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Updated doctests

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Per A Brodtkorb 8 years ago
parent 838675a55b
commit 601b282ca0
1 changed files with 118 additions and 96 deletions
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wafo/polynomial.py
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@ -21,7 +21,7 @@ from __future__ import absolute_import
import warnings # @UnusedImport import warnings # @UnusedImport
from functools import reduce from functools import reduce
from numpy.polynomial import polyutils as pu from numpy.polynomial import polyutils as pu
from .plotbackend import plotbackend as plt from wafo.plotbackend import plotbackend as plt
import numpy as np import numpy as np
from numpy import (newaxis, arange, pi) from numpy import (newaxis, arange, pi)
from scipy.fftpack import dct, idct as _idct from scipy.fftpack import dct, idct as _idct
@ -70,33 +70,33 @@ def polyint(p, m=1, k=None):
Examples Examples
-------- --------
The defining property of the antiderivative: The defining property of the antiderivative:
>>> import wafo.polynomial as wp
>>> p = np.poly1d([1,1,1]) >>> p = wp.poly1d([1,1,1])
>>> P = np.polyint(p) >>> P = wp.polyint(p)
>>> P >>> P
poly1d([ 0.33333333, 0.5 , 1. , 0. ]) poly1d([ 0.33333333, 0.5 , 1. , 0. ])
>>> np.polyder(P) == p >>> wp.polyder(P) == p
True True
The integration constants default to zero, but can be specified: The integration constants default to zero, but can be specified:
>>> P = np.polyint(p, 3) >>> P = wp.polyint(p, 3)
>>> P(0) >>> P(0)
0.0 0.0
>>> np.polyder(P)(0) >>> wp.polyder(P)(0)
0.0 0.0
>>> np.polyder(P, 2)(0) >>> wp.polyder(P, 2)(0)
0.0 0.0
>>> P = np.polyint(p, 3, k=[6, 5, 3]) >>> P = wp.polyint(p, 3, k=[6, 5, 3])
>>> P >>> P
poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ]) poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ])
Note that 3 = 6 / 2!, and that the constants are given in the order of Note that 3 = 6 / 2!, and that the constants are given in the order of
integrations. Constant of the highest-order polynomial term comes first: integrations. Constant of the highest-order polynomial term comes first:
>>> np.polyder(P, 2)(0) >>> wp.polyder(P, 2)(0)
6.0 6.0
>>> np.polyder(P, 1)(0) >>> wp.polyder(P, 1)(0)
5.0 5.0
>>> P(0) >>> P(0)
3.0 3.0
@ -160,9 +160,9 @@ def polyder(p, m=1):
Examples Examples
-------- --------
The derivative of the polynomial :math:`x^3 + x^2 + x^1 + 1` is: The derivative of the polynomial :math:`x^3 + x^2 + x^1 + 1` is:
>>> import wafo.polynomial as wp
>>> p = np.poly1d([1,1,1,1]) >>> p = wp.poly1d([1,1,1,1])
>>> p2 = np.polyder(p) >>> p2 = wp.polyder(p)
>>> p2 >>> p2
poly1d([3, 2, 1]) poly1d([3, 2, 1])
@ -179,11 +179,11 @@ def polyder(p, m=1):
The fourth-order derivative of a 3rd-order polynomial is zero: The fourth-order derivative of a 3rd-order polynomial is zero:
>>> np.polyder(p, 2) >>> wp.polyder(p, 2)
poly1d([6, 2]) poly1d([6, 2])
>>> np.polyder(p, 3) >>> wp.polyder(p, 3)
poly1d([6]) poly1d([6])
>>> np.polyder(p, 4) >>> wp.polyder(p, 4)
poly1d([ 0.]) poly1d([ 0.])
