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QuickStart Samples

Advanced Polynomials QuickStart Sample (IronPython)

Illustrates more advanced uses of the Polynomial class, including real and complex root finding, calculating least squares polynomials and polynomial arithmetic in IronPython.

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import numerics

from math import *
from System import Array

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

#/ Illustrates the more advanced uses of the Polynomial class 
#/ in the Extreme.Mathematics.Curve namespace of the Extreme Optimization 
#/ Mathematics Library for .NET.

# Basic operations on polynomials are covered in the
# BasicPolynomials QuickStart Sample. This QuickStart
# Sample focuses on more advanced topics, including
# finding complex roots, calculating least-squares
# polynomials, and polynomial arithmetic.

#
# Complex numbers and polynomials
#

polynomial = Polynomial(Array[float]([ -2, 0, 1, 1 ]))

# The Polynomial class supports complex numbers
# as arguments for polynomials. It does not support
# polynomials with complex coefficients.
#
# For more about complex numbers, see the
# ComplexNumbers QuickStart Sample.
z1 = DoubleComplex(1, 2)

# Polynomial provides variants of ValueAt and
# SlopeAt for complex arguments:
print "polynomial.ComplexValueAt({0}) = {1}", z1, polynomial.ComplexValueAt(z1)
print "polynomial.ComplexSlopeAt({0}) = {1}", z1, polynomial.ComplexSlopeAt(z1)

#
# Real and complex roots
#
# Our polynomial has only one real root:
roots = polynomial.FindRoots()
print "Number of roots of polynomial1:", roots.Length
print "Value of root 1 =", roots[0]
# The FindComplexRoots method returns all three
# roots, two of which are complex:
complexRoots = polynomial.FindComplexRoots()
print "Number of complex roots:", complexRoots.Length
print "Value of root 1 =", complexRoots[0]
print "Value of root 2 =", complexRoots[1]
print "Value of root 3 =", complexRoots[2]

#
# Least squares polynomials
#
			
# Let's approximate 7 points on the unit circle
# by a fourth degree polynomial in the least squares
# sense.
# First, we create two arrays containing the x and
# y values of our data points:
xValues = Array[float]([ cos(i * Constants.Pi / 6) for i in range(7) ])
yValues = Array[float]([ -sin(i * Constants.Pi / 6) for i in range(7) ])

# Now we can find the least squares polynomial
# by calling the ststic LeastSquaresFit method.
# The last parameter is the degree of the desired
# polynomial.
lsqPolynomial = Polynomial.LeastSquaresFit(xValues, yValues, 4)
# Note that, as expected, the odd coefficients
# are close to zero.
print "Least squares fit:", lsqPolynomial.ToString()
			
#
# Polynomial arithmetic
#

# We can add, subtract, multiply and divide
# polynomials using overloaded operators:
a = Polynomial(Array[float]([ 4, -2, 4 ]))
b = Polynomial(Array[float]([ -3, 1 ]))

print "a =", a.ToString()
print "b =", b.ToString()
c = a + b
print "a + b =", c.ToString()
c = a - b
print "a - b =", c.ToString()
c = a * b
print "a * b =", c.ToString()
c = a / b
print "a / b =", c.ToString()
c = a % b
print "a % b =", c.ToString()