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Polynomials

Extreme Optimization Numerical Libraries for .NET Professional

A polynomial is a linear function that is a combination of the argument raised to different integer powers. Polynomials are among the most common mathematical objects. The Extreme Optimization Numerical Libraries for .NET has extensive support for common polynomials and Chebyshev polynomials.

The PolynomialBase class

All polynomial classes are derived from PolynomialBase This class defines a number of properties shared by all polynomial classes. PolynomialBase is itself derived from LinearCombination.

The Parameters of a polynomial are the coefficients of the polynomial.

The Degree of a polynomial is the highest power that appears in the sum. The number of parameters of the polynomial equals the degree plus one.

General polynomials are implemented by the Polynomial class.

Constructing polynomials

The Polynomial class has three constructors: The first constructor takes the degree of the polynomial as its only argument. All coefficients are initially set to zero. The second constructor takes an array of Double values containing the coefficients. The third constructor takes a single VectorT argument. This vector should contain the coefficients of the polynomial, starting with the constant term. The following example illustrates three different ways of constructing the polynomial

x³ - 5x² + 9x + 2

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Polynomial p1 = new Polynomial(3);
p1[3] = 1;
p1[2] = -5;
p1[1] = 9;
p1[0] = 2;
var coefficients = Vector.Create(2.0, 9.0, -5.0, 1.0);
Polynomial p2 = new Polynomial(coefficients);
Polynomial p3 = new Polynomial(new[] { 1.0, -5.0, 9.0, 2.0 }, true);

Dim p1 As Polynomial = New Polynomial(3)
p1(3) = 1
p1(2) = -5
p1(1) = 9
p1(0) = 2
Dim coefficients = Vector.Create(2.0, 9.0, -5.0, 1.0)
Dim p2 As Polynomial = New Polynomial(coefficients)
Dim p3 As Polynomial = New Polynomial({1.0, -5.0, 9.0, 2.0}, True)

let p1 = Polynomial(3)
p1.[3] <- 1.0
p1.[2] <- -5.0
p1.[1] <- 9.0
p1.[0] <- 2.0
let coefficients = Vector.Create(2.0, 9.0, -5.0, 1.0)
let p2 = Polynomial(coefficients)
let p3 = Polynomial([| 1.0; -5.0; 9.0; 2.0 |], true)

Notice that the coefficients in the constructors for p2 and p3 are listed in order of degree. The nth element of the array or VectorT is the coefficient of degree n.

You can also create a polynomial using one of several static methods.

The GetInterpolatingPolynomial method returns the polynomial that interpolates a set of points. The x-coordinates of the points must be distinct. The x and y-coordinates must be given as two separate Double arrays. The following creates the interpolating polynomial through a set of four points:

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double[] xValues = new double[] { 1, 2, 3, 4 };
double[] yValues = new double[] { 1, 4, 10, 8 };
Polynomial p4;
p4 = Polynomial.GetInterpolatingPolynomial(xValues, yValues);

Dim xValues = New Double() {1, 2, 3, 4}
Dim yValues = New Double() {1, 4, 10, 8}
Dim p4 = Polynomial.GetInterpolatingPolynomial(xValues, yValues)

let xValues = [| 1.0; 2.0; 3.0; 4.0 |]
let yValues = [| 1.0; 4.0; 10.0; 8.0 |]
let p4 = Polynomial.GetInterpolatingPolynomial(xValues, yValues)

The FromRoots method constructs a Polynomial whose roots are given by the arguments passed to the method. The roots are provided as a Double array. They can be in any order. The same root can appear multiple times. The following constructs a polynomial with roots 1, 2 (twice), and 4:

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double[] roots = new double[] { 1, 2, 2, 4 };
Polynomial p5 = Polynomial.FromRoots(roots);

Dim roots = New Double() {1, 2, 2, 4}
Dim p5 As Polynomial = Polynomial.FromRoots(roots)

let roots = [| 1.0; 2.0; 2.0; 4.0 |]
let p5 = Polynomial.FromRoots(roots)

The LeastSquaresFit method constructs a Polynomial that is the least squares fit of a specified degree through a given set of points.