""" """
@ -234,15 +234,16 @@ def polydeg(x, y):
Example: Example:
------- -------
>>> import wafo.polynomial as wp
>>> x = np.linspace(0,10,300) >>> x = np.linspace(0,10,300)
>>> noise = 0.05 * np.random.randn(x.size) >>> noise = 0.05 * np.random.randn(x.size)
>>> noise = 0.05 * np.sin(100*x) >>> noise = 0.05 * np.sin(100*x)
>>> y = np.sin(x ** 3 / 100) ** 2 + noise >>> y = np.sin(x ** 3 / 100) ** 2 + noise
>>> n = polydeg(x,y) >>> n = wp.polydeg(x,y)
>>> n >>> n
21 21
ys = orthofit(x,y,n); ys = wp.orthofit(x,y,n);
plt.plot(x, y, '.', x, ys, 'k') plt.plot(x, y, '.', x, ys, 'k')
See also See also
@ -350,16 +351,17 @@ def ortho2poly(p):
Examples Examples
-------- --------
>>> import numpy as np >>> import numpy as np
>>> import wafo.polynomial as wp
>>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0]) >>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0]) >>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> p = orthofit(x, y, 3) >>> p = wp.orthofit(x, y, 3)
>>> p >>> p
array([[ 0. , -0.30285714, -0.16071429, 0.08703704], array([[ 0. , -0.30285714, -0.16071429, 0.08703704],
[ 0. , 2.5 , 2.5 , 2.5 ], [ 0. , 2.5 , 2.5 , 2.5 ],
[ 0. , 0. , 2.91666667, 2.13333333]]) [ 0. , 0. , 2.91666667, 2.13333333]])
>>> ortho2poly(p) >>> wp.ortho2poly(p)
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
>>> np.polyfit(x, y, 3) >>> wp.polyfit(x, y, 3)
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
References References
@ -410,10 +412,11 @@ def orthofit(x, y, n):
Example: Example:
------- -------
>>> import wafo.polynomial as wp
>>> x = np.linspace(0,10,300); >>> x = np.linspace(0,10,300);
>>> y = np.sin(x**3/100)**2 + 0.05*np.random.randn(x.size) >>> y = np.sin(x**3/100)**2 + 0.05*np.random.randn(x.size)
>>> p = orthofit(x, y, 25) >>> p = wp.orthofit(x, y, 25)
>>> ys = orthoval(p, x) >>> ys = wp.orthoval(p, x)
plot(x, y,'.',x, ys, 'k') plot(x, y,'.',x, ys, 'k')
@ -481,15 +484,16 @@ def polyreloc(p, x, y=0.0):
Example Example
------- -------
>>> import numpy as np >>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(6); p.shape = (2,-1) >>> p = np.arange(6); p.shape = (2,-1)
>>> np.polyval(p,0) >>> wp.polyval(p,0)
array([3, 4, 5]) array([3, 4, 5])
>>> np.polyval(p,1) >>> wp.polyval(p,1)
array([3, 5, 7]) array([3, 5, 7])
>>> r = polyreloc(p,-1) # move to the left along x-axis >>> r = polyreloc(p,-1) # move to the left along x-axis
>>> np.polyval(r,-1) # = polyval(p,0) >>> wp.polyval(r,-1) # = polyval(p,0)
array([3, 4, 5]) array([3, 4, 5])
>>> np.polyval(r,0) # = polyval(p,1) >>> wp.polyval(r,0) # = polyval(p,1)
array([3, 5, 7]) array([3, 5, 7])
""" """
@ -534,15 +538,16 @@ def polyrescl(p, x, y=1.0):
Example Example
------- -------
>>> import numpy as np >>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(6); p.shape = (2,-1) >>> p = np.arange(6); p.shape = (2,-1)
>>> np.polyval(p,0) >>> wp.polyval(p,0)
array([3, 4, 5]) array([3, 4, 5])
>>> np.polyval(p,1) >>> wp.polyval(p,1)
array([3, 5, 7]) array([3, 5, 7])
>>> r = polyrescl(p,2) # scale by 2 along x-axis >>> r = wp.polyrescl(p,2) # scale by 2 along x-axis
>>> np.polyval(r,0) # = polyval(p,0) >>> wp.polyval(r,0) # = polyval(p,0)
array([ 3., 4., 5.]) array([ 3., 4., 5.])
