Finding e^x with more number of significant digits using a previously computed value of the same (having lesser number of significant digits)

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...)

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Possible duplicate: math.stackexchange.com/questions/5420/… – Hans Lundmark Jan 22 '11 at 11:21
It probably depends on how you calculated the first $m$ decimal places. If you did it using the series you could just continue adding smaller terms to increase your accuracy. – Listing Jan 22 '11 at 11:39
@user3123.. how can I continue adding terms, because the previously computed terms won`t be of the required accuracy... – Guanidene Jan 22 '11 at 11:41

1 Answer

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: $$a_{n+1} = a_n - \frac{f(a_n)}{f'(a_n)} = a_n - \frac{x^2-2}{2x}$$

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