(*Newton iteration ... *)

dig:=50.

APSPRE(dig). 

AbsErr:=APFRN(RNRED(1,IEXP(10,dig/2))).
MaxIter:=100.


PROCEDURE Newton(f,fp,x);
(*Newton iteration. f and fp are functions.
x is the starting value for iteration. A fix point
of x-f(x)/fp(x) is returned. *)
VAR   i, y, z, zp, w: ANY;
BEGIN 
(*1*) i:=0; y:=x;
(*2*) WHILE i < MaxIter DO i:=i+1;
            z:=f(y); zp:=fp(y);
            IF APCMPR(APABS(zp),AbsErr) <= 0 THEN
               CLOUT("Derivation becommes zero."); BLINES(1);
               RETURN(y) END;
            w:=APQ(z,zp); y:=APDIFF(y,w);
            IF APCMPR(APABS(z),AbsErr) <= 0 THEN
               RETURN(y) END;
            END;
      CLOUT("Maximal number of iterations reached."); BLINES(1);
      RETURN(y);
(*9*) END Newton. 


zwei:=APFINT(2).

PROCEDURE E(x);
(*Expression function. An expression is evaluated at x. *)
VAR   y: ANY;
BEGIN 
(*1*) y:=APEXP(x,2); y:=APDIFF(y,zwei);
      (*RETURN(y);*)
(*9*) END E. 


PROCEDURE Ep(x);
(*Expression function derivation. The derivation of
an expression is evaluated at x. *)
VAR   y: ANY;
BEGIN 
(*1*) y:=APPROD(x,zwei);
      (*RETURN(y);*)
(*9*) END Ep. 

start:=APFINT(1).

b:=Newton(E,Ep,start).
BEGIN CLOUT("AbsErr = "); APWRIT(AbsErr); BLINES(0) END. 
BEGIN CLOUT("Result = "); APWRIT(b); BLINES(0) END. 
BEGIN CLOUT("W2     = "); APWRIT(APROOT(zwei,2)); BLINES(0) END.