>>> np.polyval(r,2) # = polyval(p,1) >>> wp.polyval(r,2) # = polyval(p,1)
array([ 3., 5., 7.]) array([ 3., 5., 7.])
""" """
@ -577,11 +582,12 @@ def polytrim(p):
Example Example
------- -------
>>> import wafo.polynomial as wp
>>> p = [0,1,2] >>> p = [0,1,2]
>>> polytrim(p) >>> wp.polytrim(p)
array([1, 2]) array([1, 2])
>>> p1 = [[0,0],[1,2],[3,4]] >>> p1 = [[0,0],[1,2],[3,4]]
>>> polytrim(p1) >>> wp.polytrim(p1)
array([[1, 2], array([[1, 2],
[3, 4]]) [3, 4]])
""" """
@ -620,7 +626,8 @@ def poly2hstr(p, variable='x'):
Examples Examples
-------- --------
>>> poly2hstr([1, 1, 2], 's' ) >>> import wafo.polynomial as wp
>>> wp.poly2hstr([1, 1, 2], 's' )
'(s + 1)*s + 2' '(s + 1)*s + 2'
See also See also
@ -710,7 +717,8 @@ def poly2str(p, variable='x'):
Examples Examples
-------- --------
>>> poly2str([1, 1, 2], 's' ) >>> import wafo.polynomial as wp
>>> wp.poly2str([1, 1, 2], 's' )
's**2 + s + 2' 's**2 + s + 2'
""" """
thestr = "0" thestr = "0"
@ -787,11 +795,12 @@ def polyshift(py, a=-1, b=1):
Example Example
------- -------
>>> import wafo.polynomial as wp
>>> py = [1, 0] >>> py = [1, 0]
>>> px = polyshift(py,0,5) >>> px = wp.polyshift(py,0,5)
>>> polyval(px,[0, 2.5, 5]) #% This is the same as the line below >>> wp.polyval(px,[0, 2.5, 5]) #% This is the same as the line below
array([-1., 0., 1.]) array([-1., 0., 1.])
>>> polyval(py,[-1, 0, 1 ]) >>> wp.polyval(py,[-1, 0, 1 ])
array([-1, 0, 1]) array([-1, 0, 1])
""" """
@ -830,11 +839,12 @@ def polyishift(px, a=-1, b=1):
Example Example
------- -------
>>> import wafo.polynomial as wp
>>> px = [1, 0] >>> px = [1, 0]
>>> py = polyishift(px,0,5); >>> py = wp.polyishift(px,0,5);
>>> polyval(px,[0, 2.5, 5]) #% This is the same as the line below >>> wp.polyval(px,[0, 2.5, 5]) #% This is the same as the line below
array([ 0. , 2.5, 5. ]) array([ 0. , 2.5, 5. ])
>>> polyval(py,[-1, 0, 1]) >>> wp.polyval(py,[-1, 0, 1])
array([ 0. , 2.5, 5. ]) array([ 0. , 2.5, 5. ])
""" """
if (a == -1) & (b == 1): if (a == -1) & (b == 1):
@ -892,9 +902,10 @@ def poly2cheb(p, a=-1, b=1):
Examples Examples
-------- --------
>>> import numpy as np >>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(5) >>> p = np.arange(5)
>>> ck = poly2cheb(p) >>> ck = wp.poly2cheb(p)
>>> cheb2poly(ck) >>> wp.cheb2poly(ck)
array([ 1., 2., 3., 4.]) array([ 1., 2., 3., 4.])
Reference Reference
@ -934,12 +945,12 @@ def cheb2poly(ck, a=-1, b=1):
Examples Examples
-------- --------
>>> import numpy as np >>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(5) >>> p = np.arange(5)
>>> ck = poly2cheb(p) >>> ck = wp.poly2cheb(p)
>>> cheb2poly(ck) >>> wp.cheb2poly(ck)
array([ 1., 2., 3., 4.]) array([ 1., 2., 3., 4.])
