/// <summary>
/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
/// </summary>
public static CubicSpline InterpolateBoundaries(IEnumerable <double> x, IEnumerable <double> y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
// note: we must make a copy, even if the input was arrays already
return(InterpolateBoundariesInplace(x.ToArray(), y.ToArray(), leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary));
}
/// <summary>
/// Initialize the interpolation method with the given spline coefficients (sorted by the sample points t).
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
public void Initialize(
IList <double> samplePoints, IList <double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
double[] derivatives = EvaluateSplineDerivatives(
samplePoints, sampleValues,
leftBoundaryCondition, leftBoundary,
rightBoundaryCondition, rightBoundary);
_spline.Initialize(samplePoints, sampleValues, derivatives);
}
/// <summary>
/// Evaluate the spline coefficients as used
/// internally by this interpolation algorithm.
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
/// <returns>Spline Coefficient Vector</returns>
public static double[] EvaluateSplineCoefficients(
IList <double> samplePoints, IList <double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
double[] derivatives = EvaluateSplineDerivatives(
samplePoints, sampleValues,
leftBoundaryCondition, leftBoundary,
rightBoundaryCondition, rightBoundary);
return(CubicHermiteSplineInterpolation.EvaluateSplineCoefficients(samplePoints, sampleValues, derivatives));
}
/// <summary>
/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
/// WARNING: Works in-place and can thus causes the data array to be reordered.
/// </summary>
public static CubicSpline InterpolateBoundariesInplace(double[] x, double[] y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
if (x.Length != y.Length)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
Sorting.Sort(x, y);
return(InterpolateBoundariesSorted(x, y, leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary));
}
/// <summary>
/// Initializes a new instance of the CubicSplineInterpolation class.
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
public CubicSplineInterpolation(
IList <double> samplePoints, IList <double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
_spline = new CubicHermiteSplineInterpolation();
Initialize(
samplePoints, sampleValues,
leftBoundaryCondition, leftBoundary,
rightBoundaryCondition, rightBoundary);
}
/// <summary>
/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
/// WARNING: Works in-place and can thus causes the data array to be reordered.
/// </summary>
public static CubicSpline InterpolateBoundariesInplace(double[] x, double[] y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
if (x.Length != y.Length)
{
throw new ArgumentException("All vectors must have the same dimensionality.");
}
Sorting.Sort(x, y);
return(InterpolateBoundariesSorted(x, y, leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary));
}
/// <summary>
/// Initializes a new instance of the CubicSplineInterpolation class.
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
public CubicSplineInterpolation(
IList<double> samplePoints,
IList<double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary)
{
_spline = new CubicHermiteSplineInterpolation();
Initialize(
samplePoints,
sampleValues,
leftBoundaryCondition,
leftBoundary,
rightBoundaryCondition,
rightBoundary);
}
CreateCubicSpline(
IList <double> points,
IList <double> values,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary)
{
CubicSplineInterpolation method = new CubicSplineInterpolation();
method.Init(
points,
values,
leftBoundaryCondition,
leftBoundary,
rightBoundaryCondition,
rightBoundary);
return(method);
}
/// <summary>
/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
/// </summary>
public static CubicSpline InterpolateBoundaries(IEnumerable<double> x, IEnumerable<double> y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
// note: we must make a copy, even if the input was arrays already
return InterpolateBoundariesInplace(x.ToArray(), y.ToArray(), leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary);
}
/// <summary>
/// Create a cubic spline interpolation from an unsorted set of (x,y) value pairs and custom boundary/termination conditions.
/// WARNING: Works in-place and can thus causes the data array to be reordered.
/// </summary>
public static CubicSpline InterpolateBoundariesInplace(double[] x, double[] y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
if (x.Length != y.Length)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
Sorting.Sort(x, y);
return InterpolateBoundariesSorted(x, y, leftBoundaryCondition, leftBoundary, rightBoundaryCondition, rightBoundary);
}
/// <summary>
/// Create a cubic spline interpolation from a set of (x,y) value pairs, sorted ascendingly by x,
/// and custom boundary/termination conditions.
