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Re: "Cascaded sums"?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg59196] Re: [mg59190] "Cascaded sums"?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Tue, 2 Aug 2005 00:42:21 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 1 Aug 2005, at 07:05, AES wrote:

> A Mathematic calculation is yielding a result of the form
>
>    (x^4 + a (x^3 + a (x^2 + a ( x + a ))))
>
> for the case n = 4, and analogous results for larger n; and  
> Mathematica
> seems to be able to do some interesting simplifications and  
> factoring of
> these results.
>
> I'm sure I learned about this form in a math class somewhere along the
> line, but this was 50+ years ago.  Could some kind soul remind me of
> what this form is called, and maybe some of its interesting  
> properties?
>
> Thanks . . .
>

This is known as the Horner method of evaluating a polynomial.

<< Algebra`Horner`


Horner[a^4 + x*a^3 + x^2*a^2 + x^3*a + x^4, a]


x^4 + a*(x^3 + a*(x^2 + a*(a + x)))

The main point is that to evaluate a polynomial expressed in Horner  
form takes fewer multiplications hence for computers this method is  
more efficient (computers are much faster at adding numbers than at  
multiplying). The Horner method is also the basis of an obvious but   
efficient numerical procedure called deflation for  computing the  
remaining roots of a polynomial once you have evaluated one of them  
by some approximate method (e.g. Newton's method).

Andrzej Kozlowski



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