spline3.cs

三样条插值函数算法

/*************************************************************************
Copyright (c) 2007, Sergey Bochkanov (ALGLIB project).

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modification, are permitted provided that the following conditions are
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- Redistributions in binary form must reproduce the above copyright
  notice, this list of conditions and the following disclaimer listed
  in this license in the documentation and/or other materials
  provided with the distribution.

- Neither the name of the copyright holders nor the names of its
  contributors may be used to endorse or promote products derived from
  this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*************************************************************************/

using System;

class spline3
{
    /*************************************************************************
    This subroutine builds linear spline coefficients table.

    Input parameters:
        X   -   spline nodes, array[0..N-1]
        Y   -   function values, array[0..N-1]
        N   -   points count, N>=2
        
    Output parameters:
        C   -   coefficients table.  Used  by  SplineInterpolation  and  other
                subroutines from this file.

      -- ALGLIB PROJECT --
         Copyright 24.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void buildlinearspline(double[] x,
        double[] y,
        int n,
        ref double[] c)
    {
        int i = 0;
        int tblsize = 0;

        x = (double[])x.Clone();
        y = (double[])y.Clone();

        System.Diagnostics.Debug.Assert(n>=2, "BuildLinearSpline: N<2!");
        
        //
        // Sort points
        //
        heapsortpoints(ref x, ref y, n);
        
        //
        // Fill C:
        //  C[0]            -   length(C)
        //  C[1]            -   type(C):
        //                      3 - general cubic spline
        //  C[2]            -   N
        //  C[3]...C[3+N-1] -   x[i], i = 0...N-1
        //  C[3+N]...C[3+N+(N-1)*4-1] - coefficients table
        //
        tblsize = 3+n+(n-1)*4;
        c = new double[tblsize-1+1];
        c[0] = tblsize;
        c[1] = 3;
        c[2] = n;
        for(i=0; i<=n-1; i++)
        {
            c[3+i] = x[i];
        }
        for(i=0; i<=n-2; i++)
        {
            c[3+n+4*i+0] = y[i];
            c[3+n+4*i+1] = (y[i+1]-y[i])/(x[i+1]-x[i]);
            c[3+n+4*i+2] = 0;
            c[3+n+4*i+3] = 0;
        }
    }


    /*************************************************************************
    This subroutine builds cubic spline coefficients table.

    Input parameters:
        X           -   spline nodes, array[0..N-1]
        Y           -   function values, array[0..N-1]
        N           -   points count, N>=2
        BoundLType  -   boundary condition type for the left boundary
        BoundL      -   left boundary condition (first or second derivative,
                        depending on the BoundLType)
        BoundRType  -   boundary condition type for the right boundary
        BoundR      -   right boundary condition (first or second derivative,
                        depending on the BoundRType)

    Output parameters:
        C           -   coefficients table.  Used  by  SplineInterpolation and
                        other subroutines from this file.
                        
    The BoundLType/BoundRType parameters can have the following values:
        * 0,   which  corresponds  to  the  parabolically   terminated  spline
          (BoundL/BoundR are ignored).
        * 1, which corresponds to the first derivative boundary condition
        * 2, which corresponds to the second derivative boundary condition

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void buildcubicspline(double[] x,
        double[] y,
        int n,
        int boundltype,
        double boundl,
        int boundrtype,
        double boundr,
        ref double[] c)
    {
        double[] a1 = new double[0];
        double[] a2 = new double[0];
        double[] a3 = new double[0];
        double[] b = new double[0];
        double[] d = new double[0];
        int i = 0;
        int tblsize = 0;
        double delta = 0;
        double delta2 = 0;
        double delta3 = 0;

        x = (double[])x.Clone();
        y = (double[])y.Clone();

        System.Diagnostics.Debug.Assert(n>=2, "BuildCubicSpline: N<2!");
        System.Diagnostics.Debug.Assert(boundltype==0 | boundltype==1 | boundltype==2, "BuildCubicSpline: incorrect BoundLType!");
        System.Diagnostics.Debug.Assert(boundrtype==0 | boundrtype==1 | boundrtype==2, "BuildCubicSpline: incorrect BoundRType!");
        a1 = new double[n-1+1];
        a2 = new double[n-1+1];
        a3 = new double[n-1+1];
        b = new double[n-1+1];
        
        //
        // Special case:
        // * N=2
        // * parabolic terminated boundary condition on both ends
        //
        if( n==2 & boundltype==0 & boundrtype==0 )
        {
            
            //
            // Change task type
            //
            boundltype = 2;
            boundl = 0;
            boundrtype = 2;
            boundr = 0;
        }
        
        //
        //
        // Sort points
        //
        heapsortpoints(ref x, ref y, n);
        
        //
        // Left boundary conditions
        //
        if( boundltype==0 )
        {
            a1[0] = 0;
            a2[0] = 1;
            a3[0] = 1;
            b[0] = 2*(y[1]-y[0])/(x[1]-x[0]);
        }
        if( boundltype==1 )
        {
            a1[0] = 0;
            a2[0] = 1;
            a3[0] = 0;
            b[0] = boundl;
        }
        if( boundltype==2 )
        {
            a1[0] = 0;
            a2[0] = 2;
            a3[0] = 1;
            b[0] = 3*(y[1]-y[0])/(x[1]-x[0])-0.5*boundl*(x[1]-x[0]);
        }
        
        //
        // Central conditions
        //
        for(i=1; i<=n-2; 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 conditions
        //
        if( boundrtype==0 )
        {
            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]);
        }
        if( boundrtype==1 )
        {
            a1[n-1] = 0;
            a2[n-1] = 1;
            a3[n-1] = 0;
            b[n-1] = boundr;
        }
        if( boundrtype==2 )
        {
            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*boundr*(x[n-1]-x[n-2]);
        }
        
        //
        // Solve
        //
        solvetridiagonal(a1, a2, a3, b, n, ref d);
        
        //
        // Now problem is reduced to the cubic Hermite spline
        //
        buildhermitespline(x, y, d, n, ref c);
    }


    /*************************************************************************
    This subroutine builds cubic Hermite spline coefficients table.