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double[] xValues = new double[7];
double[] yValues = new double[7];
double angle = 0;
for (int index = 0; index < 7; index++)
{
    xValues[index] = Math.Cos(angle);
    yValues[index] = -Math.Sin(angle);
    angle = angle + Constants.Pi / 6;
}
Polynomial p6 = Polynomial.LeastSquaresFit(xValues, yValues, 4);

Dim xValues(6) As Double
Dim yValues(6) As Double
Dim angle As Double = 0
For index As Integer = 0 To 6
    xValues(index) = Math.Cos(angle)
    yValues(index) = -Math.Sin(angle)
    angle = angle + Constants.Pi / 6
Next

Dim p6 As Polynomial = Polynomial.LeastSquaresFit(xValues, yValues, 4)

let xValues = Vector.Create(7,
                fun i -> cos((float i) * Constants.Pi / 6.0))
let yValues = Vector.Create(7,
                fun i -> cos((float i) * Constants.Pi / 6.0))
let p6 = Polynomial.LeastSquaresFit(xValues, yValues, 4)

Working with polynomials

Polynomial implements all methods and properties of the Curve class, and has some extras.

The ValueAt method returns the value of the polynomial at a specified point. SlopeAt returns the derivative. Separate methods, ComplexValueAt and ComplexSlopeAt, take ComplexT arguments and return a complex number.

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Console.WriteLine("p1.ValueAt(2) = {0}", p1.ValueAt(2));
Console.WriteLine("p1.SlopeAt(2) = {0}", p1.SlopeAt(2));
Complex<double> z = new Complex<double>(1, -2);
Console.WriteLine("p1.ValueAt(1-2i) = {0}", p1.ComplexValueAt(z));

Console.WriteLine("p1.ValueAt(2) = {0}", p1.ValueAt(2))
Console.WriteLine("p1.SlopeAt(2) = {0}", p1.SlopeAt(2))
Dim z = New Complex(Of Double)(1, -2)
Console.WriteLine("p1.ValueAt(1-2i) = {0}", p1.ComplexValueAt(z))

Console.WriteLine("p1.ValueAt(2) = {0}", p1.ValueAt(2.0))
Console.WriteLine("p1.SlopeAt(2) = {0}", p1.SlopeAt(2.0))
let z = Complex<double>(1.0, -2.0)
Console.WriteLine("p1.ValueAt(1-2i) = {0}", p1.ComplexValueAt(z))

The GetDerivative method returns the polynomial that is the derivative of a polynomial. If the derivative is a constant, a line, or a quadratic curve, an instance of the corresponding type is returned.

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Curve derivative = p1.GetDerivative();
Console.WriteLine("Slope at 2 (derivative) = {0}",
    derivative.ValueAt(2));
Console.WriteLine("Type of derivative: {0}",
    derivative.GetType().FullName);

Dim derivative As Curve = p1.GetDerivative()
Console.WriteLine("Slope at 2 (derivative) = {0}",
    derivative.ValueAt(2))
Console.WriteLine("Type of derivative: {0}",
    derivative.GetType().FullName)

let derivative = p1.GetDerivative()
Console.WriteLine("Slope at 2 (derivative) = {0}",
    derivative.ValueAt(2.0))
Console.WriteLine("Type of derivative: {0}",
    derivative.GetType().FullName)

Integral evaluates the definite integral over a specified interval. The integral is calculated directly using the coefficients of the polynomial. No numerical approximations are used.

The parameters of a polynomial correspond to the coefficients of the polynomial. To make it easier to work with polynomial coefficients, an indexer property has been defined, which can be used to get and set individual coefficients directly. In Visual Basic .NET and other languages that don't support indexers, the indexed property Coefficient should be used instead.

