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Homework "Splines"

Objective

Implement functions for interpolation of tabulated data points {xi, yi}i=1…n .

Hints

You can use either our "vector" class or "double[]" to keep the x and y vectors.

The exercise (parts B and C) can be solved using different programming styles:

  1. Procedural programming: One makes two functions (procedures). The first one, say 

    static void qspline_build(vector x,vector y,vector b,vector c)
    builds the (quadratic) spline-coefficients b and c. The second one, say 
    static double qspline_evaluate(vector x,vector y,vector b,vector c,double z)
    evaluates the spline at point "z" using the vectors x, y, b, c. The disadvantage is that one has to explicitly allocate and pass around the spline-coefficients: 
    vector b=new vector(n-1);
vector c=new vector(n-1);
qspline_build(x,y,b,c);
double fz = qspline_evaluate(x,y,b,c,z);

  2. Object oriented programming: One makes an object that holds both the data and the appropriate functions (now called member functions or methods). The methods have direct access to the data in the structure, for example, 

    public class qspline{
	vector x,y,b,c;
	public qspline(vector xs, vector ys){ /* copy xs,ys into x,y and build b,c */}
	public double evaluate(double z){/* evaluate the spline using x,y,b,c */}
}
    Now the procedurial style, 
    vector b=new vector(n-1);
vector c=new vector(n-1);
qspline_build(x,y,b,c);
double fz = qspline_evaluate(x,y,b,c,z);
    is replaced by the object-oriented style, 
    qspline myspline = new qspline(x,y);
double fz=myspline.evaluate(z);
    No need to pass the vectors x,y,b,c to "myspline" – they are captured in the (environment of the) function "myspline".

  3. Functional programming: The function for spline-evaluation is created and returned. This returned function captures all the needed information in itself (in its environment, to be precise). For example, 

    static Func<double,double> qspline(vector xs,vector ys){
	vector x, y, b, c; /* will be captured when qspline returns */
	/* copy xs,ys into x,y and calculate b and c */
	return delegate(double z){/* evaluate and return spline(z) */};
}
    Now the procedurial style, 
    vector b=new vector(n-1);
vector c=new vector(n-1);
qspline_build(x,y,b,c);
double fz = qspline_evaluate(x,y,b,c,z);
    is replaced by the functional style, 
    Func<double,double> myspline = qspline(x,y);
double fz=myspline(z);
    No need to pass the vectors x,y,b,c to "myspline.evaluate" – they are kept in the "myspline" instance of the class.

Tasks

  1. (6 points) Linear spline (linear interpolation)

    1. Implement a function that makes linear spline interpolation from a table {x[i], y[i]} at a given point z. Here you don't need to either pre-calculate anything or use any structs. The signature of your function should be something like 
      public static double linterp(double[] x, double[] y, double z){
        int i=binsearch(x,z);
        double dx=x[i+1]-x[i]; if(!(dx>0)) throw new Exception("uups...");
        double dy=y[i+1]-y[i];
        return y[i]+dy/dx*(z-x[i]);
        }
      Location of the index i of the interval containing z, x[i]<z<x[i+1], must(!) be done using binary search, like this: 
      public static int binsearch(double[] x, double z)
	{/* locates the interval for z by bisection */ 
	if(!(x[0]<=z && z<=x[x.Length-1])) throw new Exception("binsearch: bad z");
	int i=0, j=x.Length-1;
	while(j-i>1){
		int mid=(i+j)/2;
		if(z>x[mid]) i=mid; else j=mid;
		}
	return i;
	}

    2. Implement a function that calculates the intergral of the linear spline from the point x[0] to the given point z. The integral must be calculated analytically as it is the integral of a linear function. The signature could be something like 

      double linterpInteg(double[] x, double[] y, double z)

    3. Make some indicative plots to prove that your linear spline and your integrator work as intended.

  2. (3 points) Quadratic spline

    Implement quadratic spline with derivative and integral. Note that quadratic spline is only intended for learning, for practical applications always use the cubic spline.

    In C-sharp a quadratic spline can be kept in a class like the following, 

    class qspline {
	vector x,y,b,c;
	public qspline(vector xs,vector ys){
		/* x=xs.copy(); y=ys.copy(); calculate b and c */
		}
	public double evaluate(double z){/* evaluate the spline */}
	public double derivative(double z){/* evaluate the derivative */}
	public double integral(double z){/* evaluate the integral */}
	}

    Alternatively, you can use functional programming style: 

    static Func<double, int, double> qspline(vector xs,vector ys){
	/* x=xs.copy(); y=ys.copy(); calculate b and c */
	return delegate(double z,int deriv){
		if(deriv==1) /* return derivative */
		if(deriv==-1) /* return integral */
		else /* return spline */
		}; /* x,y,b,c are captured here */
}

    You can test your quadratic-spline by considering the following {xi, yi} tables,
    {xi=i, yi=1} , i=1,...,5
    {xi=i, yi=xi} , i=1,...,5
    {xi=i, yi=xi2} , i=1,...,5

    Calcullate manually the parameters {bi, ci} of the corresponding quadratic-splines, and compare the results with your quadratic-spline program.

  3. (1 points) Cubic spline