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Advanced Polynomials QuickStart Sample (F#)
Illustrates more advanced uses of the Polynomial class, including real and complex root finding, calculating least squares polynomials and polynomial arithmetic in F#.
Visual Basic code
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// Illustrates the more advanced uses of the Polynomial class
// in the Extreme.Mathematics.Curve namespace of the Extreme Optimization
// Mathematics Library for .NET.
// The Complex<float> structure resides in the Extreme.Mathematics namespace.
// The Polynomial class resides in the Extreme.Mathematics.Curves namespace.
// 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.
// Complex<float> numbers and polynomials
let polynomial = Polynomial([| -2.0; 0.0; 1.0; 1.0 |])
// 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.
let z1 = Complex<float>(1.0, 2.0)
// Polynomial provides variants of ValueAt and
// SlopeAt for complex arguments:
printfn "polynomial.ComplexValueAt(%A) = %A" z1 (polynomial.ComplexValueAt(z1))
printfn "polynomial.ComplexSlopeAt(%A) = %A" z1 (polynomial.ComplexSlopeAt(z1))
// Real and complex roots
// Our polynomial has only one real root:
let roots = polynomial.FindRoots()
printfn "Number of roots of polynomial1: %d" roots.Length
printfn "Value of root 1 = %A" roots.
// The FindComplexRoots method returns all three
// roots, two of which are complex:
let complexRoots = polynomial.FindComplexRoots()
printfn "Number of complex roots: %d" complexRoots.Length
printfn "Value of root 1 = %A" complexRoots.
printfn "Value of root 2 = %A" complexRoots.
printfn "Value of root 3 = %A" complexRoots.
// Least squares polynomials
// Let's approximate 7 points on the unit circle
// by a fourth degree polynomial in the least squares
// First, we create two arrays containing the x and
// y values of our data points:
let xValues = Vector.Create([| for i in 0..6 -> Math.Cos(float i * Constants.Pi / 6.0) |])
let yValues = Vector.Create([| for i in 0..6 -> -Math.Sin(float i * Constants.Pi / 6.0) |])
// Now we can find the least squares polynomial
// by calling the ststic LeastSquaresFit method.
// The last parameter is the degree of the desired
let lsqPolynomial = Polynomial.LeastSquaresFit(xValues, yValues, 4)
// Note that, as expected, the odd coefficients
// are close to zero.
printfn "Least squares fit: %O" lsqPolynomial
// Polynomial arithmetic
// We can add, subtract, multiply and divide
// polynomials using overloaded operators:
let a = Polynomial([| 4.0; -2.0; 4.0 |])
let b = Polynomial([| -3.0; 1.0 |])
printfn "a = %O" a
printfn "b = %O" b
printfn "a + b = %O" (a + b)
printfn "a - b = %O" (a - b)
printfn "a * b = %O" (a * b)
printfn "a / b = %O" (a / b)
printfn "a %% b = %O" (a % b)
// You can also calculate quotient and remainder
// at the same time by calling the overloaded Divide
let (c,d) = Polynomial.Divide(a, b)
printfn "Using Divide method:"
printfn " a / b = %O" c
printfn " a %% b = %O" d
printf "Press Enter key to exit..."
Console.ReadLine() |> ignore
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