'*******************************************************
'*  Program to demonstrate the Newton root subroutine  *
'*             in the complex domain                   *
'* --------------------------------------------------- *
'*  Reference: BASIC Scientific Subroutines, Vol. II   *
'*  By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].  *
'*                                                     *
'* --------------------------------------------------- *
'* SAMPLE RUN:                                         *
'*                                                     *
'* (Example: find a complex root of  z^2 + 1 = 0)      *
'*                                                     *
'* What is the initial guess:                          *
'*                                                     *
'*    X0 = .2                                          *
'*    Y0 = 1.2                                         *
'*                                                     *
'* Convergence criterion: 1e-8                         *
'* Maximum number of iterations: 10                    *
'*                                                     *
'* The root estimate is:                               *
'*                                                     *
'*    X0 = -0.000000                                   *
'*    Y0 =  1.000000                                   *
'*                                                     *
'* The number of iterations performed was: 5           *
'*                                                     *
'*******************************************************
defint i-n
defdbl a-h,o-z
cls
print
print " What is the initial guess: "
print
input "    X0 = ", x0
input "    Y0 = ", y0
print
input " Convergence criterion: ", e
input " Maximum number of iterations: ", n
print
gosub 2000  'Call ZNewton subroutine
print
print " The root estimate is:"
print
print using "    X0 = ##.######"; x
print using "    Y0 = ##.######"; y
print
print " The number of iterations performed was: "; k
print
end
'****************************
1000 ' Functions subroutine
  u=x*x-y*y+1# : v=2#*x*y
  u1=2#*x : u2=-2#*y
  v1=2#*y : v2=2#*x
return
'****************************

'*************************************************
'*  Complex root seeking using Newton's method   *
'* --------------------------------------------- *
'* This routine uses the complex domain form of  *
'* Newton's method for iteratively searching     *
'* for roots. The complex function and its first *
'* partial derivatives must be available in the  *
'* form: F(Z) = U(X,Y) + I V(X,Y). The required  *
'* derivatives are DU/DX and DU/DY.              *
'* --------------------------------------------- *
'* INPUTS: initial guess, x0 and y0, convergence *
'* criteria, e, maximum number of iterations, n. *
'* OUTPUTS: approximation to the root, x and y,  *
'* number of performed iterations, k.            *
'*************************************************
2000 k=0
2100 k=k+1
  'Get u,v and the derivatives
  x=x0 : y=y0 : gosub 1000
  a=u1*u1+u2*u2
  x=x0+(v*u2-u*u1)/a
  y=y0-(v*u1+u*u2)/a
  'Check for convergence in euclidean space
  if (x0-x)*(x0-x)+(y0-y)*(y0-y)<=e*e then return
  if k>=n then return
  x0=x : y0=y
  goto 2100
return

'End of file Znewton.bas