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// We create the polynomial x^3 + x^2 - x - 2:
Polynomial p = new Polynomial(new[] { 1.0, 1.0, -1.0, -2.0 }, true);
// The coefficient of x^1=x is '-1':
Console.WriteLine(p[1]);
// We can change the value of this coefficient.
p[1] = 0;

' We create the polynomial x^3 + x^2 - x - 2:
Dim p As Polynomial = New Polynomial({1, 1, -1, -2}, True)
' The coefficient of x^1=x is '-1':
Console.WriteLine(p(1))
' We can change the value of this coefficient.
p(1) = 0

// We create the polynomial x^3 + x^2 - x - 2:
let p = Polynomial([| 1.0; 1.0; -1.0; -2.0 |], true)
// The coefficient of x^1=x is '-1':
Console.WriteLine(p.[1])
// We can change the value of this coefficient.
p.[1] <- 0.0

Operations on polynomials

The result of adding or subtracting two polynomials is a new polynomial. Similarly, the result of multiplying two polynomials is a new polynomial.

The Polynomial class implements these operations through overloaded operators. Static methods are available for languages that don't support operator overloading. The following table summarizes these operations:

Polynomial operators and their static (Shared) method equivalents

Operator	Static method equivalent	Description
+p	(no equivalent)	Returns the polynomial p.
-p	Negate	Returns the negation of the polynomial p.
p1 + p2	Add	Adds the polynomials p1 and p2.
p1 - p2	Subtract	Subtracts the polynomials p1 and p2.
p1 * p2	Multiply	Multiplies the polynomials p1 and p2.
a * p	Multiply	Multiplies the polynomial p by the real number a.
p1 / p2	Divide	Divides the polynomial p1 by p2.
p / a	Divide	Divides the polynomial p by the real number a.
p1 % p2	Modulus	Returns the remainder when dividing the polynomial p1 by p2.

The following example shows how to use these operators:

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Polynomial a = new Polynomial(new[] { 2.0, -2.0, 4.0 }, true);
Polynomial b = new Polynomial(new[] { 1.0, -3.0 }, true);
Polynomial c;
Console.WriteLine("a = {0}", a.ToString());
Console.WriteLine("b = {0}", b.ToString());
c = a + b;
Console.WriteLine("a + b = {0}", c.ToString());
c = a - b;
Console.WriteLine("a - b = {0}", c.ToString());
c = a * b;
Console.WriteLine("a * b = {0}", c.ToString());
c = a / b;
Console.WriteLine("a / b = {0}", c.ToString());
c = a % b;
Console.WriteLine("a % b = {0}", c.ToString());

Dim a As Polynomial = New Polynomial({2, -2, 4}, True)
Dim b As Polynomial = New Polynomial({1, -3}, True)
Dim c As Polynomial
Console.WriteLine("a = {0}", a.ToString())
Console.WriteLine("b = {0}", b.ToString())
c = a + b
Console.WriteLine("a + b = {0}", c.ToString())
c = a - b
Console.WriteLine("a - b = {0}", c.ToString())
c = a * b
Console.WriteLine("a * b = {0}", c.ToString())
c = a / b
Console.WriteLine("a / b = {0}", c.ToString())
c = a Mod b
Console.WriteLine("a % b = {0}", c.ToString())

let a = Polynomial([| 2.0; -2.0; 4.0 |], true)
let b = Polynomial([| 1.0; -3.0 |], true)
let mutable c : Polynomial = a
Console.WriteLine("a = {0}", a.ToString())
Console.WriteLine("b = {0}", b.ToString())
c <- a + b
Console.WriteLine("a + b = {0}", c.ToString())
c <- a - b
Console.WriteLine("a - b = {0}", c.ToString())
c <- a * b
Console.WriteLine("a * b = {0}", c.ToString())
c <- a / b
Console.WriteLine("a / b = {0}", c.ToString())
c <- a % b
Console.WriteLine("a % b = {0}", c.ToString())

As with integers, division of polynomials is not always exact. The division operator returns a polynomial such that the degree of the remainder of the division is less than the degree of the divisor.

