MaplePrimes - answers and comments on Question, How to find the roots of Bessel's function between -4 and 4 using the newton raphson method in maple?

plot it

Axel Vogt — Wed, 20 Mar 2013 22:26:38 Z

g:= x -> BesselJ(0,x);
plot(g(x), x = -4 .. 4);

Use floats

Adri van der Meer — Mon, 15 Apr 2013 12:00:11 Z

The errormessage is

Error, (in newton_raphson) cannot determine if this expression is true or false: 
   0.5000000000e-5 < BesselJ(0, 2)/BesselJ(1, 2)

This means that f and D(f) are not evaluated to numeric values.
Of course you can insert some evalf(...) in the procedure, but it is easier to call it with floating point arguments:

 low:=-5.:
up:=5.:
step:=1.:
dp:=5.:

Compare the different results of

 BesselJ(0,2);
                         BesselJ(0, 2)
BesselJ(0,2.);
                          0.2238907791

addition

Adri van der Meer — Thu, 21 Mar 2013 05:18:37 Z

... and you must change

if g(a)*g(b) < 0

if evalf(g(a)*g(b)) < 0

in the loop

@Adri van der Meer Any idea what's wrong with this...

snack365 — Sun, 14 Apr 2013 16:56:15 Z

@Adri van der Meer

Any idea what's wrong with this code?

> f:=x->BesselJ(0,x);

f := x -> BesselJ(0, x)

> low:=-5;

low := -5

> up:=5;

up := 5

> step:=1;

step := 1

> dp:=5;

dp := 5

> #Newton Raphson on x-cos(x)=0
> newton_raphson:=proc(eqn,x,dp)
> local x_old, x_new, counter, not_converged:
> x_old:=x:
> x_new:=x_old-(eqn(x_old)/(D(eqn)(x_old))):
> counter:=0:
> not_converged:=abs(x_old-x_new)>5*10.0^(-(dp+1)):
> while not_converged do
> x_old:=x_new:
> x_new:=evalf(x_old-(eqn(x_old)/(D(eqn)(x_old)))):
> counter:=counter+1:
> not_converged:=abs(x_old-x_new)>5*10.0^(-(dp+1)):
> od:
> print(`The root lies at the point `,x_new);
> print(`It took `,counter,` iterations to reach this root.`);
> print(`The correct value according to Maple is`,evalf(solve(BesselJ(0,z),z)));
> end:
>
> for i from low to up-step by step do
>
> a:=i:
> b:=i+step:
>
> if evalf(f(a)*f(b)) < 0 then
>
> print(`root in interval`,a,b):
> newton_raphson(f,b,5);
>
> fi:
>
> od: