Basic Polynomials in IronPython QuickStart Sample

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

from System import Array

# The Polynomial class resides in the Extreme.Mathematics.Curves namespace.
from Extreme.Mathematics.Curves import *

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

# All curves inherit from the Curve abstract base
# class. The Polynomial class overrides implements all
# the methods and properties of the Curve class, # and adds a few more.

#
# Polynomial constructors
#

# The Polynomial class has multiple constructors. Each
# constructor derives from a different way to define
# a polynomial or parabola.

# 1st option: a polynomial of a specified degree.
polynomial1 = Polynomial(3)
# Now set the coefficients individually. The coefficients
# can be set using the indexer property. The constant term 
# has index 0:
polynomial1[3] = 1
polynomial1[2] = 1
polynomial1[1] = 0
polynomial1[0] = -2

# 2nd option: specify the coefficients in the constructor
# as an array of doubles:
coefficients = Array[float] ([ -2, 0, 1, 1 ])
polynomial2 = Polynomial(coefficients)

# In addition, you can create a polynomial that
# has certain roots using the static FromRoots
# method:
roots = Array[float] ([ 1, 2, 3, 4 ])
polynomial3 = Polynomial.FromRoots(roots)
# Or you can construct the interpolating polynomial
# by calling the static GetInterpolatingPolynomial
# method. The parameters are two double arrays 
# containing the x values and y values respectively.
xValues = Array[float] ([ 1, 2, 3, 4 ])
yValues = Array[float] ([ 1, 4, 10, 8 ])
polynomial4 = Polynomial.GetInterpolatingPolynomial(xValues, yValues)

# The ToString method gives a common string
# representation of the polynomial:
print "polynomial3 =", polynomial3.ToString()

#
# Curve Parameters
#

# The shape of any curve is determined by a set of parameters.
# These parameters can be retrieved and set through the
# Parameters collection. The number of parameters for a curve 
# is given by this collection's Count property.
#
# For polynomials, the parameters are the coefficients
# of the polynomial. The constant term has index 0:
print "polynomial1.Parameters.Count =", polynomial1.Parameters.Count
# Parameters can easily be retrieved:			
print "polynomial1 parameters:"
for p in polynomial1.Parameters:
	print p,
print 
# We can see that polynomial2 defines the same polynomial 
# curve as polynomial1:
print "polynomial2 parameters:"
for p in polynomial2.Parameters:
	print p,
print 
# Parameters can also be set:
polynomial2.Parameters[0] = 1

# For polynomials and other classes that inherit from
# the LinearCombination class, the parameters are also
# available through the indexer property of Polynomial.
# The following is equivalent to the line above:
polynomial2[0] = 1

# The degree of the polynomial is returned by
# the Degree property:
print "Degree of polynomial3 =", polynomial3.Degree

#
# Curve Methods
#

# The ValueAt method returns the y value of the
# curve at the specified x value:
print "polynomial1.ValueAt(2) =", polynomial1.ValueAt(2)

# The SlopeAt method returns the slope of the curve
# a the specified x value:
print "polynomial1.SlopeAt(2) =", polynomial1.SlopeAt(2)

# You can also create a new curve that is the 
# derivative of the original:
derivative = polynomial1.GetDerivative()
print "Slope at 2 (derivative) =", derivative.ValueAt(2)
# For a polynomial, the derivative is a Quadratic curve
# if the degree is equal to three:
print "Type of derivative:", derivative.GetType().FullName
print "Derivative parameters: "
for p in derivative.Parameters:
	print p,
print 
# If the degree is 4 or higher, the derivative is
# once again a polynomial:
print "Type of derivative for polynomial3:", polynomial3.GetDerivative().GetType().FullName

# You can get a Line that is the tangent to a curve
# at a specified x value using the TangentAt method:
tangent = polynomial1.TangentAt(2)
print "Tangent line at 2:"
print "  Y-intercept =", tangent.Parameters[0]
print "  Slope =", tangent.Parameters[1]

# For many curves, you can evaluate a definite
# integral exactly:
print "Integral of polynomial1 between 0 and 1 =", polynomial1.Integral(0, 1)

# You can find the zeroes or roots of the curve
# by calling the FindRoots method. Note that this
# method only returns the real roots.
roots = polynomial1.FindRoots()
print "Number of roots of polynomial1:", roots.Length
print "Value of root 1 =", roots[0]
# Let's find polynomial3's roots again:
roots = polynomial3.FindRoots()
print "Number of roots of polynomial3:", roots.Length
print "Value of root =", roots[0]
print "Value of root =", roots[1]
# Root finding isn't an exact science. Note the 
# round-off error in these values:
print "Value of root =", roots[2]
print "Value of root =", roots[3]

# For more advanced uses of the Polynomial class, # see the AdvancedPolynomials QuickStart sample.
```