References References
---------- ----------
http://en.wikipedia.org/wiki/Chebyshev_polynomials http://en.wikipedia.org/wiki/Chebyshev_polynomials
@ -990,8 +1001,9 @@ def chebextr(n):
Examples Examples
-------- --------
>>> x = chebextr(4) >>> import wafo.polynomial as wp
>>> chebpoly(4,x) >>> x = wp.chebextr(4)
>>> wp.chebpoly(4,x)
array([ 1., -1., 1., -1., 1.]) array([ 1., -1., 1., -1., 1.])
@ -1021,15 +1033,16 @@ def chebroot(n, kind=1):
Examples Examples
-------- --------
>>> import numpy as np >>> import numpy as np
>>> x = chebroot(3) >>> import wafo.polynomial as wp
>>> np.abs(chebpoly(3,x))<1e-15 >>> x = wp.chebroot(3)
array([ True, True, True], dtype=bool) >>> np.allclose(wp.chebpoly(3,x), [0, 0, 0])
>>> chebpoly(3) True
>>> wp.chebpoly(3)
array([ 4., 0., -3., 0.]) array([ 4., 0., -3., 0.])
>>> x2 = chebroot(4,kind=2) >>> x2 = wp.chebroot(4, kind=2)
>>> np.abs(chebpoly(4,x2,kind=2))<1e-15 >>> np.allclose(wp.chebpoly(4,x2,kind=2), [0, 0, 0, 0])
array([ True, True, True, True], dtype=bool) True
>>> chebpoly(4,kind=2) >>> wp.chebpoly(4,kind=2)
array([ 16., 0., -12., 0., 1.]) array([ 16., 0., -12., 0., 1.])
@ -1071,15 +1084,16 @@ def chebpoly(n, x=None, kind=1):
Examples Examples
-------- --------
>>> import numpy as np >>> import numpy as np
>>> x = chebroot(3) >>> import wafo.polynomial as wp
>>> np.abs(chebpoly(3,x))<1e-15 >>> x = wp.chebroot(3)
array([ True, True, True], dtype=bool) >>> np.allclose(wp.chebpoly(3,x), [0, 0, 0])
>>> chebpoly(3) True
>>> wp.chebpoly(3)
array([ 4., 0., -3., 0.]) array([ 4., 0., -3., 0.])
>>> x2 = chebroot(4,kind=2) >>> x2 = wp.chebroot(4,kind=2)
>>> np.abs(chebpoly(4,x2,kind=2))<1e-15 >>> np.allclose(wp.chebpoly(4,x2,kind=2), [0, 0, 0, 0])
array([ True, True, True, True], dtype=bool) True
>>> chebpoly(4,kind=2) >>> wp.chebpoly(4,kind=2)
array([ 16., 0., -12., 0., 1.]) array([ 16., 0., -12., 0., 1.])
@ -1131,14 +1145,15 @@ def chebfit(fun, n=10, a=-1, b=1, trace=False):
Fit exp(x) Fit exp(x)
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> a = 0; b = 2 >>> a = 0; b = 2
>>> ck = chebfit(np.exp,7,a,b); >>> ck = wp.chebfit(np.exp,7,a,b);
>>> x = np.linspace(0,4); >>> x = np.linspace(0,4);
>>> x1 = chebroot(9)*(b-a)/2+(b+a)/2 >>> x1 = wp.chebroot(9)*(b-a)/2+(b+a)/2
>>> ck1 = chebfit(np.exp(x1)) >>> ck1 = wp.chebfit(np.exp(x1))
h=plt.plot(x, np.exp(x), 'r', x, chebval(x,ck,a,b), 'g.') h=plt.plot(x, np.exp(x), 'r', x, wp.chebval(x,ck,a,b), 'g.')
h = plt.plot(x,np.exp(x), 'r', x, chebval(x,ck1,a,b),'g.') h = plt.plot(x,np.exp(x), 'r', x, wp.chebval(x,ck1,a,b),'g.')
