop of ufunc does not support argument 0 of type mpf which has no callable sin method

You can convert the mpmath normal python number with:

In [44]: float(one)                                                             
Out[44]: 1.0

In [45]: jn(float(one),float(two))                                              
Out[45]: 0.5767248077568736

np.float64(one) also works, but jn does not like np.float128(one). Evidently jn has been compiled for C doubles, but not higher precision floats.

Does mpmath does talk about using it with numpy? I've seen mpmath used with sympy, but not with numpy.

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