/// </summary>
public static CubicSpline InterpolateBoundariesSorted(double[] x, double[] y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
if (x.Length != y.Length)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (x.Length < 2)
{
throw new ArgumentException(string.Format(Resources.ArrayTooSmall, 2), "x");
}
int n = x.Length;
// normalize special cases
if ((n == 2)
&& (leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated)
&& (rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
var a1 = new double[n];
var a2 = new double[n];
var a3 = new double[n];
var b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2*(y[1] - y[0])/(x[1] - x[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = (3*((y[1] - y[0])/(x[1] - x[0]))) - (0.5*leftBoundary*(x[1] - x[0]));
break;
default:
throw new NotSupportedException(Resources.InvalidLeftBoundaryCondition);
}
// Central Conditions
for (int i = 1; i < x.Length - 1; i++)
{
a1[i] = x[i + 1] - x[i];
a2[i] = 2*(x[i + 1] - x[i - 1]);
a3[i] = x[i] - x[i - 1];
b[i] = (3*(y[i] - y[i - 1])/(x[i] - x[i - 1])*(x[i + 1] - x[i])) + (3*(y[i + 1] - y[i])/(x[i + 1] - x[i])*(x[i] - x[i - 1]));
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2*(y[n - 1] - y[n - 2])/(x[n - 1] - x[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = (3*(y[n - 1] - y[n - 2])/(x[n - 1] - x[n - 2])) + (0.5*rightBoundary*(x[n - 1] - x[n - 2]));
break;
default:
throw new NotSupportedException(Resources.InvalidRightBoundaryCondition);
}
// Build Spline
double[] dd = SolveTridiagonal(a1, a2, a3, b);
return InterpolateHermiteSorted(x, y, dd);
}
/// <summary>
/// Evaluate the spline derivatives as used
/// internally by this interpolation algorithm.
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
/// <returns>Spline Derivative Vector</returns>
public static double[] EvaluateSplineDerivatives(
IList<double> samplePoints,
IList<double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary)
{
if (null == samplePoints)
{
throw new ArgumentNullException("samplePoints");
}
if (null == sampleValues)
{
throw new ArgumentNullException("sampleValues");
}
if (samplePoints.Count < 2)
{
throw new ArgumentOutOfRangeException("samplePoints");
}
if (samplePoints.Count != sampleValues.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
int n = samplePoints.Count;
// normalize special cases
if ((n == 2)
&& (leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated)
&& (rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
double[] a1 = new double[n];
double[] a2 = new double[n];
double[] a3 = new double[n];
double[] b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2 * (sampleValues[1] - sampleValues[0]) / (samplePoints[1] - samplePoints[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = (3 * ((sampleValues[1] - sampleValues[0]) / (samplePoints[1] - samplePoints[0]))) - (0.5 * leftBoundary * (samplePoints[1] - samplePoints[0]));
break;
default:
throw new NotSupportedException(Resources.InvalidLeftBoundaryCondition);
}
// Central Conditions
for (int i = 1; i < samplePoints.Count - 1; i++)
{
a1[i] = samplePoints[i + 1] - samplePoints[i];
a2[i] = 2 * (samplePoints[i + 1] - samplePoints[i - 1]);
a3[i] = samplePoints[i] - samplePoints[i - 1];
b[i] = (3 * (sampleValues[i] - sampleValues[i - 1]) / (samplePoints[i] - samplePoints[i - 1]) * (samplePoints[i + 1] - samplePoints[i])) + (3 * (sampleValues[i + 1] - sampleValues[i]) / (samplePoints[i + 1] - samplePoints[i]) * (samplePoints[i] - samplePoints[i - 1]));
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2 * (sampleValues[n - 1] - sampleValues[n - 2]) / (samplePoints[n - 1] - samplePoints[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = (3 * (sampleValues[n - 1] - sampleValues[n - 2]) / (samplePoints[n - 1] - samplePoints[n - 2])) + (0.5 * rightBoundary * (samplePoints[n - 1] - samplePoints[n - 2]));
break;
default:
throw new NotSupportedException(Resources.InvalidRightBoundaryCondition);
}
// Build Spline
return SolveTridiagonal(a1, a2, a3, b);
}
/// <summary>
/// Initialize the interpolation method with the given samples.
