Talk:Newton polynomial

message: Can we get an algorithm here?  —Preceding unsigned comment added by 24.199.203.189 (talk) 20:57, 17 February 2008 (UTC)

message: User http://en.wikipedia.org/wiki/Special:Contributions/P000010110 deleted some Octave code on this page a while ago. The reason was that wikipedia is not a programming tutorial. However, for computer scientists learning about this topic, this kind of contributions greatly increase the understanding in the topic. We are used to program every day and some programming code will help us to understand the topic much faster than mathematical symbols.  — Preceding unsigned comment added by 91.176.97.127 (talk) 12:15, 4 August 2012 (UTC)
thread_title:  Adding programming example 

message: It's probably worth pointing out that the example provided is only worked out to very low precision.  If someone takes that example and works it out on a computer with double precision (or probably even single precision), they will quickly see that the constant and quadratic terms in the interpolating polynomial are 0.  Intuitively, this should make sense because the tangent function is odd and the data points are symmetric about x = 0.  —Preceding unsigned comment added by 151.200.114.163 (talk) 21:15, 24 October 2007 (UTC)
replies: text: Right. I added a comment that the computation uses only six digits. -- Jitse Niesen (talk) 12:08, 1 April 2008 (UTC)

text: If the interpolation points are close together, then the coefficient for x^1 is indeed close to 1. However, here the space between the points is fairly wide so there is a large error. I computed the interpolating polynomial myself to check and the end result is correct (to the limited precision of the computation, of course). -- Jitse Niesen (talk) 12:08, 1 April 2008 (UTC)
thread_title:  Numerical Precision in Example 

message: Contrary to frequent claims, Lagrange interpolation (in a "barycentric" form) _does_ allow calculation of the next degree of polynomial interpolation without re-doing previous calculations. This requires recording the value of each term in the calculation.
Though it's often said that Lagrange interpolation isn't used as a practical method, I've read that some _do_ use it for practical interpolation, and some advocate it for that purpose. Speaking for myself, I've used Lagrange for problems in which the interpolation is always at the same X-value, with only the y-values changing, and in which it's known in advance, from experience, how many terms are needed for sufficient accuracy. 
Of course, when it it isn't known how many terms will be needed for sufficient accuracy, divided-difference methods are the ones of choice.
A frequent use of polynomial interpolation is to determine whether or not linear interpolation will be accurate enough.  ...by evaluating the quadratic term of a divided-difference method (to find out if it's big enough to matter in the problem). Of course only a divided-difference method can be used for that determination.
Among the divided-difference formulas, Netwon's usually isn't as practical as those of Stirling, Bessel and Gauss, because the more points it adds at one end, the farther the center of the points moves. Since, by assumption, you don't know in advance how many points you'll need, you also don't know where their center will eventually be. If the center turns out to be far from the evaluated point, then accuracy will be lost, because polynomial interpolation's accuracy is best when the interpolated point is near the middle of the data points being used. Therefore, Newton's formula is of interest mainly as the finite-differences version of Taylor's polynomial, from which the other divided-difference methods can be derived.  —Preceding unsigned comment added by JoinedOct1309 (talk • contribs) 01:39, 6 November 2009 (UTC)
thread_title:  Comments on article 

message: On 1.1/2, "Newton forward/backward divided difference formula", what is the justification for using binomial coefficients, when "s" is, in general, not an integer? Pasmao (talk) 05:12, 30 April 2022 (UTC)
thread_title:  Discussion 

message: The definition section covers more than the definition and is certainly too long. The long text part following the formulas needs to be structured!82.130.71.230 (talk) 19:28, 1 February 2010 (UTC)
thread_title:  Definition section too long 

message: My algorithm is more compact
function [fp,ko] = newton_interpolation(TA)
n = size(TA);
n = n(2);
FX = zeros(n,n);
FX(1,:) = TA(2,:);
H=TA(2,1);
syms x;
k1=n;
for i = 2:n
    for j = 1:n+1-i
        FX(i,j) = (FX(i-1,j+1)-FX(i-1,j))/(TA(1,j+i-1)-TA(1,j));
        k1 = k1 + 1;
    end
    k = 1;
    for j = 1:i-1
        k = k * (x - TA(1, j));
    end
    H = H + k * FX(i,1);
end
y = simplify(H);
fp = sym2poly(y);
ko = k1;
--Ignotusp (talk) 21:40, 20 December 2010 (UTC)
thread_title:  Matlab algorithm 

message: A proof of the recursion formula for the coefficient might make this article complete. As with other Wikipedia articles I was thinking the proof can be hidden. Here's an elegant proof I found: http://www.maths.lancs.ac.uk/~gilbert/m243a/node6.html — Preceding unsigned comment added by 128.111.185.31 (talk) 23:14, 12 September 2011 (UTC)
thread_title:  Proof 

message: At the top of the article, it says:
"This article needs additional citations for verification. (September 2014")
Answer: Yes, when I initially posted my material here, I didn't realize the need to cite my source. But recently I've added a citation of the source of the information that I posted (at the place where "citation needed" was written).
"This article is written like a personal reflection or opinion essay that states a Wikipedia editor's personal feelings about a topic."
Answer: My material here, about the differences among, relative advantages of, and particular uses of, various polynomial interpolation formulas, consists of objective facts supported by the source that I cited (Numerical methods for Scientists and Engineers, by R.W. Hamming).
It isn't a personal reflection, opinion essay, or expression of personal feelings.  — Preceding unsigned comment added by 68.186.18.145 (talk) 18:32, 30 April 2018 (UTC)
thread_title:  Answers to template messages at top of article