Cubic Splines in Visual Basic QuickStart Sample

Illustrates using natural and clamped cubic splines for interpolation using classes in the Extreme.Mathematics.LinearAlgebra namespace in Visual Basic.

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Option Infer On

' The Constant and Line classes resides in the 
' Extreme.Mathematics.Curves namespace.
Imports Extreme.Mathematics.Curves

' Illustrates the use of the Constant class in the
' Extreme.Mathematics.Curve namespace of Extreme Numerics.NET.
Module CubicSplines

    Sub Main()
        ' The license is verified at runtime. We're using
        ' a demo license here. For more information, see
        Extreme.License.Verify("Demo license")

        ' A cubic spline is a piecewise curve that is made up
        ' of pieces of cubic polynomials. Its value as well as its first
        ' derivative are continuous, giving it a smooth appearance.
        ' Cubic splines are implemented by the CubicSpline class, 
        ' which inherits from PiecewiseCurve.
        ' For an example of piecewise constant and piecewise 
        ' linear curves, see the PiecewiseCurves QuickStart
        ' Sample.

        ' Creating Cubic Splines

        ' In order to define a spline curve completely, two extra
        ' conditions must be imposed.

        ' 'Natural' splines have zero second derivatives at the
        ' end points. This is the default.

        ' The data points are specified as double arrays containing
        ' the x and y values:
        Dim xValues As Double() = {1, 2, 3, 4, 5, 6}
        Dim yValues As Double() = {1, 3, 4, 3, 4, 2}
        Dim naturalSpline As New CubicSpline(xValues, yValues)

        ' 'Clamped' splines have a fixed slope or first derivative at the 
        ' leftmost and rightmost points. The slopes are specified as
        ' two extra parameters in the constructor:
        Dim clampedSpline As New CubicSpline(xValues, yValues, -1, 1)

        ' 'Akima' splines minimize the oscillations in the interpolating
        ' curve. The constructor takes an extra argument of type
        ' CubicSplineKind:
        Dim akimaSpline As New CubicSpline(xValues, yValues, CubicSplineKind.Akima)
        ' The factory method does not require the extra parameter:
        akimaSpline = CubicSpline.CreateAkima(xValues, yValues)

        ' Hermite splines have fixed values for the first derivative at each
        ' data point. The first derivatives must be supplied as an array
        ' or list:
        Dim yPrimeValues As Double() = {0, 1, -1, 1, 0, -1}
        Dim hermiteSpline As CubicSpline = New CubicSpline(xValues, yValues, yPrimeValues)
        ' Likewise for the factory method:
        hermiteSpline = CubicSpline.CreateHermiteInterpolant(xValues, yValues, yPrimeValues)

        ' Curve Parameters

        ' The shape of any curve is determined by a set of parameters.
        ' These parameters can be retrieved and set through the
        ' Parameters collection. The number of parameters for a curve 
        ' is given by this collection's Count property.
        ' Cubic splines have 2n+2 parameters, where n is the number of
        ' data points. The first n parameters are the x-values. The next
        ' n parameters are the y-values. The last two parameters are
        ' the values of the derivative at the first and last point. For natural
        ' splines, these parameters are unused.

        Console.WriteLine("naturalSpline.Parameters.Count = {0}",
        ' Parameters can easily be retrieved:
        Console.WriteLine("naturalSpline.Parameters(0) = {0}",
        ' Parameters can also be set:
        naturalSpline.Parameters(0) = 1

        ' Piecewise curve methods and properties

        ' The NumberOfIntervals property returns the number of subintervals
        ' on which the curve has unique definitions.
        Console.WriteLine("Number of intervals: {0}",

        ' The IndexOf method returns the index of the interval
        ' that contains a specific value.
        Console.WriteLine("naturalSpline.IndexOf(1.4) = {0}", naturalSpline.IndexOf(1.4))
        ' The method returns -1 when the value is smaller than the lower bound
        ' of the first interval, and NumberOfIntervals if the value is equal to or larger than
        ' the upper bound of the last interval.

        ' Curve Methods

        ' The ValueAt method returns the y value of the
        ' curve at the specified x value:
        Console.WriteLine("naturalSpline.ValueAt(2.4) = {0}", naturalSpline.ValueAt(2.4))

        ' The SlopeAt method returns the slope of the curve
        ' a the specified x value:
        Console.WriteLine("naturalSpline.SlopeAt(2) = {0}", naturalSpline.SlopeAt(2))
        ' You can verify that the clamped spline has the correct slope at the end points:
        Console.WriteLine("clampedSpline.SlopeAt(1) = {0}", clampedSpline.SlopeAt(1))
        Console.WriteLine("clampedSpline.SlopeAt(6) = {0}", clampedSpline.SlopeAt(6))

        ' Cubic splines do not have a defined derivative. The GetDerivative method
        ' returns a GeneralCurve:
        Dim derivative = naturalSpline.GetDerivative()
        Console.WriteLine("Type of derivative: {0}", derivative.GetType().ToString())
        Console.WriteLine("derivative(2) = {0}", derivative.ValueAt(2))

        ' You can get a Line that is the tangent to a curve
        ' at a specified x value using the TangentAt method:
        Dim tangent = clampedSpline.TangentAt(2)
        Console.WriteLine("Slope of tangent line at 2 = {0}", tangent.Parameters(1))

        ' The integral of a spline curve can be calculated exactly. This technique is
        ' often used to approximate the integral of a tabulated function:
        Console.WriteLine("Integral of naturalSpline between 1.4 and 4.6 = {0}",
                naturalSpline.Integral(1.4, 4.6))

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

    End Sub

End Module