/// </summary>
/// <param name="t">Points t</param>
/// <param name="x">Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
public void Init(
IList<double> t,
IList<double> x,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary
)
{
if(null == t)
{
throw new ArgumentNullException("t");
}
if(null == x)
{
throw new ArgumentNullException("x");
}
if(t.Count < 2)
{
throw new ArgumentOutOfRangeException("t");
}
if(t.Count != x.Count)
{
throw new ArgumentException(Properties.Resources.ArgumentVectorsSameLengths, "x");
}
int n = t.Count;
double[] tt = new double[n];
t.CopyTo(tt, 0);
double[] xx = new double[n];
x.CopyTo(xx, 0);
Sorting.Sort(tt, xx);
// normalize special cases
if((n == 2)
&& (leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated)
&& (rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if(leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if(rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
double[] a1 = new double[n];
double[] a2 = new double[n];
double[] a3 = new double[n];
double[] b = new double[n];
// Left Boundary
switch(leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2 * (xx[1] - xx[0]) / (tt[1] - tt[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = 3 * (xx[1] - xx[0]) / (tt[1] - tt[0]) - 0.5 * leftBoundary * (tt[1] - tt[0]);
break;
default:
throw new NotSupportedException(Properties.Resources.InvalidLeftBoundaryCondition);
}
// Central Conditions
for(int i = 1; i < tt.Length - 1; i++)
{
a1[i] = tt[i + 1] - tt[i];
a2[i] = 2 * (tt[i + 1] - tt[i - 1]);
a3[i] = tt[i] - tt[i - 1];
b[i] = 3 * (xx[i] - xx[i - 1]) / (tt[i] - tt[i - 1]) * (tt[i + 1] - tt[i]) + 3 * (xx[i + 1] - xx[i]) / (tt[i + 1] - tt[i]) * (tt[i] - tt[i - 1]);
}
// Right Boundary
switch(rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2 * (xx[n - 1] - xx[n - 2]) / (tt[n - 1] - tt[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = 3 * (xx[n - 1] - xx[n - 2]) / (tt[n - 1] - tt[n - 2]) + 0.5 * rightBoundary * (tt[n - 1] - tt[n - 2]);
break;
default:
throw new NotSupportedException(Properties.Resources.InvalidRightBoundaryCondition);
}
// Build Spline
double[] dd = SolveTridiagonal(a1, a2, a3, b);
_hermiteSpline.InitInternal(tt, xx, dd);
}
/// <summary>
/// Create a cubic spline interpolation from a set of (x,y) value pairs and custom boundary/termination conditions.
/// </summary>
/// <remarks>
/// The value pairs do not have to be sorted, but if they are not sorted ascendingly
/// and the passed x and y arguments are arrays, they will be sorted inplace and thus modified.
