'****************************************************
'*    Program to demonstrate Evaluating elliptic    *
'*  integrals of first and second kinds (complete)  *
'* ------------------------------------------------ *
'* Reference: BASIC Scientific Subroutines, Vol. II *
'* By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].*
'*                                                  *
'* ------------------------------------------------ *
'* SAMPLE RUN:                                      *
'*                                                  *
'*    K         K(K)          E(K)      STEPS       *
'*  -------------------------------------------     *
'*   0.00     1.5707963     1.5707963     1         *
'*   0.05     1.5717795     1.5698141     3         *
'*   0.10     1.5747456     1.5668619     3         *
'*   0.15     1.5797457     1.5619230     3         *
'*   0.20     1.5868678     1.5549685     3         *
'*   0.25     1.5962422     1.5459573     3         *
'*   0.30     1.6080486     1.5348335     3         *
'*   0.35     1.6225281     1.5215252     3         *
'*   0.40     1.6399999     1.5059416     4         *
'*   0.45     1.6608862     1.4879683     4         *
'*   0.50     1.6857504     1.4674622     4         *
'*   0.55     1.7153545     1.4442435     4         *
'*   0.60     1.7507538     1.4180834     4         *
'*   0.65     1.7934541     1.3886864     4         *
'*   0.70     1.8456940     1.3556611     4         *
'*   0.75     1.9109898     1.3184721     4         *
'*   0.80     1.9953028     1.2763499     4         *
'*   0.85     2.1099355     1.2281083     4         *
'*   0.90     2.2805490     1.1716970     4         *
'*   0.95     2.5900112     1.1027216     5         *
'*   1.00     INFINITY      1.0000000     0         *
'*                                                  *
'****************************************************
defint i-n
defdbl a-h,o-z
cls
print
dim A(100), B(100)
e=1e-7
print " K         K(K)          E(K)      STEPS "
print "-----------------------------------------"
xk=0.0
for i=1 to 20
  gosub 1000
  print using "#.##     #.#######     #.#######     #"; xk; e1; e2; n
  xk = xk + 0.05
next i
print "1.00     INFINITY      1.0000000     0"
end

'******************************************************
'* Complete elliptic integral of the first and second *
'* kind. The input parameter is xk, which should be   *
'* between 0 and 1. Technique uses Gauss' formula for *
'* the arithmogeometrical mean. e is a measure of the *
'* convergence accuracy. The returned values are e1,  *
'* the elliptic integral of the first kind, and e2,   *
'* the elliptic integral of the second kind.          *
'* -------------------------------------------------- *
'* Reference: Ball, algorithms for RPN calculators.   *
'******************************************************
1000 A(0)=1+xk : B(0)=1-xk
  n=0 : pi=4#*ATN(1#)
  if xk < 0 then return
  if xk > 1 then return
  if e <= 0 then return
1100 n=n+1
  'Generate improved values
  A(n)=(A(n-1)+B(n-1))/2#
  B(n)=SQR(A(n-1)*B(n-1))
  if ABS(A(n)-B(n)) > e then goto 1100
  e1=pi/2#/A(n)
  e2=2#
  m=1
  for j=1 to n
    e2=e2-m*(A(j)*A(j)-B(j)*B(j))
    m=m*2
  next j
  e2=e2*e1/2#
return

'End of file CLIPTIC.BAS