from __future__ import division
import warnings
import numpy as np
import scipy.integrate
from scipy import exp, pi
[docs]class ComplexPath(object):
"""A base class for paths in the complex plane."""
def __init__(self):
self._integralCache = {}
self._trapValuesCache = {}
[docs] def trap_values(self, f, k, useCache=True):
"""
Compute or retrieve (if cached) the values of the functions f
at :math:`2^k+1` points along the contour which are evenly
spaced with respect to the parameterisation of the contour.
Parameters
----------
f : function
A function of a single complex variable.
k : int
Defines the number of points along the curve that f is to be
evaluated at as :math:`2^k+1`.
useCache : bool, optional
If True then use, if available, the results of any previous
calls to this function for the same f and save any new
results so that they can be reused later.
Returns
-------
:class:`numpy.ndarray`
The values of f at :math:`2^k+1` points along the contour
which are evenly spaced with respect to the parameterisation
of the contour.
"""
if f in self._trapValuesCache.keys() and useCache:
vals = self._trapValuesCache[f]
vals_k = int(np.log2(len(vals)-1))
if vals_k == k:
return vals
elif vals_k > k:
return vals[::2**(vals_k-k)]
else:
t = np.linspace(0, 1, 2**k+1)
vals = np.empty(2**k+1, dtype=np.complex128)
vals.fill(np.nan)
vals[::2**(k-vals_k)] = self._trapValuesCache[f]
vals[np.isnan(vals)] = f(self(t[np.isnan(vals)]))
# cache values
self._trapValuesCache[f] = vals
return vals
else:
t = np.linspace(0, 1, 2**k+1)
vals = f(self(t))
if useCache:
self._trapValuesCache[f] = vals
return vals
[docs] def plot(self, N=100, linecolor='C0', linestyle='-'):
"""
Uses matplotlib to plot, but not show, the path as a 2D plot in
the Complex plane.
Parameters
----------
N : int, optional
The number of points to use when plotting the path.
linecolor : optional
The colour of the plotted path, passed to the
:func:`matplotlib.pyplot.plot` function as the keyword
argument of 'color'. See the matplotlib tutorial on
`specifying colours <https://matplotlib.org/users/colors.html#>`_.
linestyle : str, optional
The line style of the plotted path, passed to the
:func:`matplotlib.pyplot.plot` function as the keyword
argument of 'linestyle'. The default corresponds to a solid
line. See :meth:`matplotlib.lines.Line2D.set_linestyle` for
other acceptable arguments.
"""
import matplotlib.pyplot as plt
t = np.linspace(0,1,N)
path = self(t)
plt.plot(path.real, path.imag, color=linecolor, linestyle=linestyle)
plt.xlabel('Re[$z$]', size=16)
plt.ylabel('Im[$z$]', size=16)
plt.gca().set_aspect(1)
# add arrow to indicate direction of path
arrow_direction = (self(0.51) - self(0.5))/abs(self(0.51) - self(0.5))
arrow_extent = 1e-6*arrow_direction
ymin, ymax = plt.gca().get_ylim()
xmin, xmax = plt.gca().get_xlim()
head_length = max(abs(ymax - ymin), abs(xmax - xmin))/40.
plt.arrow(self(0.5).real, self(0.5).imag,
arrow_extent.real, arrow_extent.imag,
head_width=head_length*2/3., head_length=head_length,
fc=linecolor, ec=linecolor)
[docs] def show(self, saveFile=None, **plotKwargs):
"""
Shows the path as a 2D plot in the complex plane. Requires
Matplotlib.
Parameters
----------
saveFile : str (optional)
If given then the plot will be saved to disk with name
'saveFile'. If saveFile=None the plot is shown on-screen.
**plotKwargs
Other key word arguments are passed to :meth:`~cxroots.Paths.ComplexPath.plot`.
"""
import matplotlib.pyplot as plt
self.plot(**plotKwargs)
if saveFile is not None:
plt.savefig(saveFile, bbox_inches='tight')
plt.close()
else:
plt.show()
[docs] def integrate(self, f, absTol=0, relTol=1e-12, divMax=15, intMethod='quad', verbose=False):
"""
Integrate the function f along the path. The value of the
integral is cached and will be reused if the method is called
with same arguments (ignoring verbose).
Parameters
----------
f : function
A function of a single complex variable.
absTol : float, optional
The absolute tolerance for the integration.
relTol : float, optional
The realative tolerance for the integration.
divMax : int, optional
If the Romberg integration method is used then divMax is the
maximum number of divisions before the Romberg integration
routine of a path exits.
intMethod : {'quad', 'romb'}, optional
If 'quad' then :func:`scipy.integrate.quad` is used to
compute the integral. If 'romb' then Romberg integraion,
using :func:`scipy.integrate.romberg`, is used instead.
verbose : bool, optional
Passed ass the `show` argument of :func:`scipy.integrate.romberg`.
Returns
-------
complex
The integral of the function f along the path.