/// </remarks>
public static CubicSpline InterpolateBoundaries(IEnumerable <double> x, IEnumerable <double> y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
var xx = (x as double[]) ?? x.ToArray();
var yy = (y as double[]) ?? y.ToArray();
if (xx.Length != yy.Length)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (xx.Length < 2)
{
throw new ArgumentOutOfRangeException("x");
}
Sorting.Sort(xx, yy);
int n = xx.Length;
// normalize special cases
if ((n == 2) &&
(leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated) &&
(rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
var a1 = new double[n];
var a2 = new double[n];
var a3 = new double[n];
var b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2 * (yy[1] - yy[0]) / (xx[1] - xx[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = (3 * ((yy[1] - yy[0]) / (xx[1] - xx[0]))) - (0.5 * leftBoundary * (xx[1] - xx[0]));
break;
default:
throw new NotSupportedException(Resources.InvalidLeftBoundaryCondition);
}
// Central Conditions
for (int i = 1; i < xx.Length - 1; i++)
{
a1[i] = xx[i + 1] - xx[i];
a2[i] = 2 * (xx[i + 1] - xx[i - 1]);
a3[i] = xx[i] - xx[i - 1];
b[i] = (3 * (yy[i] - yy[i - 1]) / (xx[i] - xx[i - 1]) * (xx[i + 1] - xx[i])) + (3 * (yy[i + 1] - yy[i]) / (xx[i + 1] - xx[i]) * (xx[i] - xx[i - 1]));
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2 * (yy[n - 1] - yy[n - 2]) / (xx[n - 1] - xx[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = (3 * (yy[n - 1] - yy[n - 2]) / (xx[n - 1] - xx[n - 2])) + (0.5 * rightBoundary * (xx[n - 1] - xx[n - 2]));
break;
default:
throw new NotSupportedException(Resources.InvalidRightBoundaryCondition);
}
// Build Spline
double[] dd = SolveTridiagonal(a1, a2, a3, b);
return(InterpolateHermite(xx, yy, dd));
}
public void Init(IList <double> t, IList <double> x, SplineBoundaryCondition leftBoundaryCondition, double leftBoundary, SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
if (null == t)
{
throw new ArgumentNullException("t");
}
if (null == x)
{
throw new ArgumentNullException("x");
}
if (t.Count < 2)
{
throw new ArgumentOutOfRangeException("t");
}
if (t.Count != x.Count)
{
throw new ArgumentException("x");
}
int n = t.Count;
double[] tt = new double[n];
t.CopyTo(tt, 0);
double[] xx = new double[n];
x.CopyTo(xx, 0);
Sorting.Sort(tt, xx);
// normalize special cases
if ((n == 2) &&
(leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated) &&
(rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
double[] a1 = new double[n];
double[] a2 = new double[n];
double[] a3 = new double[n];
double[] b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2 * (xx[1] - xx[0]) / (tt[1] - tt[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = 3 * (xx[1] - xx[0]) / (tt[1] - tt[0]) - 0.5 * leftBoundary * (tt[1] - tt[0]);
break;
default:
throw new NotSupportedException("BoundaryCondition");
}
// Central Conditions
for (int i = 1; i < tt.Length - 1; i++)
{
a1[i] = tt[i + 1] - tt[i];
a2[i] = 2 * (tt[i + 1] - tt[i - 1]);
a3[i] = tt[i] - tt[i - 1];
b[i] = 3 * (xx[i] - xx[i - 1]) / (tt[i] - tt[i - 1]) * (tt[i + 1] - tt[i]) + 3 * (xx[i + 1] - xx[i]) / (tt[i + 1] - tt[i]) * (tt[i] - tt[i - 1]);
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2 * (xx[n - 1] - xx[n - 2]) / (tt[n - 1] - tt[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = 3 * (xx[n - 1] - xx[n - 2]) / (tt[n - 1] - tt[n - 2]) + 0.5 * rightBoundary * (tt[n - 1] - tt[n - 2]);
break;
default:
throw new NotSupportedException("BoundaryCondition");
}
// Build Spline
double[] dd = SolveTridiagonal(a1, a2, a3, b);
_hermiteSpline.InitInternal(tt, xx, dd);
}
/// <summary>
/// Create a cubic spline interpolation from a set of (x,y) value pairs, sorted ascendingly by x,
/// and custom boundary/termination conditions.