The Divide method has an Divide that allows you to obtain both the quotient and remainder at the same time. This method takes a third out parameter that will hold the remainder.

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Polynomial d;
c = Polynomial.Divide(a, b, out d);
Console.WriteLine("Using Divide method:");
Console.WriteLine("  a / b = {0}", c.ToString());
Console.WriteLine("  a % b = {0}", d.ToString());

Dim d As Polynomial
c = Polynomial.Divide(a, b, d)
Console.WriteLine("Imports  Divide method:")
Console.WriteLine("  a / b = {0}", c.ToString())
Console.WriteLine("  a % b = {0}", d.ToString())

let (c, d) = Polynomial.Divide(a, b)
Console.WriteLine("Using Divide method:")
Console.WriteLine("  a / b = {0}", c.ToString())
Console.WriteLine("  a % b = {0}", d.ToString())

Note also that Constant, Line, and Quadratic objects can be used in polynomial expressions, as they inherit from Polynomial.

Finding roots of polynomials

A polynomial of degree n has at most n real roots, some of which may occur several times. The FindRoots method attempts to find all the real roots of a polynomial. For example, the polynomial x³ + x² - 2 has one real root at x = 1:

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Polynomial polynomial1 = new Polynomial(new[] { -2.0, 0.0, 1.0, 1.0 });
double[] roots = polynomial1.FindRoots();
Console.WriteLine("Number of roots of p: {0}", roots.Length);
Console.WriteLine("Value of root 1 = {0}", roots[0]);

Dim polynomial1 As Polynomial = New Polynomial({1, 1, 0, -2}, True)
Dim roots = polynomial1.FindRoots()
Console.WriteLine("Number of roots of p: {0}", roots.Length)
Console.WriteLine("Value of Dim 1 As root = {0}", roots(0))

let polynomial1 = Polynomial([| 1.0; 1.0; 0.0; -2.0 |], true)
let roots = polynomial1.FindRoots()
Console.WriteLine("Number of roots of p: {0}", roots.Length)
Console.WriteLine("Value of root 1 = {0}", roots.[0])

In the complex domain, a polynomial of degree n always has n roots. The Polynomial class defines the FindComplexRoots method, which returns an array of ComplexT numbers that are the approximate roots of the polynomial.

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Polynomial polynomial2 = new Polynomial(new[] { -2.0, 0.0, 1.0, 1.0 });
Complex<double>[] roots = polynomial2.FindComplexRoots();
Console.WriteLine("Number of roots of p: {0}", roots.Length);
Console.WriteLine("Value of root 1 = {0}", roots[0]);
Console.WriteLine("Value of root 2 = {0}", roots[1]);
Console.WriteLine("Value of root 3 = {0}", roots[2]);

Dim polynomial2 As Polynomial = New Polynomial({1, 1, 0, -2}, True)
Dim roots() As Complex(Of Double) = polynomial2.FindComplexRoots()
Console.WriteLine("Number of roots of p: {0}", roots.Length)
Console.WriteLine("Value of root 1 = {0}", roots(0))
Console.WriteLine("Value of root 2 = {0}", roots(1))
Console.WriteLine("Value of root 3 = {0}", roots(2))

let polynomial2 = Polynomial([| 1.0; 1.0; -1.0; -2.0 |], true)
let roots = polynomial2.FindComplexRoots();
Console.WriteLine("Number of roots of p: {0}", roots.Length);
Console.WriteLine("Value of root 1 = {0}", roots.[0]);
Console.WriteLine("Value of root 2 = {0}", roots.[1]);
Console.WriteLine("Value of root 3 = {0}", roots.[2]);

The roots are returned in order of increasing real part and, if the real parts are equal, increasing imaginary part.

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