EXPLANATION FILE OF PROGRAM CLIPTIC
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  Elliptic IntegraIs by Recursion
  -------------------------------
 
    Elliptie integrals outwardly appear to be very esoteric functions and many
  practicing scientists and engineers avoid them. But these functions are inti-
  mately related to such very common and simple concepts as the period of a pen-
  dulum and the circumference of an ellipse.
   
    We will consider the complete elliptic integrals of the first and second
  kinds. They are defined respectively as 

                               pi/2         1
                     K(k) = Sum     ------------------- dt        (3.6.1)
                               0    sqrt(1-k^2 sin^2 t)
  and
                               pi/2         1
                     E(k) = Sum     ------------------- dt        (3.6.2)
                               0    sqrt(1+k^2 sin^2 t)

  K(k) can easily be used to accurately calculate the period of an ideal pendu-
  lum as follows. If the length of the pendulum is l, the gravitational constant
  g and the maximum amplitude of swing alpha (in radians), then the period is 

                  T = 4 K(k) sqrt(l/g)  where k = sin(alpha/2)    (3.6.3)

  ln most introductory courses and texts that discuss the pendulum, it is assu-
  med that the amplitude of swing is so small that sin(theta) can be replaced
  with theta in the defining differential equation. ln that case, k is near 0
  and T is near 2 pi sqrt(l/g). For amplitudes of vibration so small that this
  approximation is valid, the period is independent of the amplitude and is said
  to be isochronous. However, as the maximum amplitude is increased, the period
  gets longer (e.g., a church bell stuck upside down). Greenhill gives a prac-
  tical example that shows that if a pendulum's amplitude is adjusted from a 6°
  swing to a 10° swing, it will lose 26 seconds a day. As you can imagine, the-
  se calculations were particularly important in the days when the pendulum (or
  a similar dock mechanism) was the primary timepiece.
   
    The complete elliptie integral of the second kind, E(k), also has a simple
  application. If the major and minor axes of an ellipse are 2a and 2b, respec-
  tively, then the circumference of that ellipse is 

                   c = 4b E(k)   where k^2 = (b^2 - a^)/b^2       (3.6.4) 

  For the special case of a circle, a = b and c = 2 pi b.

    As you can see, elliptie integrals are involved in sorne very fundamental
  calculations. We will now consider how K(k) and E(k) can be approximated. 

    The steps in the elliptic integral recursion procedure are very similar to
  those involved in the arctangent calculation. First, the starting values are
  defined:

                           ao = 1 + k
                                                                  (3.6.5)
                           ho = 1 - k 

  Next, the recursion relation is repeatedly exercised until lan - bnl is less
  than sorne convergence criterion, E:
   
                           a    = (a + b) / 2 
                            i+1     i   i
                                              for i = 1, 2, ...   (3.6.6)

                           (a b)^(1/2)
                            i i

  Note that equation (3.6.6) is nearly identical to equation (3.4.2).) Once the
  convergence criterion has been met, K(k) is evaluated as

                           K(k) = pi/2 a
                                        n

  E(k) requires sorne further calculation:
   
       E(k) = (K(k/2)[2 - (aČ - bČ) - 2(aČ - bČ) - 4(aČ - bČ) - ... ]   (3.6.8)
                            0    0       1    1       2    2

  Technically, the form of the evaluation for E(k) is prone to round-off error.
  Convergence is usually rapid, however, and the round-off error tends to be
  small.
   
    The above recursive procedure can be very compactly programmed in BASIC as
  shown in program CLIPTIC.BAS. The inputs to CLIPTIC are simply the parameter K
  (0 <= K <= 1) and the convergence criterion, E. The outputs are El = K(K),
  E2 = E(K), and the number of steps performed, N.
   
    As is typical with this kind of recursion calculation, the accuracy is good. 
 
  From [BIBLI 01].
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End of file cliptic.txt