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I know Graeffe's method to approximate roots is great, but people always find troubles when they convert the root in n-th-root form into base 10 form by hand, for example, a large number in 64-th-root.

What is a more modern way of getting numerical solution/roots of polynomial in base 10 than this by hand? I prefer a method that you could control the accuracy in terms of significant digit

Graeffe's method -

Now a bounty is offered to address all the concerns.

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There are a lot of modern methods (in this respect, Graffe root-squaring is relatively ancient, but there have been improvements even to it, such as the one by Sebastião e Silva), but I don't believe they're that easily done by hand. On the other hand, if you have some amount of stamina, I think the quotient-difference (QD) algorithm might be something you can use manually. It's quite slow to converge, so one often uses Newton-Raphson afterwards once the results from QD start to agree to a few or so digits. (QD is definitely a more recent development than root squaring, though!) – J. M. Apr 25 '12 at 16:35

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