在SciPy / NumPy中查找复杂函数的零

Question

I've been told that the method scipy.optimize.newton() will solve complex functions so long as the first derivative is provided. I can't make it work. The documentation for newton() mentions nothing of complex functions. Could someone show me how one would find the roots of a function like f(z) = 1 + z^2 in SciPy? I need to solve something much more complex, but a simple example will help me immensely.

Answer 1

Here's an example of the use of newton with a complex function, in an IPython session:

In [1]: def func(z):
   ...:     return 1 + z*z
   ...: 

In [2]: def deriv(z):
   ...:     return 2*z
   ...: 

In [3]: from scipy.optimize import newton

In [4]: newton(func, x0=1+1j, fprime=deriv, tol=1e-12)
Out[4]: 1j

In [5]: newton(func, x0=-2j, fprime=deriv, tol=1e-12)
Out[5]: -1j

Note that newton doesn't handle x0 being a point where the derivative is zero:

In [6]: newton(func, x0=0, fprime=deriv, tol=1e-12)
/Users/warren/local_scipy/lib/python2.7/site-packages/scipy/optimize/zeros.py:119: RuntimeWarning: derivative was zero.
  warnings.warn(msg, RuntimeWarning)
Out[6]: 0.0

Also, if your function returns real values for real arguments, be sure to pass in a complex value for x0 . Otherwise the function is stuck on the real axis:

In [21]: newton(func, x0=0.5, fprime=deriv, tol=1e-12)
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
<ipython-input-21-2feb08057c57> in <module>()
----> 1 newton(func, x0=0.5, fprime=deriv, tol=1e-12)

/Users/warren/local_scipy/lib/python2.7/site-packages/scipy/optimize/zeros.pyc in newton(func, x0, fprime, args, tol, maxiter, fprime2)
    159             q1 = func(*((p1,) + args))
    160     msg = "Failed to converge after %d iterations, value is %s" % (maxiter, p)
--> 161     raise RuntimeError(msg)
    162 
    163 

RuntimeError: Failed to converge after 50 iterations, value is -0.870752774435

Increasing maxiter won't help:

In [22]: newton(func, x0=0.5, fprime=deriv, tol=1e-12, maxiter=1000)
---------------------------------------------------------------------------
RuntimeError                              Traceback (most recent call last)
[...]
RuntimeError: Failed to converge after 1000 iterations, value is -0.0895687261655