Re: Integer-Valued Polynomials
In article
<852523.1141159679225.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Maury Barbato <mauriziobarbato@xxxxxxxx> wrote:
Maury wrote:
Two remarks:
(I)Newton method works also for several
variables and for general fields?
There is a risk of confusion in terminology here. I
mentioned (1) interpolation and (2) Newton divided
differences (as a specific approach to finding an
interpolating polynomial). Newton's method often
refers to root finding, and while a multivariate form
of this (usually called Newton-Raphson) exists, it
has little to do with the question of interpolation.
Interpolating polynomials in many variables, versus
in one variable, does introduce a complication of
choosing interpolating points not "correlated" by
a polynomial of smaller degree. See this easily
read survey-style article for more information:
[Polynomial interpolation in several variables -
by Tomas Sauer]
http://www.uni-giessen.de/tomas.sauer/Publ/MAIA.pdf
(II) Even if it works, it can't be very useful. How
could
you apply it, to use later the "Identity
Polynomials
Principle"?
But your argument intended maybe to be only
intuitive.
My "argument" is an outline, but if you have specific
questions perhaps I can fill in more details.
regards, chip
The point is: let P(x_1,...,x_n) be a complex polynomial
with integer values when (x_1,...,x_n) is in Z^n. How do
you construct a rational polynomial Q(x_1,...,x_n) wich
agrees with P on Z^n?
It seems to me a weak point in your argument.
Thanks again for your hope and your interest.
My Best Regards,
Maury
Any polynomial function, as long as the number of variables and degree
are both finite, as usual for polynomials, can be totally determined by
its values at a sufficient number of points.
If there is solution at all, it is merely a matter of solving a
suitable system of linear equations with the polynomial's coefficients
as unknowns.
.
Relevant Pages
- Re: Secant method vs Newtons method
... I am presenting on my web site the NIH method, a natural generalization to higher order of the bracketed Secant Method for root finding, where the Secant Method occurs naturally as the zero order case. ... My main example is the computation of the roots of all the Legendre polynomials up to a certain given order which is a nice example which suits my purposes. ... I have chosen to call them Normalized Hermite Polynomails, but am open to suggstions and maybe Two-Point Normalized Hermite [Interpolation Polynomials] would be better. ... The two root finding methods which I am presenting use these same polynomials to interpolate the inverse of a given function (after transforming the derivatives according to the rules). ...
(sci.math.num-analysis) - Re: Secant method vs Newtons method
... It is found that the Secant Method is obtained using zero order NIH and higher order methods with increasingly higher order error terms are obtained by using First or Second Order NIH. ... The simple innovation which I am using in NIH, and which is indicated by the word Normalized, is that I am using for interpolation the two point Hermite Interpolating Polynomial interpolating the function and some of its derivatives on a normalized interval by simply mapping the bracketing interval to the normalized interval each iteration. ... This allows use of the same interpolation formula every step of the iterative search for the root by evaluating the root of the Hermite polynomial interpolating the inverse function and its first or even second derivatives on two points within the normalized interval. ... I have described this on my website www.numerical-algorithms.com under the section "Root Finding Using Inverse Normalized Hermite Interpolation" and present an example where I compute the roots of the Legendre polynomials. ...
(sci.math.num-analysis) - Re: Integer-Valued Polynomials
... variables and for general fields? ... has little to do with the question of interpolation. ... Interpolating polynomials in many variables, ... A tensor product of nonsingular ...
(sci.math) - Re: Secant method vs Newtons method
... The natural generalization of the Secant Method to higher order is ... obtained by using Normalized Inverse Hermite interpolation. ... Interpolation" and present an example where I compute the roots of the ... " It is therefore easier to compute the roots of all polynomials up ...
(sci.math.num-analysis) - Re: interpolation accuracy, oversampling and fractional interpolation
... by Vesma Jussi? ... concept of interpolating using a variety of fitted polynomials is not ... interpolation of segments of your Parks-McClellan kernel (instead ...
(comp.dsp) |
|
