R help - Jan 2010 - solving cubic/quartic equations non-iteratively

To R-helpers,

R offers the polyroot function for solving mentioned equations 
iteratively.

However, Dr Math and Mathworld (and other places) show in detail how to
solve mentioned equations non-iteratively.

Do implementations for R that are non-iterative and that solve mentioned 
equations exists?

Regards, Mads Jeppe

There are certainly formulas for solving polynomials numerically up to 4th
degree non-iteratively, but you will almost certainly get better results
using iterative methods.

Even the much more trivial formula for the 2nd degree (quadratic) is tricky
to implement correctly and accurately, see:

* George Forsythe, How do you solve a quadratic equation?
* Yves Nievergelt, How (Not) to Solve Quadratic Equations

Hope this helps.

           -s

On Tue, Jan 5, 2010 at 10:11 AM, Mads Jeppe Tarp-Johansen <
s02mjtj@math.ku.dk> wrote:
> To R-helpers,
>
> R offers the polyroot function for solving mentioned equations iteratively.
>
> However, Dr Math and Mathworld (and other places) show in detail how to
> solve mentioned equations non-iteratively.
>
> Do implementations for R that are non-iterative and that solve mentioned
> equations exists?
>
> Regards, Mads Jeppe
>
> ______________________________________________
> R-help@r-project.org mailing list
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> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
	[[alternative HTML version deleted]]

Mads Jeppe Tarp-Johansen wrote:
> To R-helpers,
> 
> R offers the polyroot function for solving mentioned equations iteratively.
> 
> However, Dr Math and Mathworld (and other places) show in detail how to
> solve mentioned equations non-iteratively.
> 
> Do implementations for R that are non-iterative and that solve mentioned
> equations exists?
As far as I know, we don't even have the quadratic...

They can't be hard to implement, though. However, you may need to take
care that non-iterative solutions are not necessarily more precise or
even faster than iterative ones. There may be cancellation issues and
the closed-form expressions can be complicated and involve slow function
calls.


-- 
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quote:
There are certainly formulas for solving polynomials numerically up to 
4th degree non-iteratively, but you will almost certainly get better 
results using iterative methods.
endquote

I must be missing something here.  Why not use the analytic formulas for 
polynomials below 5th degree?  Once you do so, your answer is as precise 
as the level of precision you enter for the coefficients.

Try,

library(polynom)
p <- polynomial(c(-8,14,-7,1))
solve(p)


Kasper