Advanced Polynomials in IronPython QuickStart Sample

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

This sample is also available in: C#, Visual Basic, F#.

Overview

This QuickStart sample demonstrates advanced capabilities of the Polynomial class in Numerics.NET, including working with complex roots, least squares polynomial fitting, and polynomial arithmetic operations.

The sample shows how to:

Evaluate polynomials with complex arguments using ComplexValueAt and ComplexSlopeAt methods
Find both real and complex roots of polynomials using FindRoots and FindComplexRoots
Perform least squares polynomial fitting to approximate a set of data points
Execute basic arithmetic operations on polynomials including addition, subtraction, multiplication, and division
Use the Divide method to simultaneously obtain both quotient and remainder when dividing polynomials

Each operation is demonstrated with clear examples and the output is displayed to the console. The sample serves as a practical guide for working with polynomials in scientific and engineering applications where both real and complex analysis is required.

The code

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 Extreme Numerics.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()