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pywafo/wafo/polynomial_old.py

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Initial import of original WAFO code.
15 years ago
#-------------------------------------------------------------------------------
# Name: polynomial
# Purpose: Functions to operate on polynomials.
#
# Author: pab
# polyXXX functions are based on functions found in the matlab toolbox polyutil written by
# Author: Peter J. Acklam
# E-mail: pjacklam@online.no
# WWW URL: http://home.online.no/~pjacklam
#
# Created: 30.12.2008
# Copyright: (c) pab 2008
# Licence: LGPL
#-------------------------------------------------------------------------------
#!/usr/bin/env python
"""
Extended functions to operate on polynomials
"""
import warnings
import numpy as np
from numpy.lib.polynomial import *
__all__ = np.lib.polynomial.__all__
__all__ = __all__ + ['polyreloc', 'polyrescl', 'polytrim', 'poly2hstr', 'poly2str',
'polyshift', 'polyishift', 'map_from_intervall', 'map_to_intervall',
'cheb2poly', 'chebextr', 'chebroot', 'chebpoly', 'chebfit', 'chebval',
'chebder', 'chebint', 'Cheb1d', 'dct', 'idct']
def polyreloc(p,x,y=0.0):
"""
Relocate polynomial
The polynomial `p` is relocated by "moving" it `x`
units along the x-axis and `y` units along the y-axis.
So the polynomial `r` is relative to the point (x,y) as
the polynomial `p` is relative to the point (0,0).
Parameters
----------
p : array-like, poly1d
vector or matrix of column vectors of polynomial coefficients to relocate.
(Polynomial coefficients are in decreasing order.)
x : scalar
distance to relocate P along x-axis
y : scalar
distance to relocate P along y-axis (default 0)
Returns
-------
r : ndarray, poly1d
vector/matrix/poly1d of relocated polynomial coefficients.
See also
--------
polyrescl
Example
-------
>>> import numpy as np
>>> p = np.arange(6); p.shape = (2,-1)
>>> np.polyval(p,0)
array([3, 4, 5])
>>> np.polyval(p,1)
array([3, 5, 7])
>>> r = polyreloc(p,-1) # move to the left along x-axis
>>> np.polyval(r,-1) # = polyval(p,0)
array([3, 4, 5])
>>> np.polyval(r,0) # = polyval(p,1)
array([3, 5, 7])
"""
truepoly = isinstance(p, poly1d)
r = np.atleast_1d(p).copy()
n = r.shape[0]
# Relocate polynomial using Horner's algorithm
for ii in range(n,1,-1):
for i in range(1,ii):
r[i] = r[i] - x*r[i-1]
r[-1] = r[-1] + y
if r.ndim>1 and r.shape[-1]==1:
r.shape = (r.size,)
if truepoly:
r = poly1d(r)
return r
def polyrescl(p, x, y=1.0):
"""
Rescale polynomial.
Parameters
----------
p : array-like, poly1d
vector or matrix of column vectors of polynomial coefficients to rescale.
(Polynomial coefficients are in decreasing order.)
x,y : scalars
defining the factors to rescale the polynomial `p` in
x-direction and y-direction, respectively.
Returns
-------
r : ndarray, poly1d
vector/matrix/poly1d of rescaled polynomial coefficients.
See also
--------
polyreloc
Example
-------
>>> import numpy as np
>>> p = np.arange(6); p.shape = (2,-1)
>>> np.polyval(p,0)
array([3, 4, 5])
>>> np.polyval(p,1)
array([3, 5, 7])
>>> r = polyrescl(p,2) # scale by 2 along x-axis
>>> np.polyval(r,0) # = polyval(p,0)
array([ 3., 4., 5.])
>>> np.polyval(r,2) # = polyval(p,1)
array([ 3., 5., 7.])
"""
truepoly = isinstance(p, poly1d)
r = np.atleast_1d(p).copy()
n = r.shape[0]
xscale =(float(x)**np.arange(1-n , 1))
if r.ndim==1:
q = y*r*xscale
else:
q = y*r*xscale[:,np.newaxis]
if truepoly:
q = poly1d(q)
return q
def polytrim(p):
"""
Trim polynomial by stripping off leading zeros.
Parameters
----------
p : array-like, poly1d
vector of polynomial coefficients in decreasing order.
Returns
-------
r : ndarray, poly1d
vector/matrix/poly1d of trimmed polynomial coefficients.
Example
-------
>>> p = [0,1,2]
>>> polytrim(p)
array([1, 2])
"""
truepoly = isinstance(p, poly1d)
if truepoly:
return p
else:
r = np.atleast_1d(p).copy()
# Remove leading zeros
is_not_lead_zeros =np.logical_or.accumulate(r != 0,axis=0)
if r.ndim==1:
r = r[is_not_lead_zeros]
else:
is_not_lead_zeros = np.any(is_not_lead_zeros,axis=1)
r = r[is_not_lead_zeros,:]
return r
def poly2hstr(p, variable='x' ):
"""
Return polynomial as a Horner represented string.
Parameters
----------
p : array-like poly1d
vector of polynomial coefficients in decreasing order.
variable : string
display character for variable
Returns
-------
p_str : string
consisting of the polynomial coefficients in the vector P multiplied
by powers of the given `variable`.
Examples
--------
>>> poly2hstr([1, 1, 2], 's' )
'(s + 1)*s + 2'
See also
--------
poly2str
"""
var = variable
coefs = polytrim(np.atleast_1d(p))
order = len(coefs)-1 # Order of polynomial.
s = '' # Initialize output string.
ix = 1;
for expon in range(order,-1,-1):
coef = coefs[order-expon]
#% There is no point in adding a zero term (except if it's the only
#% term, but we'll take care of that later).
if coef == 0:
ix = ix+1
else:
#% Append exponent if necessary.
if ix>1:
exponstr = '%.0f' % ix
s = '%s**%s' % (s,exponstr);
ix = 1
#% Is it the first term?
isfirst = s == ''
# We need the coefficient only if it is different from 1 or -1 or
# when it is the constant term.
needcoef = (( abs(coef) != 1 ) | ( expon == 0 ) & isfirst) | 1-isfirst
# We need the variable except in the constant term.
needvar = ( expon != 0 )
#% Add sign, but we don't need a leading plus-sign.
if isfirst:
if coef < 0:
s = '-' # % Unary minus.
else:
if coef < 0:
s = '%s - ' % s # % Binary minus (subtraction).
else:
s = '%s + ' % s # % Binary plus (addition).
#% Append the coefficient if it is different from one or when it is
#% the constant term.
if needcoef:
coefstr = '%.20g' % abs(coef)
s = '%s%s' % (s,coefstr)
#% Append variable if necessary.
if needvar:
#% Append a multiplication sign if necessary.
if needcoef:
if 1-isfirst:
s = '(%s)' % s
s = '%s*' % s
s = '%s%s' % ( s, var )
#% Now treat the special case where the polynomial is zero.
if s=='':
s = '0'
return s
def poly2str(p,variable='x'):
"""
Return polynomial as a string.
Parameters
----------
p : array-like poly1d
vector of polynomial coefficients in decreasing order.
variable : string
display character for variable
Returns
-------
p_str : string
consisting of the polynomial coefficients in the vector P multiplied
by powers of the given `variable`.