plt.close() plt.close()
See also See also
@ -1209,22 +1224,23 @@ def chebfit_dct(f, n=(10, ), domain=None):
Fit exponential function Fit exponential function
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> domain = (0, 2) >>> domain = (0, 2)
>>> ck = chebfit_dct(np.exp, 7, domain) >>> ck = wp.chebfit_dct(np.exp, 7, domain)
>>> np.allclose(ck, [3.44152387e+00, 3.07252345e+00, 7.38000848e-01, >>> np.allclose(ck, [3.44152387e+00, 3.07252345e+00, 7.38000848e-01,
... 1.20520053e-01, 1.48805268e-02, 1.47579673e-03, ... 1.20520053e-01, 1.48805268e-02, 1.47579673e-03,
... 1.21719524e-04]) ... 1.21719524e-04])
True True
>>> x1 = map_to_interval(chebroot(9), *domain) >>> x1 = wp.map_to_interval(wp.chebroot(9), *domain)
>>> ck1 = chebfit(np.exp(x1)) >>> ck1 = wp.chebfit(np.exp(x1))
>>> np.allclose(ck1, [5.40019009e-07, 8.69418381e-06, 1.22261037e-04, >>> np.allclose(ck1, [5.40019009e-07, 8.69418381e-06, 1.22261037e-04,
... 1.47582673e-03, 1.48805283e-02, 1.20520053e-01, ... 1.47582673e-03, 1.48805283e-02, 1.20520053e-01,
... 7.38000848e-01, 3.07252345e+00, 3.44152387e+00]) ... 7.38000848e-01, 3.07252345e+00, 3.44152387e+00])
True True
x = np.linspace(0,4) x = np.linspace(0,4)
h = plt.plot(x, np.exp(x), 'r', x, chebvalnd(ck, x,ck,a,b), 'g.') h = plt.plot(x, np.exp(x), 'r', x, wp.chebvalnd(ck, x,ck,a,b), 'g.')
h = plt.plot(x, np.exp(x), 'r', x, chebvalnd(ck1, x,ck1,a,b),'b.') h = plt.plot(x, np.exp(x), 'r', x, wp.chebvalnd(ck1, x,ck1,a,b),'b.')
plt.close() plt.close()
See also See also
@ -1280,11 +1296,12 @@ def idct(x, n=None):
Examples Examples
-------- --------
>>> import numpy as np >>> import numpy as np
>>> import wafo.polynomial as wp
>>> x = np.arange(5)*1.0 >>> x = np.arange(5)*1.0
>>> np.abs(x-idct(dct(x)))<1e-14 >>> np.allclose(idct(dct(x)), x)
array([ True, True, True, True, True], dtype=bool) True
>>> np.abs(x-dct(idct(x)))<1e-14 >>> np.allclose(dct(idct(x)), x)
array([ True, True, True, True, True], dtype=bool) True
Reference Reference
--------- ---------
@ -1358,18 +1375,19 @@ def chebval(x, ck, a=-1, b=1, kind=1, fill=None):
-------- --------
Plot Chebychev polynomial of the first kind and order 4: Plot Chebychev polynomial of the first kind and order 4:
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> x = np.linspace(-1,1) >>> x = np.linspace(-1,1)
>>> ck = np.zeros(5); ck[-1]=1 >>> ck = np.zeros(5); ck[-1]=1
>>> y = chebval(x,ck) >>> y = wp.chebval(x,ck)
h = plt.plot(x, y, x, chebpoly(4,x),'.') h = plt.plot(x, y, x, wp.chebpoly(4,x),'.')
