问题
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.
回答1:
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.
回答2:
There is no way to use scipy.special.jv without converting its arguments to floats/integers.
来源：https://stackoverflow.com/questions/59440969/relation-between-mpmath-and-scipy-type-error