def leggauss(n):
"""
For Legendre-Gaussian quadrature.
Parameters
----------
n : int
Returns
-------
roots : list of mpf
weights : list of mpf
"""
assert n >= 2
roots = []
weights = []
for k in range(n, n // 2 - 1, -1):
# Using Tricomi approximation to get high precision estimates of roots.
roots.append(polish_root(n, tricomi_root(n, k)))
weights.append(
2 / legendre(n - 1, roots[-1]) * (roots[-1]**2 - 1) / n**2 /
(roots[-1] * legendre(n, roots[-1]) - legendre(n - 1, roots[-1])))
# roots are antisymmetric about 0
for k in range(n // 2 - 2, -1, -1):
roots.append(-roots[k])
weights.append(weights[k])
return roots, weights
def make_legendre_p_vals():
from mpmath import legendre
l = range(5)
x = linspace('-0.99', '0.99', 67)
l, x = zip(*outer(l, x))
p = [legendre(*vals) for vals in zip(l, x)]
return make_special_vals('legendre_p_vals', ('l', l), ('x', x), ('p', p))
def make_legendre_p_vals():
from mpmath import legendre
l = list(range(5))
x = linspace('-0.99', '0.99', 67)
l, x = zip(*outer(l, x))
p = [legendre(*vals) for vals in zip(l, x)]
return make_special_vals('legendre_p_vals', ('l', l), ('x', x), ('p', p))
def polish_root(n, approx_root, n_iters=None, tol=None):
"""Use Newton method to get close to root.
"""
if n_iters is None:
tol = 10**(-mp.dps + 5)
n_iters = mp.dps
dx = 1.
oldRoot = approx_root
while abs(dx) > tol:
newRoot = (
oldRoot - legendre(n, oldRoot) * (oldRoot**2 - 1) / n /
(oldRoot * legendre(n, oldRoot) - legendre(n - 1, oldRoot)))
dx = newRoot - oldRoot
oldRoot = newRoot
return newRoot
else:
oldRoot = approx_root
for i in range(n_iters):
newRoot = (
oldRoot - legendre(n, oldRoot) * (oldRoot**2 - 1) / n /
(oldRoot * legendre(n, oldRoot) - legendre(n - 1, oldRoot)))
oldRoot = newRoot
return newRoot
def main():
print(__doc__)
x = symbols('x')
# a numpy array we can apply the ufuncs to
grid = np.linspace(-1, 1, 1000)
# set mpmath precision to 20 significant numbers for verification
mpmath.mp.dps = 20
print("Compiling legendre ufuncs and checking results:")
# Let's also plot the ufunc's we generate
plot1 = Plot(visible=False)
for n in range(6):
# Setup the SymPy expression to ufuncify
expr = legendre(n, x)
print("The polynomial of degree %i is" % n)
pprint(expr)
# This is where the magic happens:
binary_poly = ufuncify(x, expr)
# It's now ready for use with numpy arrays
polyvector = binary_poly(grid)
# let's check the values against mpmath's legendre function
maxdiff = 0
for j in range(len(grid)):
precise_val = mpmath.legendre(n, grid[j])
diff = abs(polyvector[j] - precise_val)
if diff > maxdiff:
maxdiff = diff
print("The largest error in applied ufunc was %e" % maxdiff)
assert maxdiff < 1e-14
# We can also attach the autowrapped legendre polynomial to a sympy
# function and plot values as they are calculated by the binary function
g = implemented_function('g', binary_poly)
plot1[n] = g(x), [200]
print(
"Here's a plot with values calculated by the wrapped binary functions")
plot1.show()
def main():
print(__doc__)
x = symbols('x')
# a numpy array we can apply the ufuncs to
grid = np.linspace(-1, 1, 1000)
# set mpmath precision to 20 significant numbers for verification
mpmath.mp.dps = 20
print("Compiling legendre ufuncs and checking results:")
# Let's also plot the ufunc's we generate
for n in range(6):
# Setup the SymPy expression to ufuncify
expr = legendre(n, x)
print("The polynomial of degree %i is" % n)
pprint(expr)
# This is where the magic happens:
binary_poly = ufuncify(x, expr)
# It's now ready for use with numpy arrays
polyvector = binary_poly(grid)
# let's check the values against mpmath's legendre function
maxdiff = 0
for j in range(len(grid)):
precise_val = mpmath.legendre(n, grid[j])
diff = abs(polyvector[j] - precise_val)
if diff > maxdiff:
maxdiff = diff
print("The largest error in applied ufunc was %e" % maxdiff)
assert maxdiff < 1e-14
# We can also attach the autowrapped legendre polynomial to a sympy
# function and plot values as they are calculated by the binary function
plot1 = plt.pyplot.plot(grid, polyvector, hold=True)
print("Here's a plot with values calculated by the wrapped binary functions")
# plt.pyplot.show()
pltshow(plt)
def eval(self, z):
return mpmath.legendre(1, z)
def f81(x):
# legendre_P3
return mpmath.legendre(3, x)
def f80(x):
# legendre_P2
return mpmath.legendre(2, x)
def f79(x):
# legendre_P1
return mpmath.legendre(1, x)
def eval(self, z):
return mpmath.legendre(1, z)
def func(channel):
channell = []
for i in range(0,len(channel)):
channell.append(float(mpmath.legendre(degree,channel[i])))
return numpy.asarray(channell)
def roots_legendre(n):
"""
Compute the roots of the Legendre polynomial, and quadrature weights.