See also
--------
poly2hstr
Examples
--------
>>> poly2str([1, 1, 2], 's' )
's**2 + s + 2'
"""
thestr = "0"
var = variable
# Remove leading zeros
coeffs = polytrim(np.atleast_1d(p))
N = len(coeffs)-1
for k in range(len(coeffs)):
coefstr = '%.4g' % abs(coeffs[k])
if coefstr[-4:] == '0000':
coefstr = coefstr[:-5]
power = (N-k)
if power == 0:
if coefstr != '0':
newstr = '%s' % (coefstr,)
else:
if k == 0:
newstr = '0'
else:
newstr = ''
elif power == 1:
if coefstr == '0':
newstr = ''
elif coefstr == 'b' or coefstr == '1':
newstr = var
else:
newstr = '%s*%s' % (coefstr, var)
else:
if coefstr == '0':
newstr = ''
elif coefstr == 'b' or coefstr == '1':
newstr = '%s**%d' % (var, power,)
else:
newstr = '%s*%s**%d' % (coefstr, var, power)
if k > 0:
if newstr != '':
if coeffs[k] < 0:
thestr = "%s - %s" % (thestr, newstr)
else:
thestr = "%s + %s" % (thestr, newstr)
elif (k == 0) and (newstr != '') and (coeffs[k] < 0):
thestr = "-%s" % (newstr,)
else:
thestr = newstr
return thestr
def polyshift(py,a=-1,b=1):
"""
Polynomial coefficient shift
Polyshift shift the polynomial coefficients by a variable shift:
Y = 2*(X-.5*(b+a))/(b-a)
i.e., the interval -1 <= Y <= 1 is mapped to the interval a <= X <= b
Parameters
----------
py : array-like
polynomial coefficients for the variable y.
a,b : scalars
lower and upper limit.
Returns
-------
px : ndarray
polynomial coefficients for the variable x.
See also
--------
polyishift
Example
-------
>>> py = [1, 0]
>>> px = polyshift(py,0,5)
>>> polyval(px,[0, 2.5, 5]) #% This is the same as the line below
array([-1., 0., 1.])
>>> polyval(py,[-1, 0, 1 ])
array([-1, 0, 1])
"""
if (a==-1) & (b ==1):
return py
L = b-a
return polyishift(py,-(2.+b+a)/L,(2.-b-a)/L)
def polyishift(px,a=-1,b=1):
"""
Inverse polynomial coefficient shift
Polyishift does the inverse of Polyshift,
shift the polynomial coefficients by a variable shift:
Y = 2*(X-.5*(b+a)/(b-a)
i.e., the interval a <= X <= b is mapped to the interval -1 <= Y <= 1
Parameters
----------
px : array-like
polynomial coefficients for the variable x.
a,b : scalars
lower and upper limit.
Returns
-------
py : ndarray
polynomial coefficients for the variable y.
See also
--------
polyishift
Example
-------
>>> px = [1, 0]
>>> py = polyishift(px,0,5);
>>> polyval(px,[0, 2.5, 5]) #% This is the same as the line below
array([ 0. , 2.5, 5. ])
>>> polyval(py,[-1, 0, 1])
array([ 0. , 2.5, 5. ])
"""
if (a==-1) & (b ==1):
return px
L = b-a
xscale = 2./L
xloc = -float(a+b)/L
return polyreloc( polyrescl(px,xscale),xloc)
def map_from_interval(x,a,b) :
"""F(x), where F: [a,b] -> [-1,1]."""
return (x - (b + a)/2.0)*(2.0/(b - a))
def map_to_interval(x,a,b) :
"""F(x), where F: [-1,1] -> [a,b]."""
return (x*(b - a) + (b + a))/2.0
def poly2cheb(p,a=-1,b=1):
"""
Convert polynomial coefficients into Chebyshev coefficients
Parameters
----------
p : array-like
polynomial coefficients
a,b : real scalars
lower and upper limits (Default -1,1)
Returns
-------
ck : ndarray
Chebychef coefficients
POLY2CHEB do the inverse of CHEB2POLY: given a vector of polynomial
coefficients AK, returns an equivalent vector of Chebyshev
coefficients CK.
This is useful for economization of power series.
The steps for doing so:
1. Convert polynomial coefficients to Chebychev coefficients, CK.
2. Truncate the CK series to a smaller number of terms, using the
coefficient of the first neglected Chebychev polynomial as an error
estimate.
3 Convert back to a polynomial by CHEB2POLY
See also
--------
cheb2poly
chebval
chebfit
Examples
--------
>>> import numpy as np
>>> p = np.arange(5)
>>> ck = poly2cheb(p)
>>> cheb2poly(ck)
array([ 1., 2., 3., 4.])