plt.close() plt.close()
Fit exponential function: Fit exponential function:
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> ck = chebfit(np.exp,7,0,2) >>> ck = wp.chebfit(np.exp,7,0,2)
>>> x = np.linspace(0,4); >>> x = np.linspace(0,4);
>>> y2 = chebval(x,ck,0,2) >>> y2 = wp.chebval(x,ck,0,2)
h=plt.plot(x, y2, 'g', x, np.exp(x)) h=plt.plot(x, y2, 'g', x, np.exp(x))
plt.close() plt.close()
@ -1417,10 +1435,11 @@ def chebder(ck, a=-1, b=1):
Fit exponential function: Fit exponential function:
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> ck = chebfit(np.exp,7,0,2) >>> import wafo.polynomial as wp
>>> ck = wp.chebfit(np.exp,7,0,2)
>>> x = np.linspace(0,4) >>> x = np.linspace(0,4)
>>> ck2 = chebder(ck,0,2) >>> ck2 = wp.chebder(ck,0,2)
>>> y = chebval(x,ck2,0,2) >>> y = wp.chebval(x,ck2,0,2)
h = plt.plot(x, y, 'g', x, np.exp(x), 'r') h = plt.plot(x, y, 'g', x, np.exp(x), 'r')
plt.close() plt.close()
@ -1470,12 +1489,13 @@ def chebint(ck, a=-1, b=1):
-------- --------
Fit exponential function: Fit exponential function:
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> ck = chebfit(np.exp,7,0,2) >>> import wafo.polynomial as wp
>>> ck = wp.chebfit(np.exp, 7, 0, 2)
>>> x = np.linspace(0,4) >>> x = np.linspace(0,4)
>>> ck2 = chebint(ck,0,2); >>> ck2 = wp.chebint(ck, 0, 2);
>>> y =chebval(x,ck2,0,2) >>> y = wp.chebval(x, ck2, 0, 2)
h=plt.plot(x,y,'g',x,np.exp(x),'r.') h = plt.plot(x, y, 'g', x, np.exp(x), 'r.')
plt.close() plt.close()
See also See also
@ -1713,14 +1733,15 @@ def padefit(c, m=None):
Pade approximation to exp(x) Pade approximation to exp(x)
>>> import scipy.special as sp >>> import scipy.special as sp
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> c = poly1d(1./sp.gamma(np.r_[6+1:0:-1])) >>> import wafo.polynomial as wp
>>> [p, q] = padefit(c) >>> c = wp.poly1d(1./sp.gamma(np.r_[6+1:0:-1]))
>>> [p, q] = wp.padefit(c)
>>> p; q >>> p; q
poly1d([ 0.00277778, 0.03333333, 0.2 , 0.66666667, 1. ]) poly1d([ 0.00277778, 0.03333333, 0.2 , 0.66666667, 1. ])
poly1d([ 0.03333333, -0.33333333, 1. ]) poly1d([ 0.03333333, -0.33333333, 1. ])
x = np.linspace(0,4) x = np.linspace(0,4)
h = plt.plot(x,c(x),x,p(x)/q(x),'g-', x,np.exp(x),'r.') h = plt.plot(x, c(x), x, p(x)/q(x), 'g-', x,np.exp(x), 'r.')
plt.close() plt.close()
See also See also
@ -1781,13 +1802,14 @@ def padefitlsq(fun, m, k, a=-1, b=1, trace=False, x=None, end_points=True):
Pade approximation to exp(x) between 0 and 2 Pade approximation to exp(x) between 0 and 2
>>> import matplotlib.pyplot as plt >>> import matplotlib.pyplot as plt
>>> [c1, c2] = padefitlsq(np.exp,3,3,0,2) >>> import wafo.polynomial as wp
>>> [c1, c2] = wp.padefitlsq(np.exp,3,3,0,2)
>>> c1; c2 >>> c1; c2
poly1d([ 0.01443847, 0.128842 , 0.55284547, 0.99999962]) poly1d([ 0.01443847, 0.128842 , 0.55284547, 0.99999962])
poly1d([-0.0049658 , 0.07610473, -0.44716929, 1. ]) poly1d([-0.0049658 , 0.07610473, -0.44716929, 1. ])
x = np.linspace(0,4) x = np.linspace(0,4)
h = plt.plot(x, polyval(c1,x)/polyval(c2,x),'g') h = plt.plot(x, wp.polyval(c1,x)/wp.polyval(c2,x),'g')
h = plt.plot(x, np.exp(x), 'r') h = plt.plot(x, np.exp(x), 'r')
See also See also

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