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# 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.

```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

' 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
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.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...")