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QuickStart Samples

Advanced Polynomials QuickStart Sample (Visual Basic)

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

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Option Infer On

' The Complex(Of Double) structure resides in the Extreme.Mathematics namespace.
Imports Extreme.Mathematics
' The Polynomial class resides in the Extreme.Mathematics.Curves namespace.
Imports Extreme.Mathematics.Curves

Namespace Extreme.Numerics.QuickStart.VB

    Module AdvancedPolynomials

        ' Illustrates the more advanced uses of the Polynomial class 
        ' in the Extreme.Mathematics.Curve namespace of the Extreme Optimization 
        ' Numerical Libraries for .NET.
        Sub Main()
            ' 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.

            ' Index variable.
            Dim index As Int32

            '
            ' Complex(Of Double) numbers and polynomials
            '

            Dim myPolynomial As Polynomial = New Polynomial(New Double() {-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.
            Dim z1 As New Complex(Of Double)(1, 2)

            ' Polynomial provides variants of ValueAt and
            ' SlopeAt for complex arguments:
            Console.WriteLine("polynomial.ComplexValueAt({0}) = {1}", _
                z1, myPolynomial.ComplexValueAt(z1))
            Console.WriteLine("polynomial.ComplexSlopeAt({0}) = {1}", _
                z1, myPolynomial.ComplexSlopeAt(z1))

            '
            ' Real and complex roots
            '
            ' Our polynomial has only one real root:
            Dim roots = myPolynomial.FindRoots()
            Console.WriteLine("Number of roots of polynomial1: {0}", _
                roots.Length)
            Console.WriteLine("Value of root 1 = {0}", roots(0))
            ' The FindComplexRoots method returns all three
            ' roots, two of which are complex:
            Dim complexRoots = myPolynomial.FindComplexRoots()
            Console.WriteLine("Number of complex roots: {0}", _
                complexRoots.Length)
            Console.WriteLine("Value of root 1 = {0}", _
                complexRoots(0))
            Console.WriteLine("Value of root 2 = {0}", _
                complexRoots(1))
            Console.WriteLine("Value of root 3 = {0}", _
                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:
            Dim xValues As Double() = New Double(6) {}
            Dim yValues As Double() = New Double(6) {}
            Dim angle As Double = 0
            For index = 0 To 6
                xValues(index) = Math.Cos(angle)
                yValues(index) = -Math.Sin(angle)
                angle = angle + Extreme.Mathematics.Constants.Pi / 6
            Next
            ' Now we can find the least squares polynomial
            ' by calling the ststic LeastSquaresFit method.
            ' The last parameter is the degree of the desired
            ' polynomial.
            Dim lsqPolynomial As Polynomial = _
                polynomial.LeastSquaresFit(xValues, yValues, 4)
            ' Note that, as expected, the odd coefficients
            ' are close to zero.
            Console.WriteLine("Least squares fit: {0}", _
                lsqPolynomial.ToString())

            '
            ' Polynomial arithmetic
            '
            ' We can add, subtract, multiply and divide
            ' polynomials using overloaded operators:
            Dim a As polynomial = New polynomial(New Double() {4, -2, 2})
            Dim b As polynomial = New polynomial(New Double() {-3, 1})
            Dim c As Polynomial

            Console.WriteLine("a = {0}", a.ToString())
            Console.WriteLine("b = {0}", b.ToString())
            c = polynomial.Add(a, b)
            Console.WriteLine("Add(a, b) = {0}", c.ToString())
            c = polynomial.Subtract(a, b)
            Console.WriteLine("Subtract(a, b) = {0}", c.ToString())
            c = polynomial.Multiply(a, b)
            Console.WriteLine("Multiply(a, b) = {0}", c.ToString())
            c = polynomial.Divide(a, b)
            Console.WriteLine("Divide(a, b) = {0}", c.ToString())
            c = polynomial.Modulus(a, b)
            Console.WriteLine("Remainder(a, b) = {0}", c.ToString())
            ' You can also calculate quotient and remainder
            ' at the same time by calling the overloaded Divide
            ' method:
            Dim d As Polynomial = Nothing
            c = Polynomial.Divide(a, b, d)
            Console.WriteLine("Using Divide method:")
            Console.WriteLine("  a / b = {0}", c.ToString())
            Console.WriteLine("  a % b = {0}", d.ToString())

            Console.Write("Press Enter key to exit...")
            Console.ReadLine()

        End Sub

    End Module

End Namespace