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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' The DoubleComplex 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

            '
            ' DoubleComplex 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 DoubleComplex = New DoubleComplex(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 As Double() = 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 As DoubleComplex() = 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
            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