2
$\begingroup$

Suppose I have found an irrational number, say e^x upto m decimal places....

Now I want to find upto say m+n decimal places. Is there any way that I can use the previous value of e^x upto m decimal places to help me find the remaining n decimal places? (I can save some number, say, like remainder, when finding upto m decimal places to help me find the later digits...)

  • 0
    @user3123.. how can I continue adding terms, because the previously computed terms won`t be of the required accuracy...2011-01-22

1 Answers 1

1

In general, it depends on $x$. There are cases where knowing some digits will give you a head start at finding more digits. An example is $\sqrt{2} = e^{0.5\ln(2)}$, which is a root of $f(x) = x^2-2 = 0$. One improves the accuracy of the approximation $a_n$ by using Newton's method: a_{n+1} = a_n - \frac{f(a_n)}{f'(a_n)} = a_n - \frac{x^2-2}{2x}