Reference
---------
William H. Press, Saul Teukolsky,
William T. Wetterling and Brian P. Flannery (1997)
"Numerical recipes in Fortran 77", Vol. 1, pp 184-194
"""
f = poly1d(p)
n = len(f.coeffs)
return chebfit(f,n,a,b)
def cheb2poly(ck,a=-1,b=1):
"""
Converts Chebyshev coefficients to polynomial coefficients
Parameters
----------
ck : array-like
Chebychef coefficients
a,b : real, scalars
lower and upper limits (Default -1,1)
Returns
-------
p : ndarray
polynomial coefficients
It is not advised to do this for len(ck)>10 due to numerical cancellations.
See also
--------
chebval
chebfit
Examples
--------
>>> import numpy as np
>>> p = np.arange(5)
>>> ck = poly2cheb(p)
>>> cheb2poly(ck)
array([ 1., 2., 3., 4.])
References
----------
http://en.wikipedia.org/wiki/Chebyshev_polynomials
http://en.wikipedia.org/wiki/Chebyshev_form
http://en.wikipedia.org/wiki/Clenshaw_algorithm
"""
n = len(ck)
b_Nmi = np.zeros(1)
b_Nmip1 = np.zeros(1)
y = np.r_[2/(b-a), -(a+b)/(b-a)]
y2 = 2.*y
# Clenshaw recurence
for ix in range(n-1,0,-1):
tmp = b_Nmi
b_Nmi = polymul(y2,b_Nmi) # polynomial multiplication
nb = len(b_Nmip1)
b_Nmip1[-1] = b_Nmip1[-1]-ck[ix]
b_Nmi[-nb::] = b_Nmi[-nb::]-b_Nmip1
b_Nmip1 = tmp
p = polymul(y,b_Nmi) # polynomial multiplication
nb = len(b_Nmip1)
b_Nmip1[-1] = b_Nmip1[-1]-ck[0]
p[-nb::] = p[-nb::]-b_Nmip1
return polytrim(p)
def chebextr(n):
"""
Return roots of derivative of Chebychev polynomial of the first kind.
All local extreme values of the polynomial are either -1 or 1. So,
CHEBPOLY( N, CHEBEXTR(N) ) ) return the same as (-1).^(N:-1:0)
except for the numerical noise in the former.
Because the extreme values of Chebychev polynomials of the first
kind are either -1 or 1, their roots are often used as starting
values for the nodes in minimax approximations.
Parameters
----------
n : scalar, integer
degree of Chebychev polynomial.
Examples
--------
>>> x = chebextr(4)
>>> chebpoly(4,x)
array([ 1., -1., 1., -1., 1.])
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://en.wikipedia.org/wiki/Chebyshev_polynomials
"""
return -np.cos((np.pi*np.arange(n+1))/n);
def chebroot(n,kind=1):
"""
Return roots of Chebychev polynomial of the first or second kind.
The roots of the Chebychev polynomial of the first kind form a particularly
good set of nodes for polynomial interpolation because the resulting
interpolation polynomial minimizes the problem of Runge's phenomenon.
Parameters
----------
n : scalar, integer
degree of Chebychev polynomial.
kind: 1 or 2, optional
kind of Chebychev polynomial (default 1)
Examples
--------
>>> import numpy as np
>>> x = chebroot(3)
>>> np.abs(chebpoly(3,x))<1e-15
array([ True, True, True], dtype=bool)
>>> chebpoly(3)
array([ 4., 0., -3., 0.])
>>> x2 = chebroot(4,kind=2)
>>> np.abs(chebpoly(4,x2,kind=2))<1e-15
array([ True, True, True, True], dtype=bool)
>>> chebpoly(4,kind=2)
array([ 16., 0., -12., 0., 1.])
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://en.wikipedia.org/wiki/Chebyshev_polynomials
"""
if kind not in (1,2):
raise ValueError('kind must be 1 or 2')
return -np.cos(np.pi*(np.arange(n)+0.5*kind)/(n+kind-1));
def chebpoly(n, x=None, kind=1):
"""
Return Chebyshev polynomial of the first or second kind.
