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Topic: cube root of a given number
Replies: 112   Last Post: Jan 10, 2013 1:39 PM

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 Iain Davidson Posts: 1,173 Registered: 12/12/04
Re: cube root of a given number
Posted: Jul 22, 2007 7:29 AM

On 22 Jul, 04:28, arithmonic <djes...@gmail.com> wrote:
> On 21 jul, 17:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
> wrote:
>

> > On 21 Jul, 21:38, arithmonic <djes...@gmail.com> wrote:
>
> > > On 16 jul, 04:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
>
> 1.- I challenge you to show such Eshbach's method in this thread,
> because both of you are trying to state that my methods --based on the
> Rational Mean-- are the same as the one you read in Eshbach's
> work ("Handbook of Engineering Fundamentals).

I'm sure the poster knows what he read.

> 2.- I challenged you in this posting to show to the sci.math audience
> a very simple numerical example on your alleged GENERAL Hurwitz's
> ROOT-SOLVING METHOD for computing, say, THE FIFHT ROOT OF 2.

There are 5 fifth roots of 2. Which one do you want ?
Are you saying you can find complex roots too ?

Finding the real root of x^5-2 =0 is simple.
s(x,y) is the sign of binary quntic x^5-2y^5

The root must lie between (0,1) = 0 and (1,0) = "inf"
calculate s(0,1) and s(1,0). Form the mediant (1,1)
At any stage in the process s(xn,yn) will be 1 or -1.
if s(xn,yn) = un The new new mediant is formed with
(xk,yk) where k is the largest index <n such that un*uk = -1

0 1 -1
1 0 1
1 1 -1
2 1 1
3 2 1
4 3 1
5 4 1
6 5 1
7 6 1
8 7 -1
15 13 1
23 20 1
31 27 -1
54 47 1
85 74 -1
139 121 1
224 195 1
309 269 1
394 343 -1
703 612 -1
1012 881 -1
1321 1150 -1
1630 1419 -1
1939 1688 -1
2248 1957 -1
2557 2226 -1
2866 2495 -1
3175 2764 -1
3484 3033 -1
3793 3302 -1
4102 3571 -1
4411 3840 -1
4720 4109 -1
5029 4378 -1
5338 4647 -1
5647 4916 -1
5956 5185 -1
6265 5454 -1
6574 5723 -1
6883 5992 -1
7192 6261 -1
7501 6530 -1
7810 6799 -1
8119 7068 1
15929 13867 -1
24048 20935 -1
32167 28003 -1
40286 35071 -1
48405 42139 1
88691 77210 1
128977 112281 1
169263 147352 -1
298240 259633 -1

Can you solve x^3 -2x^2 -x+1 =0 ?

> Notice that I am challenging you and your friend Grover Hughes with two very simple inquires.

I don't know why you think Grover Hughes is a friend of mine.

You used to claim that you could find the best simultaneous
approximations to cubrt(2), cubrt(4).
As you methods are so revolutionary, I would have thought
this would have been possible too.

In fact, you can use your methods for simultaneous approximation of
cubrt(2), cubrt(4), but it would be
sheer luck if a best approximation was found.
Shall I give you a hint ?

Do you understand the diffference between fast convergence and best
rational approximation ?

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