# Question:Newton-Raphson's method- how to improve the range of result interval?

## Question:Newton-Raphson's method- how to improve the range of result interval?

Let's consider the function f(x)= tan(x)- x and its root in the interval (Pi/2,3*Pi/2)

Firstly, we zoom the intersection point of the function with g(x)= tan(x) and h(x)= x ( or the function f(x) and the Ox axis):

`plot([tan(x),x], x= 4.3..4.55, gridlines=true);`

Now, we use the Newton-Raphson method to reach the root, like this:

```restart: f:= x -> x-tan(x);
a[-1]:= 0: a[0]:= 4.55:                # a[0]= 4.55 since we obtain the root is close to 4.55
N:= 0:
while abs(a[N]-a[N-1])>10^(-9) do
a[N+1]:= a[N]- f(a[N])/fdiff(f(x), x=a[N]) ;
N:=N+1 od:
print();
'N'=N;
'a[N]'= a[N];
'f(a[N])'= f(a[N]);
```

Then, of course, it certainly reach the root.

However the problem is that how can I find an interval, as large as possible, in which any starting value (i.e a[0]) generates a sequence that converges to the solution?

Is there any other theorem that improve such interval? Or we just use heudristics?

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