Chebyshev Series in IronPython QuickStart Sample

Illustrates the basic use of the ChebyshevSeries class in IronPython.
```Python
import numerics

from math import *
from System import Array

# The ChebyshevSeries class resides in the Extreme.Mathematics.Curves 
# namespace.
from Extreme.Mathematics.Curves import *
# The Func<double, double> delegate resides in the 
# Extreme.Mathematics namespace.
from Extreme.Mathematics import *

# Illustrates the use of the ChebyshevSeries class 
# in the Extreme.Mathematics.Curve namespace of Extreme Numerics.NET.

# Chebyshev polynomials form an alternative basis
# for polynomials. A Chebyshev expansion is a 
# polynomial expressed as a sum of Chebyshev 
# polynomials.
#
# Using the ChebyshevSeries class instead of 
# Polynomial can have two major advantages:
#   1. They are numerically more stable. Higher
#      accuracy is maintained even for large problems.
#   2. When approximating other functions with
#      polynomials, the coefficients in the
#      Chebyshev expansion will tend to decrease
#      in size, where those of the normal polynomial
#      approximation will tend to oscillate wildly.

#
# Constructing Chebyshev expansions
#

# Chebyshev expansions are defined over an interval.
# The first constructor requires you to specify the
# boundaries of the interval, and the coefficients
# of the expansion.
coefficients = Array[float]([ 1, 0.5, -0.3, 0.1 ])
chebyshev1 = ChebyshevSeries(0, 2, coefficients)
# If you omit the boundaries, they are assumed to be
# -1 and +1:
chebyshev2 = ChebyshevSeries(coefficients)

# 
# Chebyshev approximations
#

# A third constructor creates a Chebyshev 
# approximation to an arbitrary function. For more
# about the Func<double, double> delegate, see the
# FunctionDelegates QuickStart Sample.
#
# Chebyshev expansions allow us to obtain an 
# excellent approximation at minimal cost.
#
# The following creates a Chebyshev approximation
# of degree 7 to Cos(x) over the interval [0, 2]:
approximation1 = ChebyshevSeries.GetInterpolatingPolynomial(cos, 0, 2, 7)
# The coefficients of the expansion are available through
# the indexer property of the ChebyshevSeries object:
print "Chebyshev approximation of cos(x):"
for index in range(0, 8):
	print "  c{0} = {1}".format(index, approximation1[index])

# The largest errors are approximately at the
# zeroes of the Chebyshev polynomial of degree 8:
for index in range(0,9):
	zero = 1 + cos(index * Constants.Pi / 8)
	error = approximation1.ValueAt(zero) - cos(zero)
	print " Error {0} = {1}".format(index, error)

#
# Least squares approximations
#

# We will now calculate the least squares polynomial
# of degree 7 through 33 points.
# First, calculate the points:
xValues = Vector([ 1 + cos(index * Constants.Pi / 32) for index in range(0,33) ])
yValues = Vector([ cos(x) for x in xValues ])

# Next, define a ChebyshevBasis object for the
# approximation we want: interval [0,2] and degree
# is 7.
basis = ChebyshevBasis(0, 2, 7)
# Now we can calculate the least squares fit:
approximation2 = basis.LeastSquaresFit(xValues, yValues)
# We can see it is close to the original 
# approximation we found earlier:
for index in range(0,7):
	print "  c{0} = {1}".format(index, approximation2[index])

```