These polynomials are orthogonal on the interval [-1,1], with
respect to the weight function w(x) = (1-x^2)^(-1/2+kind-1).
chebpoly(n) returns the coefficients of the Chebychev polynomial of degree N.
chebpoly(n,x) returns the Chebychev polynomial of degree N evaluated in X.
Parameters
----------
n : integer, scalar
degree of Chebychev polynomial.
x : array-like, optional
evaluation points
kind: 1 or 2, optional
kind of Chebychev polynomial (default 1)
Returns
-------
p : ndarray
polynomial coefficients if x is None.
Chebyshev polynomial evaluated at x otherwise
Examples
--------
>>> import numpy as np
>>> x = chebroot(3)
>>> np.abs(chebpoly(3,x))<1e-15
array([ True, True, True], dtype=bool)
>>> chebpoly(3)
array([ 4., 0., -3., 0.])
>>> x2 = chebroot(4,kind=2)
>>> np.abs(chebpoly(4,x2,kind=2))<1e-15
array([ True, True, True, True], dtype=bool)
>>> chebpoly(4,kind=2)
array([ 16., 0., -12., 0., 1.])
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_polynomials
"""
if x is None: # Calculate coefficients.
if n == 0:
p = np.ones(1)
else:
p = np.round( pow(2,n-2+kind) * np.poly( chebroot(n,kind=kind) ) )
p[1::2] = 0;
return p
else: # Evaluate polynomial in chebychev form
ck = np.zeros(n+1)
ck[n] = 1.
return _chebval(np.atleast_1d(x),ck,kind=kind)
def chebfit(fun,n=10,a=-1,b=1,trace=False):
"""
Computes the Chebyshevs coefficients
so that f(x) can be approximated by:
n-1
f(x) = sum ck*Tk(x)
k=0
where Tk is the k'th Chebyshev polynomial of the first kind.
Parameters
----------
fun : callable
function to approximate
n : integer, scalar, optional
number of base points (abscissas). Default n=10 (maximum 50)
a,b : real, scalars, optional
integration limits
Returns
-------
ck : ndarray
polynomial coefficients in Chebychev form.
Examples
--------
Fit exp(x)
>>> import pylab as pb
>>> a = 0; b = 2
>>> ck = chebfit(pb.exp,7,a,b);
>>> x = pb.linspace(0,4);
>>> h=pb.plot(x,pb.exp(x),'r',x,chebval(x,ck,a,b),'g.')
>>> x1 = chebroot(9)*(b-a)/2+(b+a)/2
>>> ck1 = chebfit(pb.exp(x1))
>>> h=pb.plot(x,pb.exp(x),'r',x,chebval(x,ck1,a,b),'g.')
>>> pb.close()
See also
--------
chebval
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://mathworld.wolfram.com/ChebyshevApproximationFormula.html
W. Fraser (1965)
"A Survey of Methods of Computing Minimax and Near-Minimax Polynomial
Approximations for Functions of a Single Independent Variable"
Journal of the ACM (JACM), Vol. 12 , Issue 3, pp 295 - 314
William H. Press, Saul Teukolsky,
William T. Wetterling and Brian P. Flannery (1997)
"Numerical recipes in Fortran 77", Vol. 1, pp 184-194
"""
if (n>50):
warnings.warn('CHEBFIT should only be used for n<50')
if hasattr(fun,'__call__'):
x = map_to_interval(chebroot(n),a,b)
f = fun(x);
if trace:
import pylab as plb
plb.plot(x,f,'+')
else:
f = fun
n = len(f)
#raise ValueError('Function must be callable!')
# N-1
# c(k) = (2/N) sum w(n) f(n)*cos(pi*k*(2n+1)/(2N)), 0 <= k < N.
# n=0
#
# w(0) = 0.5, w(n)=1 for n>0
ck = dct(f[::-1])/n
ck[0] = ck[0]/2.
return ck
def dct(x,n=None):
"""
Discrete Cosine Transform
N-1
y[k] = 2* sum x[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
n=0
Examples
--------
>>> import numpy as np
>>> x = np.arange(5)
>>> np.abs(x-idct(dct(x)))<1e-14
array([ True, True, True, True, True], dtype=bool)
>>> np.abs(x-dct(idct(x)))<1e-14
array([ True, True, True, True, True], dtype=bool)
Reference
---------
http://en.wikipedia.org/wiki/Discrete_cosine_transform
http://users.ece.utexas.edu/~bevans/courses/ee381k/lectures/
"""
fft = np.fft.fft
x = np.atleast_1d(x)
if n is None:
n = x.shape[-1]
if x.shape[-1]<n:
n_shape = x.shape[:-1] + (n-x.shape[-1],)
xx = np.hstack((x,np.zeros(n_shape)))
else:
xx = x[...,:n]
real_x = np.all(np.isreal(xx))
if (real_x and (np.remainder(n,2) == 0)):
xp = 2 * fft(np.hstack( (xx[...,::2], xx[...,::-2]) ))
else:
xp = fft(np.hstack((xx, xx[...,::-1])))
xp = xp[...,:n]
w = np.exp(-1j * np.arange(n) * np.pi/(2*n))
y = xp*w
if real_x:
return y.real
else:
return y
def idct(x,n=None):
"""
Inverse Discrete Cosine Transform
N-1
x[k] = 1/N sum w[n]*y[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
n=0
w(0) = 1/2
w(n) = 1 for n>0
Examples
--------
>>> import numpy as np
>>> x = np.arange(5)
>>> np.abs(x-idct(dct(x)))<1e-14
array([ True, True, True, True, True], dtype=bool)
>>> np.abs(x-dct(idct(x)))<1e-14
array([ True, True, True, True, True], dtype=bool)
Reference
---------
http://en.wikipedia.org/wiki/Discrete_cosine_transform
http://users.ece.utexas.edu/~bevans/courses/ee381k/lectures/
"""
ifft = np.fft.ifft
x = np.atleast_1d(x)
if n is None:
n = x.shape[-1]
w = np.exp(1j * np.arange(n) * np.pi/(2*n))
if x.shape[-1]<n:
n_shape = x.shape[:-1] + (n-x.shape[-1],)
xx = np.hstack((x,np.zeros(n_shape)))*w
else:
xx = x[...,:n]*w
real_x = np.all(np.isreal(x))
if (real_x and (np.remainder(n,2) == 0)):
xx[...,0] = xx[...,0]*0.5
yp = ifft(xx)
y = np.zeros(xx.shape,dtype=complex)
y[...,::2] = yp[...,:n/2]
y[...,::-2] = yp[...,n/2::]
else:
yp = ifft(np.hstack((xx, np.zeros_like(xx[...,0]), np.conj(xx[...,:0:-1]))))
y = yp[...,:n]
if real_x:
return y.real
else:
return y
def _chebval(x,ck,kind=1):
"""
Evaluate polynomial in Chebyshev form.
A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form:
N
p(x) = sum ck*Tk(x)
k=0
or
N
p(x) = sum ck*Uk(x)
k=0
where Tk and Uk are the k'th Chebyshev polynomial of the first and second
kind, respectively.
References
----------
http://en.wikipedia.org/wiki/Clenshaw_algorithm
http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
"""
n = len(ck)
b_Nmi = np.zeros(x.shape) # b_(N-i)
b_Nmip1 = b_Nmi.copy() # b_(N-i+1)
x2 = 2*x
# Clenshaw reccurence
for ix in range(n-1,0,-1):
tmp = b_Nmi
b_Nmi = x2*b_Nmi - b_Nmip1 + ck[ix]
b_Nmip1 = tmp
return kind*x*b_Nmi - b_Nmip1 + ck[0]
def chebval(x,ck,a=-1,b=1,kind=1,fill=None):
"""
Evaluate polynomial in Chebyshev form at X
A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form:
N
p(x) = sum ck*Tk(x)
k=0
where Tk is the k'th Chebyshev polynomial of the first or second kind.
Paramaters
----------
x : array-like
points to evaluate
ck : array-like
polynomial coefficients in Chebyshev form.
a,b : real, scalars, optional
limits for polynomial (Default -1,1)
kind: 1 or 2, optional
kind of Chebychev polynomial (default 1)
fill : scalar, optional
If provided, define value to return for `x < a` or `b < x`.
