# NMath User's Guide

13.5 Function Interpolation (.NET, C#, CSharp, VB, Visual Basic, F#)

Abstract class TabulatedFunction extends OneVariableFunction (Section 13.1). Rather than encapsulating an arbitrary function delegate, TabulatedFunction holds paired vectors of known x- and y-values. The function can be evaluated at arbitrary points using derived function interpolation classes. As a OneVariableFunction, a TabulatedFunction can be manipulated algebraically. Numerical integrals and derivatives can also be computed.

A TabulatedFunction type is constructed from paired vectors of known x- and y-values. The values for x must be in strictly increasing order. Class TabulatedFunction inherits Function, Integrator, and Differentiator properties from OneVariableFunction (Section 13.1). Additionally, TabulatedFunction provides these properties:

X gets the vector of x-values represented by the function.

Y gets the vector of y-values represented by the function.

NumberOfTabulatedValues gets the number of tabulated values.

The X and Y properties return a copy of the tabulated data. Therefore, modifying the returned vectors does not change the TabulatedFunction.

To change the tabulated values represented by a TabulatedFunction, use the SetTabulatedValues() method. Provided GetX(), SetX(), GetY(), and SetY() methods also enable you to get and set individual tabulated values, or a range of values.

Class TabulatedFunction inherits the Evaluate() method from OneVariableFunction. This method evaluates the interpolated function at a given x-value, or vector of x-values.

Linear Spline Interpolation

Class LinearSpline extends TabulatedFunction and represents a function whose values are determined by linear interpolation between tabulated values. For example:

Code Example – C# linear spline interpolation

```var xValues = new DoubleVector(10, 0, 1);
DoubleVector yValues = xValues * xValues;
var ls = new LinearSpline( xValues, yValues );
double yInterpolated = ls.Evaluate( 3.5 );
```

Code Example – VB linear spline interpolation

```Dim XValues As New DoubleVector(10, 0, 1)
Dim YValues = XValues * XValues
Dim LS As New LinearSpline(XValues, YValues)
Dim YInterpolated = LS.Evaluate(3.5)
```

Evaluating x-values outside the range of tabulated values returns the last know y-value. In the example above, ls.Evaluate( 9.5 ) == ls.Evaluate( 9 ).

Cubic Spline Interpolation

Abstract class CubicSpline extends TabulatedFunction and represents a function whose values are determined by cubic spline interpolation between the tabulated values. NMath provides two concrete implementations of CubicSpline: NaturalCubicSpline and ClampedCubicSpline. The natural cubic spline is a cubic spline where the second derivative of the interpolating function is required to be zero at the left and right endpoints. The clamped cubic spline is a cubic spline where the first derivative of the interpolating function is specified at the left and right endpoints.

For example, this code creates a NaturalCubicSpline to resample at a fixed sampling interval a cubic spline fit constructed from data with a variable sampling interval:

Code Example – C# cubic spline interpolation

```var x = new DoubleVector( "1.0 1.3 1.4 1.8 2.0");
var y = new DoubleVector( "2.4 4.6 4.7 2.3 1.0" );

var s = new NaturalCubicSpline( x, y );

var xx = new DoubleVector( "1.0 1.25 1.5 1.75 2.0");
DoubleVector yy = s.Evaluate( xx );
```

Code Example – VB cubic spline interpolation

```Dim X As New DoubleVector("1.0 1.3 1.4 1.8 2.0")
Dim Y As New DoubleVector("2.4 4.6 4.7 2.3 1.0")

Dim S As New NaturalCubicSpline(X, Y)

Dim XX As New DoubleVector("1.0 1.25 1.5 1.75 2.0")
Dim YY As DoubleVector = S.Evaluate(XX)
```

This code creates a ClampedCubicSpline that enforces endslopes of zero for the cubic spline fit:

Code Example – C# cubic spline interpolation

```var s = new ClampedCubicSpline( x, y, 0, 0 );
```

Code Example – VB cubic spline interpolation

```Dim S As New ClampedCubicSpline(X, Y, 0, 0)
```

Class ClampedCubicSpline provides LeftEndSlope and RightEndSlope properties for getting and setting the clamped values, and method SetEndSlopes() for modifying them together.

Evaluating x-values outside the range of tabulated values in a NaturalCubicSpline returns the last know y-value. In a ClampedCubicSpline, the last fitted cubic is used, or a linear extrapolation is performed in the case of only 2 or 3 tabulated vaules.

Smooth Splines

Class derives from TabulatedFunction. The API is the same as for other cubic spline classes, with the addition of a smoothing factor, P. The smoothing factor takes values in the range 0 <= p <= 1, where 0 results in zero curvature (linear interpolation), and 1 results to a conventional cubic spline.

Creating Your Own Interpolation Classes

The NMath interpolation class framework is easily extensible. To create your own interpolation class, simply extend TabulatedFunction. Specify a delegate function of type Func<double, double> for the instance variable function in the base class OneVariableFunction. This delegate computes and returns values for arbitrary x-values.

In addition, deriving classes may override the virtual method ProcessTabulatedValues(). This method is invoked by TabulatedFunction instances whenever the tabulated values are changed.

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