    Input parameters:
        X           -   spline nodes, array[0..N-1]
        Y           -   function values, array[0..N-1]
        D           -   derivatives, array[0..N-1]
        N           -   points count, N>=2

    Output parameters:
        C           -   coefficients table.  Used  by  SplineInterpolation and
                        other subroutines from this file.

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void buildhermitespline(double[] x,
        double[] y,
        double[] d,
        int n,
        ref double[] c)
    {
        int i = 0;
        int tblsize = 0;
        double delta = 0;
        double delta2 = 0;
        double delta3 = 0;

        x = (double[])x.Clone();
        y = (double[])y.Clone();
        d = (double[])d.Clone();

        System.Diagnostics.Debug.Assert(n>=2, "BuildHermiteSpline: N<2!");
        
        //
        // Sort points
        //
        heapsortdpoints(ref x, ref y, ref d, n);
        
        //
        // Fill C:
        //  C[0]            -   length(C)
        //  C[1]            -   type(C):
        //                      3 - general cubic spline
        //  C[2]            -   N
        //  C[3]...C[3+N-1] -   x[i], i = 0...N-1
        //  C[3+N]...C[3+N+(N-1)*4-1] - coefficients table
        //
        tblsize = 3+n+(n-1)*4;
        c = new double[tblsize-1+1];
        c[0] = tblsize;
        c[1] = 3;
        c[2] = n;
        for(i=0; i<=n-1; i++)
        {
            c[3+i] = x[i];
        }
        for(i=0; i<=n-2; i++)
        {
            delta = x[i+1]-x[i];
            delta2 = AP.Math.Sqr(delta);
            delta3 = delta*delta2;
            c[3+n+4*i+0] = y[i];
            c[3+n+4*i+1] = d[i];
            c[3+n+4*i+2] = (3*(y[i+1]-y[i])-2*d[i]*delta-d[i+1]*delta)/delta2;
            c[3+n+4*i+3] = (2*(y[i]-y[i+1])+d[i]*delta+d[i+1]*delta)/delta3;
        }
    }


    /*************************************************************************
    This subroutine builds Akima spline coefficients table.

    Input parameters:
        X           -   spline nodes, array[0..N-1]
        Y           -   function values, array[0..N-1]
        N           -   points count, N>=5

    Output parameters:
        C           -   coefficients table.  Used  by  SplineInterpolation and
                        other subroutines from this file.

      -- ALGLIB PROJECT --
         Copyright 24.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static void buildakimaspline(double[] x,
        double[] y,
        int n,
        ref double[] c)
    {
        int i = 0;
        double[] d = new double[0];
        double[] w = new double[0];
        double[] diff = new double[0];

        x = (double[])x.Clone();
        y = (double[])y.Clone();

        System.Diagnostics.Debug.Assert(n>=5, "BuildAkimaSpline: N<5!");
        
        //
        // Sort points
        //
        heapsortpoints(ref x, ref y, n);
        
        //
        // Prepare W (weights), Diff (divided differences)
        //
        w = new double[n-2+1];
        diff = new double[n-2+1];
        for(i=0; i<=n-2; i++)
        {
            diff[i] = (y[i+1]-y[i])/(x[i+1]-x[i]);
        }
        for(i=1; i<=n-2; i++)
        {
            w[i] = Math.Abs(diff[i]-diff[i-1]);
        }
        
        //
        // Prepare Hermite interpolation scheme
        //
        d = new double[n-1+1];
        for(i=2; i<=n-3; i++)
        {
            if( Math.Abs(w[i-1])+Math.Abs(w[i+1])!=0 )
            {
                d[i] = (w[i+1]*diff[i-1]+w[i-1]*diff[i])/(w[i+1]+w[i-1]);
            }
            else
            {
                d[i] = ((x[i+1]-x[i])*diff[i-1]+(x[i]-x[i-1])*diff[i])/(x[i+1]-x[i-1]);
            }
        }
        d[0] = diffthreepoint(x[0], x[0], y[0], x[1], y[1], x[2], y[2]);
        d[1] = diffthreepoint(x[1], x[0], y[0], x[1], y[1], x[2], y[2]);
        d[n-2] = diffthreepoint(x[n-2], x[n-3], y[n-3], x[n-2], y[n-2], x[n-1], y[n-1]);
        d[n-1] = diffthreepoint(x[n-1], x[n-3], y[n-3], x[n-2], y[n-2], x[n-1], y[n-1]);
        
        //
        // Build Akima spline using Hermite interpolation scheme
        //
        buildhermitespline(x, y, d, n, ref c);
    }


    /*************************************************************************
    This subroutine calculates the value of the spline at the given point X.

    Input parameters:
        C           -   coefficients table. Built by BuildLinearSpline,
                        BuildHermiteSpline, BuildCubicSpline, BuildAkimaSpline.
        X           -   point

    Result:
        S(x)

      -- ALGLIB PROJECT --
         Copyright 23.06.2007 by Bochkanov Sergey
    *************************************************************************/
    public static double splineinterpolation(ref double[] c,
        double x)
    {
        double result = 0;
        int n = 0;
        int l = 0;
        int r = 0;
        int m = 0;

        System.Diagnostics.Debug.Assert((int)Math.Round(c[1])==3, "SplineInterpolation: incorrect C!");
        n = (int)Math.Round(c[2]);
        
        //
        // Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
        //
        l = 3;
        r = 3+n-2+1;