numerical integration

• well behaved integrand

• trapezoidal rule

• $\ x_0=a$, $\ x_n=b$
  $\Delta x=x_n-x_{n-1}\ \ $ constant
• convergence with the number $n$ of trapezoids      
  $lim_{n\to\infty}\, T_n \ =\ I$

trapezoidal rule at work

$$ y_m\ =\ \int_0^1\, \frac{x^m}{x+a}\, dx$$

• $a=10$, $m=19$, $n=2^p$

       n     p                   T(p)        (4T(p)-T(p-1)/3
       1     0     0.0454545454545455
       2     1     0.0227273635533981     0.0151516362530156
       4     2     0.0114620139299330     0.0077068973887780
       8     3     0.0066417103644638     0.0050349425093074
      16     4     0.0051140844181988     0.0046048757694438
      32     5     0.0047045044781142     0.0045679778314194
      64     6     0.0046002267297100     0.0045654674802419
     128     7     0.0045740370560200     0.0045653071647900
     256     8     0.0045674820820493     0.0045652970907257
     512     9     0.0045658428656951     0.0045652964602438
    1024    10     0.0045654330320428     0.0045652964208253
    2048    11     0.0045653305717818     0.0045652964183615
    4096    12     0.0045653049566010     0.0045652964182075
    8192    13     0.0045652985527986     0.0045652964181978
   16384    14     0.0045652969518476     0.0045652964181972
                             ^                             ^
                             9                            16

• inverse recursion ($a=10$, $m=19$): $\quad 0.0045652964181972$
• convergence acceleration with $$ T_p^* \ =\ \frac{1}{3}\,\Big(4\,T(p)-T(p-1)\Big)$$ for $n=2^p$. Why?

convergence scaling

• scaling of accuracy as a function of effort
  accuracy of trapezoidal approximation improves
  quadratically with the number of trapezoids

$$ T_n\ =\ \frac{\Delta x}{2}\,\sum_{k=1}^n\, \Big(f(x_{k-1})+f(x_{k})\Big), \qquad \Delta x\,=\, \frac{b-a}{n} $$

• can be derived via recursive partial integration $$ \int_{x_{k-1}}^{x_k} 1\cdot f(x)\,dx \ =\ (x+c_1)f(x)\Big|_{x_{k-1}}^{x_k}\, - \, \int_{x_{k-1}}^{x_k} dx\, (x+c_1)\, \frac{d}{dx}f(x) $$ valid for well behaved (analytic) integrands
  leading to the Euler–Maclaurin formula

convergence acceleration

• exploit scaling for performance improvements

$$ T_n\ =\ I \,+\,\frac{c}{n^2} \,+\, \,O(n^{-4}),\quad\qquad T_{2n}\ =\ I \,+\,\frac{c}{4n^2} \,+\, \,O(n^{-4}) $$ $$ \frac{4}{3}T_{2n}-\frac{1}{3}T_{n} \ =\ \left(\frac{4}{3}-\frac{1}{3}\right)\, I \,+\, \left(\frac{4}{3}\frac{1}{4}-\frac{1}{3}\right)\, \frac{c}{n^2} \,+\, \,O(n^{-4}) $$

• qualitative convergence improvement
  $$ \frac{4}{3}T_{2n}-\frac{1}{3}T_{n} \ =\ I \,+\,O(n^{-4}) $$
  at essentially constant numerical effort
• generalization to Romberg's method
• how can we speed an algorithm scaling like $\ \ n^{-z}\ \ $, for any $\ \ z\le1$?

accelerated integration

• simple to use, never hurts
• quadratic convergence accleration

/** Integrating
  * y_m = \int_0^1 x^m/(x+a)dx
  * for various m and n of the trapezoidal integration method
  */
public class InteTrapezoidal {

public static void main(String[] args) {
  int    m = 19;            
  double a = 10.0;
// *** **************************** ***
// *** loop of numbers of trapzoids ***
// *** **************************** ***
  double Tn = 0.0;
  double lastTn = 0.0;
  int     n = 1;
 System.out.printf("%8s %5s %22s %22s\n",
                   "n","p","T(p)","(4T(p)-T(p-1)/3");
  for (int jj=0; jj<15; jj++)
    {
    Tn =  sumTrapezoids(n, m, a);
    if (jj==0)
      System.out.printf("%8d %5d %22.16f\n", 
                        n, jj, Tn);
    else
      System.out.printf("%8d %5d %22.16f %22.16f\n", 
                        n, jj, Tn,(4*Tn-lastTn)/3.0);
    n = n*2;
    lastTn = Tn;
    }
  System.out.println(" ");
}  // end of InteTrapezoidal.main()

/** Straight summing trapezoids
  */
public static double sumTrapezoids(int n, int m, double a) {
  double result = 0.0;
  double x0, x1;
  double DeltaX = 1.0/n;
  for (int i=0; i<n; i++)
    {
    x0  = i*DeltaX;
    x1 = (i+1)*DeltaX;
    result +=  0.5*( Math.pow(x0,1.0*m)/(x0+a)
                   + Math.pow(x1,1.0*m)/(x1+a) );
    }
  return result*DeltaX;
  }    //  end of summationUP

}    //  end of class InteTrapezoidal