def graph(func, f_prime, a, b, n):
"""Graphs both a functions vectorized approximation and analytical derrivative."""
x = np.linspace(a, b, n)
f_prime = f_prime(x)
diff_mesh, f_diff = differentiator.diff(func, a, b, 1000)[0], differentiator.diff(func, a, b, 1000)[2]
mp.plot(x, f_prime, 'ko')
mp.plot(diff_mesh, f_diff, 'b-')
mp.xlim([a - 0.01, b + 0.01])
mp.xlabel('x')
mp.ylabel('y')
def arclength(g, a, b, n):
xlist = np.linspace(a, b, n + 1)
f = np.vectorize(g)
ylist = f(xlist)
difflist = calculus.diff(f, a, b, n)[1]
s = np.zeros(n + 1)
#s will hold the incremental values of each indivual arc length at first
il = np.vectorize(incremental_lengther)
s[1:] = il(difflist[1:])
#now traverse backwards to sum individual points
for i in xrange(n, -1, -1):
s[i] = np.sum(s[:i + 1])
return s
def arclength(g, a, b, n):
xlist = np.linspace(a, b, n+1)
f = np.vectorize(g)
ylist = f(xlist)
difflist = calculus.diff(f, a, b, n)[1]
s = np.zeros(n+1)
#s will hold the incremental values of each indivual arc length at first
il = np.vectorize(incremental_lengther)
s[1:] = il(difflist[1:])
#now traverse backwards to sum individual points
for i in xrange(n, -1, -1):
s[i] = np.sum(s[:i+1])
return s
def multiplicity(f, root, tol=eps, maxsteps=10, **kwargs):
"""
Return the multiplicity of a given root of f.
Internally, numerical derivatives are used. This might be inefficient for
higher order derviatives. Due to this, ``multiplicity`` cancels after
evaluating 10 derivatives by default. You can be specify the n-th derivative
using the dnf keyword.
>>> from mpmath import *
>>> multiplicity(lambda x: sin(x) - 1, pi/2)
2
"""
kwargs['d0f'] = f
for i in xrange(maxsteps):
dfstr = 'd' + str(i) + 'f'
if dfstr in kwargs:
df = kwargs[dfstr]
else:
df = lambda x: diff(f, x, i)
if not abs(df(root)) < tol:
break
return i
def multiplicity(f, root, tol=eps, maxsteps=10, **kwargs):
"""
Return the multiplicity of a given root of f.
Internally, numerical derivatives are used. This might be inefficient for
higher order derviatives. Due to this, ``multiplicity`` cancels after
evaluating 10 derivatives by default. You can be specify the n-th derivative
using the dnf keyword.
>>> from mpmath import *
>>> multiplicity(lambda x: sin(x) - 1, pi/2)
2
"""
kwargs['d0f'] = f
for i in xrange(maxsteps):
dfstr = 'd' + str(i) + 'f'
if dfstr in kwargs:
df = kwargs[dfstr]
else:
df = lambda x: diff(f, x, i)
if not abs(df(root)) < tol:
break
return i
def d2f(x):
return diff(df, x)
def df(x):
return diff(f, x)
def test_derrivatives():
"""Ensures that the value of the derrivative of x^3 is good at x = 3 as per the vectorized derrivative function."""
apt = np.fabs(differentiator.diff(h, 2, 3.01, 1000)[1][-1] - 56.6625)
msg = 'Derrivative is imporperly calculated.'
assert apt, msg
