Exercise ODEs

1. Implement a Runge-Kutta "stepper"-routine rkstep wich advances the solution by a step h and estimates the error, e.g.:
   function rkstep(f,x,y,h){ // Runge-Kutta midpoint step var k0 = f(x,y) var y12 = [y[i]+k0[i]*h/2 for(i in y)] var k = f(x+h/2,y12) var y1 = [y[i]+k[i]*h for(i in y)] var dy = [(k[i]-k0[i])*h/2 for(i in y)] return [y1, dy] }
2. Implement an adaptive-step-size Runge-Kutta "driver"-routine wchich advances the solution from a to b (by calling rkstep with appropriate step-sizes) keeping the specified relative, eps, and absolute, acc, precision:
   function rkdrive(f,xlist,ylist,b,acc,eps,h) { // ODE driver: integrates y'=f(x,y) with // absolute accuracy acc and relative // accuracy eps until x=b with initial // step h storing the results // in arrays xlist and ylist var norm = function(v) Math.sqrt(v.reduce(function(a,b)a+b*b,0)) var a = xlist[0] var x = xlist[xlist.length-1]; var y = ylist[ylist.length-1]; if(x>=b) return; if(x+h>b) h=b-x; var [y1,dy]=rkstep(f, x, y, h); var err=norm(dy); var tol=(norm(y1)*eps+acc)* Math.sqrt(h/(b-a)); if(err>0) var new_h = h*Math.pow(tol/err,.25)*.95; else var new_h = 2*h; if(tol>err){ // accept step xlist.push(x+h); ylist.push(y1); rkdrive(f,xlist,ylist,b,acc,eps,new_h);} else // reject step rkdrive(f,xlist,ylist,b,acc,eps,new_h); }
3. Implement a Prediction-Correction method with adaptive step-size for a single first order differential equation. Here is a JavaScript example.
4. Your driver must remember the points (x,y) it calculated along the way.