'*****************************************************
'*        Program to demonstrate the complex         *
'*             root counting subroutine              *
'* ------------------------------------------------- *
'* Reference: BASIC Scientific Subroutines, Vol. II  *
'* By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1]. *
'*                                                   *
'* ------------------------------------------------- *
'* Example: Find the number of complex roots of      *
'*          F(z) = z^2 + 1                           *
'*                                                   *
'* SAMPLE RUN:                                       *
'* Where is the center of the search circle (x0,y0): *
'*                                                   *
'*    X0 = 0                                         *
'*    Y0 = 0                                         *
'*                                                   *
'* What is the radius of this circle: 4              *
'* How many evaluation points per quadrant: 4        *
'*                                                   *
'* Number of complete cycles found:  2               *
'* Residual:  0                                      *
'*****************************************************
defint i-n
defdbl a-h,o-z
cls
print
print " Where is the center of the search circle (x0,y0): "
print
input "    X0 = ",x0
input "    Y0 = ",y0
print
input " What is the radius of this circle: ", w
print
input " How many evaluation points per quadrant: ", m
print
dim N(4*m)  'Table of integers
gosub 2000  'Call RootNum subroutine
print
print " Number of complete cycles found: "; nn
print
print " Residual: "; a
print
end

'*******************************************
1000 ' Complex function(x,y) subroutine
  u=x*x-y*y+1#
  v=2#*x*y
return
'*******************************************

'*************************************************
'*       Complex root counting subroutine        *
'* --------------------------------------------- *
'* This routine calculates the number of complex *
'* roots within a circle of radius w centered on *
'* (x0,y0) by counting (u,v) transitions around  *
'* the circumference. The input parameters are:  *
'*     w      radius of the circle               *
'*     x0,y0  center of the circle               *
'*     m      evaluation points per quadrant     *
'* The routine returns nn, the number of roots   *
'* found, and a, where a<>0 indicates a failure. *
'*************************************************
2000 a=3.1415926#/(2#*m)
  'Establish the N(i) array
  for i=1 to 4*m
    x=w*COS(a*(i-1))+x0
    y=w*SIN(a*(i-1))+y0
    gosub 1000  'Call complex function subroutine
    if u>=0 then if v>=0 then N(i)=1
    if u< 0 then if v>=0 then N(i)=2
    if u< 0 then if v< 0 then N(i)=3
    if u>=0 then if v< 0 then N(i)=4
  next i
  'Count complete cycles counterclockwise
  nn=N(1) : a=0
  for i=2 to 4*m
    if nn=N(i) then goto 2100
    if nn<>4 then if nn=N(i)+1 then a=a-1
    if nn= 1 then if N(i)=4 then a=a-1
    if nn=4  then if N(i)=1 then a=a+1
    if nn+1=N(i) then a=a+1
    nn=N(i)
2100 next i
  'Complete circle
    if nn<>4 then if nn=N(1)+1 then a=a-1
    if nn= 1 then if N(1)=4 then a=a-1
    if nn=4  then if N(1)=1 then a=a+1
    if nn+1=N(1) then a=a+1
    a=ABS(a) : nn=INT(a/4)
    a=a-4#*INT(a/4)
return

'End of file Rootnum.bas