> NewtonItr := proc()

#Here the convergence criterion is much more severe than usual. See comments.

local F,DF, p,deri, f1,d1, f2,d2, r,eps;

printf(`Enter a function:\n`);

F :=scanf("%a")[1];

DF := unapply(diff(F,x),x);

F := unapply(F,x);

printf(`Enter an initial point:\n`);

p := scanf(`%f`)[1];

deri := DF(p);

if abs(deri)< eps then printf(" deri=%6.3f\n",deri); RETURN(false); fi;

f1 := F(p);

d1 := -f1/deri;

printf("f(%20.15f)=%20.15f, deri=%6.3f, dx=%20.15f\n",

p,f1,deri,d1);

p := p+d1;

r := 0.5; #smaller r suggests higher threshold

eps := 10^(-7);

do

f2 := F(p);

if abs(f2/f1)< r then #function values tend to decrease

deri := DF(p);

if abs(deri)< eps then printf(" deri=%6.3f\n",deri); RETURN(false); fi;

d2 := -f2/deri; #iterator

printf("f(%20.15f)=%20.15f, deri=%6.3f, dx=%20.15f\n",

p,f2,deri,d2);

p := p+d2;

if abs(d2^2/(d1-d2))< eps then #d2 is much smaller than d1

printf("f(%20.15f)=%20.15f, deri=%6.3f, dx=%20.15f\n",

p,F(p),DF(p),-F(p)/DF(p));

printf("Converge to approximate root x=%20.15f\n",p);

RETURN(true); #success exit

else

f1:=f2; d1:=d2;

fi;

else

printf("f(%20.15f)=%20.15f, deri=%6.3f, dx=%20.15f\n",

p,f2,DF(p),-F(p)/DF(p));

printf( " f1 : f2 = %f : %f\n",f1,f2);

RETURN(false); #failure exit

fi;

od;

end: