Homework "Interpolation"

Implement functions for interpolation of tabulated data.

1. (6 points) Linear spline (linear interpolation)
   1. Implement a function that interpolates the given tabulated function, {x[i], y[i]}, at a given point z using linear spline interpolation.

      You can use either arrays, or gsl_vector's (or gsl_matrix) to keep your tabulated data. Depending on that the signature of your function might be something like

      double linterp(int n, double x[], double y[], double z);
      double linterp(gsl_vector x, gsl_vector y, double z);
      double linterp(gsl_matrix xy, double z);
      

      The function must first find (by binary search) the interval where z is located, x[i]≤z≤x[i+1], and then calculate the interpolated value using linear spline,

      double slope=(y[i+1]-y[i])/(x[i+1]-x[i]);
      double value=y[i]+slope*(z-x[i]);
      return value;
      	

   2. Implement a function that calculates the intergral of the linear spline of the tabulated data from the point x[0] to the given point z. The integral must be calculated analytically,

      ∫dx (yi+pi(x-xi)) = yi(x-xi)+pi(x-xi)2/2 + const .
      

      The signature could be something like

      double linterp_integ(gsl_vector x, gsl_vector y, double z);
      

   3. Make some indicative plots to prove that your linear spline and your integrator work as intended. Compare with GSL's [gsl_interp_linear] and [gsl_interp_eval_integ] (see this example: "gsl_interp").

   4. Location of the index i of the interval containing z, x[i]<z<x[i+1], must(!) be done using binary search, like this:

      int binsearch(int n, double* x, double z){/* locates the interval for z by bisection */ 
      	assert(n>1 && x[0]<=z && z<=x[n-1]);
      	int i=0, j=n-1;
      	while(j-i>1){
      		int mid=(i+j)/2;
      		if(z>x[mid]) i=mid; else j=mid;
      		}
      	return i;
      	}
      

2. (3 points) Quadratic spline

   Implement quadratic spline with derivative and integral. Make some illustrative plots.

   Note that quadratic splines should only be used for learning, for practical applications always use cubic splines instead.

   Hints:

   • When debugging your quadratic-spline routine you might consider the following data-sets,
     • {x_i=i, y_i=1} , i=1,...,5
     • {x_i=i, y_i=i} , i=1,...,5
     • {x_i=i, y_i=i²} , i=1,...,5
     Calculate manually the parameters {b_i, c_i} of the corresponding quadratic-splines, and compare the results with your quadratic-spline program.

3. (1 points) Cubic spline

   • Implement cubic spline with derivative and integral.

   • Compare with the cubic spline routines from GSL.