Examples
--------
Plot Chebychev polynomial of the first kind and order 4:
>>> import pylab as pb
>>> x = pb.linspace(-1,1)
>>> ck = pb.zeros(5); ck[-1]=1
>>> h = pb.plot(x,chebval(x,ck),x,chebpoly(4,x),'.')
>>> pb.close()
Fit exponential function:
>>> import pylab as pb
>>> ck = chebfit(pb.exp,7,0,2)
>>> x = pb.linspace(0,4);
>>> h=pb.plot(x,chebval(x,ck,0,2),'g',x,pb.exp(x))
>>> pb.close()
See also
--------
chebfit
References
----------
http://en.wikipedia.org/wiki/Clenshaw_algorithm
http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
"""
y = map_from_interval(np.atleast_1d(x),a,b)
if fill is None:
f = _chebval(y,ck,kind=kind)
else:
cond = (np.abs(y)<=1)
f = np.where(cond,0,fill)
if np.any(cond):
yk = np.extract(cond,y)
f[cond] = _chebval(yk,ck,kind=kind)
return f
def chebder(ck,a=-1,b=1):
"""
Differentiate Chebyshev polynomial
Parameters
----------
ck : array-like
polynomial coefficients in Chebyshev form of function to differentiate
a,b : real, scalars
limits for polynomial(Default -1,1)
Return
------
cder : ndarray
polynomial coefficients in Chebyshev form of the derivative
Examples
--------
Fit exponential function:
>>> import pylab as pb
>>> ck = chebfit(pb.exp,7,0,2)
>>> x = pb.linspace(0,4)
>>> ck2 = chebder(ck,0,2);
>>> h=pb.plot(x,chebval(x,ck,0,2),'g',x,pb.exp(x),'r')
>>> pb.close()
See also
--------
chebint
chebfit
"""
n = len(ck)
cder = np.zeros(n)
# n and n-1 are special cases.
# cder(n-1)=0;
cder[-2] = 2*(n-1)*ck[-1]
for j in range(n-3,-1,-1):
cder[j] = cder[j+2]+2*j*ck[j+1]
return cder*2./(b-a) # Normalize to the interval b-a.
def chebint(ck,a=-1,b=1):
"""
Integrate Chebyshef polynomial
Parameters
----------
ck : array-like
polynomial coefficients in Chebyshev form of function to integrate.
a,b : real, scalars
limits for polynomial(Default -1,1)
Return
------
cint : ndarray
polynomial coefficients in Chebyshev form of the integrated function
Examples
--------
Fit exponential function:
>>> import pylab as pb
>>> ck = chebfit(pb.exp,7,0,2)
>>> x = pb.linspace(0,4)
>>> ck2 = chebint(ck,0,2);
>>> h=pb.plot(x,chebval(x,ck,0,2),'g',x,pb.exp(x),'r.')
>>> pb.close()
See also
--------
chebder
chebfit
"""
n = len(ck)
cint = np.zeros(n);
con = 0.25*(b-a);
dif1 = np.diff(ck[::2])
ix1= np.r_[1:n-1:2]
cint[ix1] = -(con*dif1)/(ix1-1)
if n>3:
dif2 = np.diff(ck[1::2])
ix2=np.r_[2:n-1:2]
cint[ix2] = -(con*dif2)/(ix2-1)
#% cint(n) is a special case
cint[-1] = (con*ck[n-2])/(n-1)
cint[0] = np.sum((-1)**np.r_[1:n]*cint[1::]) # Set constant of integration
return cint
class Cheb1d(object):
coeffs = None
order = None
a = None
b = None
def __init__(self,ck,a=-1,b=1):
if isinstance(ck, poly1d):
for key in ck.__dict__.keys():
self.__dict__[key] = ck.__dict__[key]
return
cki = np.trim_zeros(np.atleast_1d(ck),'b')
if len(cki.shape) > 1:
raise ValueError, "Polynomial must be 1d only."