/// </summary>
public static CubicSpline InterpolateBoundariesSorted(double[] x, double[] y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
if (x.Length != y.Length)
{
throw new ArgumentException("All vectors must have the same dimensionality.");
}
if (x.Length < 2)
{
throw new ArgumentException("The given array is too small. It must be at least 2 long.", nameof(x));
}
int n = x.Length;
// normalize special cases
if ((n == 2) &&
(leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated) &&
(rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
var a1 = new double[n];
var a2 = new double[n];
var a3 = new double[n];
var b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2 * (y[1] - y[0]) / (x[1] - x[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = (3 * ((y[1] - y[0]) / (x[1] - x[0]))) - (0.5 * leftBoundary * (x[1] - x[0]));
break;
default:
throw new NotSupportedException("Invalid Left Boundary Condition.");
}
// Central Conditions
for (int i = 1; i < x.Length - 1; i++)
{
a1[i] = x[i + 1] - x[i];
a2[i] = 2 * (x[i + 1] - x[i - 1]);
a3[i] = x[i] - x[i - 1];
b[i] = (3 * (y[i] - y[i - 1]) / (x[i] - x[i - 1]) * (x[i + 1] - x[i])) + (3 * (y[i + 1] - y[i]) / (x[i + 1] - x[i]) * (x[i] - x[i - 1]));
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2 * (y[n - 1] - y[n - 2]) / (x[n - 1] - x[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = (3 * (y[n - 1] - y[n - 2]) / (x[n - 1] - x[n - 2])) + (0.5 * rightBoundary * (x[n - 1] - x[n - 2]));
break;
default:
throw new NotSupportedException("Invalid Right Boundary Condition.");
}
// Build Spline
double[] dd = SolveTridiagonal(a1, a2, a3, b);
return(InterpolateHermiteSorted(x, y, dd));
}
/// <summary>
/// Evaluate the spline derivatives as used
/// internally by this interpolation algorithm.
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
/// <returns>Spline Derivative Vector</returns>
public static double[] EvaluateSplineDerivatives(
IList <double> samplePoints,
IList <double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary)
{
if (null == samplePoints)
{
throw new ArgumentNullException("samplePoints");
}
if (null == sampleValues)
{
throw new ArgumentNullException("sampleValues");
}
if (samplePoints.Count < 2)
{
throw new ArgumentOutOfRangeException("samplePoints");
}
if (samplePoints.Count != sampleValues.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
for (var i = 1; i < samplePoints.Count; ++i)
{
if (samplePoints[i] <= samplePoints[i - 1])
{
throw new ArgumentException(Resources.Interpolation_Initialize_SamplePointsNotStrictlyAscendingOrder, "samplePoints");
}
}
int n = samplePoints.Count;
// normalize special cases
if ((n == 2) &&
(leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated) &&
(rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
double[] a1 = new double[n];
double[] a2 = new double[n];
double[] a3 = new double[n];
double[] b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2 * (sampleValues[1] - sampleValues[0]) / (samplePoints[1] - samplePoints[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = (3 * ((sampleValues[1] - sampleValues[0]) / (samplePoints[1] - samplePoints[0]))) - (0.5 * leftBoundary * (samplePoints[1] - samplePoints[0]));
break;
default:
throw new NotSupportedException(Resources.InvalidLeftBoundaryCondition);
}
// Central Conditions
for (int i = 1; i < samplePoints.Count - 1; i++)
{
a1[i] = samplePoints[i + 1] - samplePoints[i];
a2[i] = 2 * (samplePoints[i + 1] - samplePoints[i - 1]);
a3[i] = samplePoints[i] - samplePoints[i - 1];
b[i] = (3 * (sampleValues[i] - sampleValues[i - 1]) / (samplePoints[i] - samplePoints[i - 1]) * (samplePoints[i + 1] - samplePoints[i])) + (3 * (sampleValues[i + 1] - sampleValues[i]) / (samplePoints[i + 1] - samplePoints[i]) * (samplePoints[i] - samplePoints[i - 1]));
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2 * (sampleValues[n - 1] - sampleValues[n - 2]) / (samplePoints[n - 1] - samplePoints[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = (3 * (sampleValues[n - 1] - sampleValues[n - 2]) / (samplePoints[n - 1] - samplePoints[n - 2])) + (0.5 * rightBoundary * (samplePoints[n - 1] - samplePoints[n - 2]));
break;
default:
throw new NotSupportedException(Resources.InvalidRightBoundaryCondition);
}
// Build Spline
return(SolveTridiagonal(a1, a2, a3, b));
}
/// <summary>
/// Create a cubic spline interpolation based on arbitrary points, with specified boundary conditions.