Warning: not tested beyond n=21.
Examples
--------
>>> import mpmath
>>> mpmath.mp.dps = 40
>>> from mpsci.fun import roots_legendre
>>> roots, weights = roots_legendre(7)
>>> roots
[mpf('-0.9491079123427585245261896840478512624007709'),
mpf('-0.7415311855993944398638647732807884070741476'),
mpf('-0.405845151377397166906606412076961463347382'),
mpf('0.0'),
mpf('0.405845151377397166906606412076961463347382'),
mpf('0.7415311855993944398638647732807884070741476'),
mpf('0.9491079123427585245261896840478512624007709')]
>>> weights
[mpf('0.1294849661688696932706114326790820183285874'),
mpf('0.2797053914892766679014677714237795824869251'),
mpf('0.3818300505051189449503697754889751338783651'),
mpf('0.4179591836734693877551020408163265306122449'),
mpf('0.3818300505051189449503697754889751338783651'),
mpf('0.2797053914892766679014677714237795824869251'),
mpf('0.1294849661688696932706114326790820183285874')]
"""
n = operator.index(n)
if n < 1:
raise ValueError('n must be a positive integer.')
if n == 1:
return [mpmath.mp.zero], [mpmath.mpf(2)]
elif n == 2:
x = 1 / mpmath.sqrt(3)
w = mpmath.mp.one
return [-x, x], [w, w]
elif n == 3:
x = mpmath.sqrt('0.6')
wx = mpmath.mpf('5/9')
w0 = mpmath.mpf('8/9')
return [-x, mpmath.mp.zero, x], [wx, w0, wx]
approx_roots = [_root_approx(n, k) for k in range(1, n // 2 + 1)]
with mpmath.extradps(5):
roots = [
mpmath.findroot(lambda x: mpmath.legendre(n, x), x0)
for x0 in approx_roots
]
derivs = [
mpmath.diff(lambda x: mpmath.legendre(n, x), root)
for root in roots
]
weights = [
2 / ((1 - root**2) * deriv**2)
for root, deriv in zip(roots, derivs)
]
if n & 1:
z = mpmath.mp.zero
root0 = [z]
deriv = mpmath.diff(lambda x: mpmath.legendre(n, x), z)
weight0 = [2 / deriv**2]
else:
root0 = []
weight0 = []
return ([-r for r in roots] + root0 + roots[::-1],
weights + weight0 + weights[::-1])
import sympy as sp
sp.init_printing()
#%%
v = sp.symbols('v', real=True, positive=True)
gamma = 1 / sp.sqrt(1 - v**2)
#%%
x = sp.sqrt(v**2 + 1 / gamma**2)
sp.simplify(x)
#%%
from mpmath import legendre
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn')
from mpmath import *
f = lambda x: legendre(2, x)
plot(f)
nprint(polyroots(taylor(f, 0, 2)[::-1]))
print(np.sqrt(1 / 3))
def leg_poly(value):
"""Legendre polynomial :math:`P_n(x)`."""
return mpmath.legendre(num_points - 1, value)