self.__dict__['coeffs'] = cki
self.__dict__['order'] = len(cki) - 1
self.__dict__['a'] = a
self.__dict__['b'] = b
def __call__(self,x):
return chebval(x,self.coeffs,self.a,self.b)
def __array__(self, t=None):
if t:
return np.asarray(self.coeffs, t)
else:
return np.asarray(self.coeffs)
def __repr__(self):
vals = repr(self.coeffs)
vals = vals[6:-1]
return "Cheb1d(%s)" % vals
def __len__(self):
return self.order
def __str__(self):
pass
def __neg__(self):
return Cheb1d(-self.coeffs,self.a,self.b)
def __pos__(self):
return self
def __add__(self, other):
other = poly1d(other)
return poly1d(polyadd(self.coeffs, other.coeffs))
def __radd__(self, other):
other = poly1d(other)
return poly1d(polyadd(self.coeffs, other.coeffs))
def __sub__(self, other):
other = poly1d(other)
return poly1d(polysub(self.coeffs, other.coeffs))
def __rsub__(self, other):
other = poly1d(other)
return poly1d(polysub(other.coeffs, self.coeffs))
def __eq__(self, other):
return np.alltrue(self.coeffs == other.coeffs)
def __ne__(self, other):
return np.any(self.coeffs != other.coeffs) or (self.a!=other.a) or (self.b !=other.b)
def __setattr__(self, key, val):
raise ValueError, "Attributes cannot be changed this way."
def __getattr__(self, key):
if key in ['c','coef','coefficients']:
return self.coeffs
elif key in ['o']:
return self.order
elif key in ['a']:
return self.a
elif key in ['b']:
return self.b
else:
try:
return self.__dict__[key]
except KeyError:
raise AttributeError("'%s' has no attribute '%s'" % (self.__class__, key))
def __getitem__(self, val):
if val > self.order:
return 0
if val < 0:
return 0
return self.coeffs[val]
def __setitem__(self, key, val):
ind = self.order - key
if key < 0:
raise ValueError, "Does not support negative powers."
if key > self.order:
zr = NX.zeros(key-self.order, self.coeffs.dtype)
self.__dict__['coeffs'] = NX.concatenate((self.coeffs,zr))
self.__dict__['order'] = key
self.__dict__['coeffs'][key] = val
return
def __iter__(self):
return iter(self.coeffs)
def integ(self, m=1):
"""
Return an antiderivative (indefinite integral) of this polynomial.
Refer to `chebint` for full documentation.
See Also
--------
chebint : equivalent function
"""
return Cheb1d(chebint(self.coeffs, self.a,self.b))
def deriv(self, m=1):
"""
Return a derivative of this polynomial.
Refer to `chebder` for full documentation.
See Also
--------
chebder : equivalent function
"""
return Cheb1d(chebder(self.coeffs,self.a,self.b))
Ongoing work
12 years ago
def test_doctstrings():
Initial import of original WAFO code.
15 years ago
if False: #True: #
x = np.arange(4)
dx = dct(x)
idx = idct(dx)
import pylab as plb
a = 0;
b = 2;
ck = chebfit(np.exp,6,a,b);
t = chebval(0,ck,a,b)
x=np.linspace(0,2,6);
plb.plot(x,np.exp(x),'r', x,chebval(x,ck,a,b),'g.')
#x1 = chebroot(9).'*(b-a)/2+(b+a)/2 ;
#ck1 =chebfit([x1 exp(x1)],9,a,b);
#plot(x,exp(x),'r'), hold on
#plot(x,chebval(x,ck1,a,b),'g'), hold off
t = poly2hstr([1,1,2])
py = [1, 0]
px = polyshift(py,0,5);
t1=polyval(px,[0, 2.5, 5]) #% This is the same as the line below
t2=polyval(py,[-1, 0, 1 ])
px = [1, 0]
py = polyishift(px,0,5);
t1 = polyval(px,[0, 2.5, 5]) #% This is the same as the line below
t2 = polyval(py,[-1, 0, 1 ])
print(t1,t2)
else:
import doctest
doctest.testmod()
if __name__== '__main__':
Ongoing work
12 years ago
test_doctstrings()