/// </summary>
/// <param name="points">The sample points t. Supports both lists and arrays.</param>
/// <param name="values">The sample point values x(t). Supports both lists and arrays.</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
/// <returns>
/// An interpolation scheme optimized for the given sample points and values,
/// which can then be used to compute interpolations and extrapolations
/// on arbitrary points.
/// </returns>
public static IInterpolationMethod CreateCubicSpline(
IList<double> points,
IList<double> values,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary
)
{
CubicSplineInterpolation method = new CubicSplineInterpolation();
method.Init(
points,
values,
leftBoundaryCondition,
leftBoundary,
rightBoundaryCondition,
rightBoundary
);
return method;
}
/// <summary>
/// Evaluate the spline coefficients as used
/// internally by this interpolation algorithm.
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
/// <returns>Spline Coefficient Vector</returns>
public static double[] EvaluateSplineCoefficients(
IList<double> samplePoints,
IList<double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary)
{
double[] derivatives = EvaluateSplineDerivatives(
samplePoints,
sampleValues,
leftBoundaryCondition,
leftBoundary,
rightBoundaryCondition,
rightBoundary);
return CubicHermiteSplineInterpolation.EvaluateSplineCoefficients(
samplePoints,
sampleValues,
derivatives);
}
/// <summary>
/// Create a cubic spline interpolation from a set of (x,y) value pairs and custom boundary/termination conditions.
/// </summary>
/// <remarks>
/// The value pairs do not have to be sorted, but if they are not sorted ascendingly
/// and the passed x and y arguments are arrays, they will be sorted inplace and thus modified.
/// </remarks>
public static CubicSpline InterpolateBoundaries(IEnumerable<double> x, IEnumerable<double> y,
SplineBoundaryCondition leftBoundaryCondition, double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition, double rightBoundary)
{
var xx = (x as double[]) ?? x.ToArray();
var yy = (y as double[]) ?? y.ToArray();
if (xx.Length != yy.Length)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (xx.Length < 2)
{
throw new ArgumentOutOfRangeException("x");
}
Sorting.Sort(xx, yy);
int n = xx.Length;
// normalize special cases
if ((n == 2)
&& (leftBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated)
&& (rightBoundaryCondition == SplineBoundaryCondition.ParabolicallyTerminated))
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
if (leftBoundaryCondition == SplineBoundaryCondition.Natural)
{
leftBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
leftBoundary = 0d;
}
if (rightBoundaryCondition == SplineBoundaryCondition.Natural)
{
rightBoundaryCondition = SplineBoundaryCondition.SecondDerivative;
rightBoundary = 0d;
}
var a1 = new double[n];
var a2 = new double[n];
var a3 = new double[n];
var b = new double[n];
// Left Boundary
switch (leftBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[0] = 0;
a2[0] = 1;
a3[0] = 1;
b[0] = 2*(yy[1] - yy[0])/(xx[1] - xx[0]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[0] = 0;
a2[0] = 1;
a3[0] = 0;
b[0] = leftBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[0] = 0;
a2[0] = 2;
a3[0] = 1;
b[0] = (3*((yy[1] - yy[0])/(xx[1] - xx[0]))) - (0.5*leftBoundary*(xx[1] - xx[0]));
break;
default:
throw new NotSupportedException(Resources.InvalidLeftBoundaryCondition);
}
// Central Conditions
