# VB Polynomial Example

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```Imports System

Imports CenterSpace.NMath.Core

Namespace CenterSpace.NMath.Examples.VisualBasic

A .NET example in Visual Basic showing how to create and manipulate polynomial objects.
Module PolynomialExample

Sub Main()

Console.WriteLine()

Module Polynomial represents a polynomial by its coefficients, arranged in
ascending orderâ€”-that is, a vector of coefficients a0, a1, ... an such that
f(x) = a0*x^0 + a1*X^1 + ... + an*x^n.

A Polynomial instance can be constructed in two ways. If you know the
exact form of the polynomial, simply pass in a vector of coefficients.
Dim Coef As New DoubleVector("3 1 -2 0 5")
Dim F As New Polynomial(Coef)

f(x) = 5x^4 - 2x^2 + x + 3
Console.WriteLine( "f(x) = {0}", f.ToString() )
Console.WriteLine( "Degree = {0}", f.Degree )
Console.WriteLine("Coefficients = {0}" & Environment.NewLine, F.Coeff)

You can also interpolate a polynomial through a set of points. If the
number of points is n, then the constructed polynomial will have degree
n - 1 and pass through the interpolation points. For example, this code
interpolates a polynomial through the points (1,6), (2,11), and (3,20):
Dim X As New DoubleVector("1 2 3")
Dim Y As New DoubleVector("6 11 20")
Dim G As New Polynomial(X, Y)

g(x) = 2x^2 - x + 5
Console.WriteLine( "g(x) = {0}", g.ToString( "G3" ))
Console.WriteLine( "Degree = {0}", g.Degree )
Console.WriteLine("Coefficients = {0}" & Environment.NewLine, G.Coeff)

The Evaluate() method evaluates a polynomial at a given x-value, or
vector of x-values. This code evaluates f at ten points between 0 and 1:
x = new DoubleVector( 10, 0.1, 1.0/10 )
y = f.Evaluate( x )
Console.WriteLine( "x = {0}", x )
Console.WriteLine("y = {0}" & Environment.NewLine, Y)

Module Polynomial provides overloads of the arithmetic operators (and
equivalent named methods) that work with either with two polynomials, or
with a polynomial and a scalar. For example:
Dim H As Polynomial = (F + G) * G / 2
Console.WriteLine( "h(x) = {0}", h.ToString() )
Console.WriteLine( "h(2) = {0}", h.Evaluate( 2 ) )
Console.WriteLine()

The Integrate() method computes the integral of a polynomial over a given
interval.
Console.WriteLine( "Integral of h(x) over 0 to 1 = {0}", h.Integrate( 0, 1 ).ToString( "G3" ))
Console.WriteLine()

The AntiDerivative() method returns a new polynomial encapsulating
the antiderivative (indefinite integral) of the current polynomial.
The constant of integration is assumed to be zero.
Dim hAntiDeriv As Polynomial = H.AntiDerivative()
Console.WriteLine( "Antiderivative of h(x)..." )
Console.WriteLine( hAntiDeriv.ToString( "G3" ))
Console.WriteLine()

The Differentiate() method computes the derivative of a polynomial at a
given x-value.
Console.WriteLine( "Derivative of h(x) at 1 = {0}", h.Differentiate( 1 ).ToString( "G3" ))
Console.WriteLine()

Derivative() returns a new polynomial that is the first derivative of
the current polynomial.
Console.WriteLine( "First derivative of h(x)..." )
Console.WriteLine( h.Derivative().ToString( "G3" ))
Console.WriteLine()

Console.WriteLine( "Second derivative of h(x)..." )
Console.WriteLine( h.Derivative().Derivative().ToString( "G3" ))
Console.WriteLine()

Check that the derivative of the antiderivative of h(x) == h(x)
Console.WriteLine( "Derivative of the antiderivative of h(x)..." )
Console.WriteLine( h.AntiDerivative().Derivative().ToString( "G3" ))

Console.WriteLine()
Console.WriteLine( "Press Enter Key" )
Console.Read()

End Sub
End Module
End Namespace

```
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