Notes
-----
This function is only used when checking the
multiplicity of roots. The bulk of the integration for
rootfinding is done with :func:`cxroots.CountRoots.prod`.
"""
args = (f, absTol, relTol, divMax, intMethod)
if args in self._integralCache.keys():
integral = self._integralCache[args]
elif hasattr(self, '_reversePath') and args in self._reversePath._integralCache:
# if we have already computed the reverse of this path
integral = -self._reversePath._integralCache[args]
else:
integrand = lambda t: f(self(t))*self.dzdt(t)
if intMethod == 'romb':
integral = scipy.integrate.romberg(integrand, 0, 1, tol=absTol, rtol=relTol, divmax=divMax, show=verbose)
elif intMethod == 'quad':
integrand_real = lambda t: np.real(integrand(t))
integrand_imag = lambda t: np.imag(integrand(t))
integral_real, abserr_real = scipy.integrate.quad(integrand_real, 0, 1, epsabs=absTol, epsrel=relTol)
integral_imag, abserr_imag = scipy.integrate.quad(integrand_imag, 0, 1, epsabs=absTol, epsrel=relTol)
integral = integral_real + 1j*integral_imag
else:
raise ValueError("intMethod must be either 'romb' or 'quad'")
if np.isnan(integral):
raise RuntimeError('The integral along the segment %s is NaN.\
\nThis is most likely due to a root being on or very close to the path of integration.'%self)
self._integralCache[args] = integral
return integral
[docs]class ComplexLine(ComplexPath):
"""
A straight line :math:`z` in the complex plane from a to b
parameterised by
..math::
z(t) = a + (b-a)t, \quad 0\leq t \leq 1
Parameters
----------
a : float
b : float
"""
def __init__(self, a, b):
self.a, self.b = a, b
self.dzdt = lambda t: self.b-self.a
super(ComplexLine, self).__init__()
def __str__(self):
return 'ComplexLine from %.3f+%.3fi to %.3f+%.3fi' % (self.a.real, self.a.imag, self.b.real, self.b.imag)
[docs] def __call__(self, t):
"""
The function :math:`z(t) = a + (b-a)t`.
Parameters
----------
t : float
A real number :math:`0\leq t \leq 1`.
Returns
-------
complex
A point on the line in the complex plane.
"""
return self.a + t*(self.b-self.a)
[docs] def distance(self, z):
"""
Distance from the point z to the closest point on the line.
Parameters
----------
z : complex
Returns
-------
float
The distance from z to the point on the line which is
closest to z.
"""
# convert complex numbers to vectors
A = np.array([self.a.real, self.a.imag])
B = np.array([self.b.real, self.b.imag])
Z = np.array([z.real, z.imag])
# the projection of the point z onto the line a -> b is where
# the parameter t is
t = (Z-A).dot(B-A)/abs((B-A).dot(B-A))
# but the line segment only has 0 <= t <= 1
t = t.clip(0,1)
# so the point on the line segment closest to z is
c = self(t)
return abs(c-z)
[docs]class ComplexArc(ComplexPath):
"""
A circular arc :math:`z` with center z0, radius R, initial angle t0
and change of angle dt. The arc is parameterised by
..math::
z(t) = R e^{i(t0 + t dt)} + z0, \quad 0\leq t \leq 1
Parameters
----------
z0 : complex
R : float
t0 : float
dt : float
"""
def __init__(self, z0, R, t0, dt):
self.z0, self.R, self.t0, self.dt = z0, R, t0, dt
self.dzdt = lambda t: 1j*self.dt*self.R*exp(1j*(self.t0 + t*self.dt))
super(ComplexArc, self).__init__()
def __str__(self):
return 'ComplexArc: z0=%.3f, R=%.3f, t0=%.3f, dt=%.3f' % (self.z0, self.R, self.t0, self.dt)
[docs] def __call__(self, t):
"""
The function :math:`z(t) = R e^{i(t_0 + t dt)} + z_0`.
Parameters
----------
t : float
A real number :math:`0\leq t \leq 1`.
Returns
-------
complex
A point on the arc in the complex plane.
"""
return self.R*exp(1j*(self.t0 + t*self.dt)) + self.z0
[docs] def distance(self, z):
"""
Distance from the point z to the closest point on the arc.
Parameters
----------
z : complex
Returns
-------
float
The distance from z to the point on the arc which is closest
to z.
"""
theta = np.angle(z-self.z0) # np.angle maps to (-pi,pi]
theta = (theta-self.t0)%(2*pi) + self.t0 # put theta in [t0,t0+2pi)
if ((self.dt > 0 and self.t0 < theta < self.t0+self.dt)
or (self.dt < 0 and self.t0+self.dt < theta - 2*pi < self.t0)):
# the closest point to z lies on the arc
return abs(self.R*exp(1j*theta) + self.z0 - z)
else:
# the closest point to z is one of the endpoints
return min(abs(self(0)-z), abs(self(1)-z))