for (int i = 1; i < xx.Length - 1; i++)
{
a1[i] = xx[i + 1] - xx[i];
a2[i] = 2*(xx[i + 1] - xx[i - 1]);
a3[i] = xx[i] - xx[i - 1];
b[i] = (3*(yy[i] - yy[i - 1])/(xx[i] - xx[i - 1])*(xx[i + 1] - xx[i])) + (3*(yy[i + 1] - yy[i])/(xx[i + 1] - xx[i])*(xx[i] - xx[i - 1]));
}
// Right Boundary
switch (rightBoundaryCondition)
{
case SplineBoundaryCondition.ParabolicallyTerminated:
a1[n - 1] = 1;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = 2*(yy[n - 1] - yy[n - 2])/(xx[n - 1] - xx[n - 2]);
break;
case SplineBoundaryCondition.FirstDerivative:
a1[n - 1] = 0;
a2[n - 1] = 1;
a3[n - 1] = 0;
b[n - 1] = rightBoundary;
break;
case SplineBoundaryCondition.SecondDerivative:
a1[n - 1] = 1;
a2[n - 1] = 2;
a3[n - 1] = 0;
b[n - 1] = (3*(yy[n - 1] - yy[n - 2])/(xx[n - 1] - xx[n - 2])) + (0.5*rightBoundary*(xx[n - 1] - xx[n - 2]));
break;
default:
throw new NotSupportedException(Resources.InvalidRightBoundaryCondition);
}
// Build Spline
double[] dd = SolveTridiagonal(a1, a2, a3, b);
return InterpolateHermite(xx, yy, dd);
}
/// <summary>
/// Initialize the interpolation method with the given spline coefficients (sorted by the sample points t).
/// </summary>
/// <param name="samplePoints">Sample Points t, sorted ascending.</param>
/// <param name="sampleValues">Sample Values x(t)</param>
/// <param name="leftBoundaryCondition">Condition of the left boundary.</param>
/// <param name="leftBoundary">Left boundary value. Ignored in the parabolic case.</param>
/// <param name="rightBoundaryCondition">Condition of the right boundary.</param>
/// <param name="rightBoundary">Right boundary value. Ignored in the parabolic case.</param>
public void Initialize(
IList<double> samplePoints,
IList<double> sampleValues,
SplineBoundaryCondition leftBoundaryCondition,
double leftBoundary,
SplineBoundaryCondition rightBoundaryCondition,
double rightBoundary)
{
double[] derivatives = EvaluateSplineDerivatives(
samplePoints,
sampleValues,
leftBoundaryCondition,
leftBoundary,
rightBoundaryCondition,
rightBoundary);
_spline.Initialize(samplePoints, sampleValues, derivatives);
}
/// <summary>Creates a <see cref="MklGridPointCurve.Interpolator"/> object that represents the implementation of a specified interpolation approach.
/// </summary>
/// <param name="splineOrder">The spline order.</param>
/// <param name="splineType">The type of the spline.</param>
/// <param name="boundaryConditionType">The type of the boundary condition.</param>
/// <param name="boundaryCondition">The boundary condition.</param>
/// <param name="splineCoefficientHint">The spline coefficient hint.</param>
/// <param name="internalConditionType">The internal boundary condition type.</param>
/// <param name="internalConditions">The internal boundary condition.</param>
/// <returns>A <see cref="MklGridPointCurve.Interpolator"/> object that represents the implementation of the specified interpolation approach.</returns>
public MklGridPointCurve.Interpolator Create(SplineOrder splineOrder, SplineType splineType, SplineBoundaryCondition boundaryConditionType, double[] boundaryCondition = null, SplineCoefficientStorageFormat splineCoefficientHint = SplineCoefficientStorageFormat.DF_NO_HINT, SplineInternalConditionType internalConditionType = SplineInternalConditionType.DF_NO_IC, double[] internalConditions = null)
{
return(new MklDataFitting(splineOrder, splineType, boundaryConditionType, boundaryCondition, splineCoefficientHint, internalConditionType, internalConditions));
}
