Talk:Hermite interpolation

Method: simple case

the section abruptly ends without letting me know how to get the polynomial from the divided differences table. could someone who understands it add it please :)

Untitled

I think the error section could use an example, but I'm not confident that my example is correct. Here's my understanding of it:

Using the example above, this would be ${\frac {(c^{8}+1)^{(9)}}{9!}}*\prod _{i=0}^{2}(x-x_{i})^{3}$, where x is the number at which the function is approximated and c is between -1 and 1 (the lowest and highest x-values used to produce the approximation). This function's maximum for any x is the maximum error of the approximation at that x value.

Can somebody verify/correct this?

I believe that it isn't suppose to be f to the ninth power but rather the 9th derivative of f. In which case, the error is zero. As you notice the Hermite Interpolation of that function came out to the exact same solution as the original function. You aren't suppose to use this interpolation technique when evaluating an polynomial because you are using polynomials to approximate it, so it is just more work than actually evaluating the original polynomial. This function was used to demonstrate the point that you can get very high level accuracy even though you are only using 3 points because of the extra information the derivatives provide. 159.242.239.116 (talk) 14:33, 5 November 2008 (UTC)

Wiki Education Foundation-supported course assignment

This article was the subject of a Wiki Education Foundation-supported course assignment, between 26 May 2020 and 3 July 2020. Further details are available on the course page. Student editor(s): Yifeng Li.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 23:23, 16 January 2022 (UTC)

Article Dependancy

This article can't be understood without reading the Newton polynomial article —Preceding unsigned comment added by Arbitrary18 (talk • contribs) 04:17, 29 September 2008 (UTC)

I've expanded the introduction and usage sections to hopefully clear things up and make it explicit where Newton polynomials are required. I also redid the example table showing how the first few values are obtained, so by studying that you can get a working knowledge of how the algorithm work. - from Andrew Poelstra 22:21, 21 October 2009 (UTC)

Todo at this point would be clarifying how general the interpolation is - my additions work mainly with the first derivative and don't go into detail about how to work with highers. Also, building Hermite polynomials from Larange polynomials needs to be explained. Those two things should eliminate the article dependency. - from Andrew Poelstra 22:21, 21 October 2009 (UTC) —Preceding unsigned comment added by Apoelstra (talk • contribs)

Cleaned up document so that interpolating higher derivatives is clear. Also fixed the error formula to make sense (the old one was impossible to read) and to match the one given in Faires. I removed the "may be confusing" warning because the article is understandable now. Todo: write another example (of a non-polynomial; say $e^{(}-x^{2})$ for $x\in [-1,2]$, and prove the error bound. Also, if anyone knows a better textbook, that would be awesome. Faires is awful. — Preceding unsigned comment added by Apoelstra (talk • contribs) 21:28, 30 December 2010 (UTC)

Ruby implementation doesn't work

I c&p'ed the algorithm and tried it with

x = [ -1, 0, 0, 0, 1]
y = [ 0, 1, 0, -4/2, 0]

Output is

0 1 -1 -1 2

I used these coefficients to compute the interpolation at x=1. The result is 2, which is obviously wrong. (Last coefficient should be 1)

--Pberndt (talk) 16:13, 28 May 2009 (UTC)

Here's an working version. Unfortunately I don't understand Hungarian and I don't understand the difference between the scripts (yet).

http://www.mathworks.com/matlabcentral/fileexchange/14353 --91.5.211.65 (talk) 11:07, 31 May 2009 (UTC)

I created a working example for German wikipedia (Might be moved to de:Neville-Aitken-Schema. Translation help:

• anfangsIndex means start index (If $x_{i}=x_{n}=x_{k},i\leq n\leq k$, to calculate $[x_{n}..x_{k}]$ you have to calculate $[x_{i}..x_{k-i}]$ instead)
• funktionsWert is the value of f (or f' f..)
• z contains basic function values, so if $f'(x_{i})=y_{i}$, $z_{i}=f(x_{i})$ holds.

--Pberndt (talk) 19:59, 31 May 2009 (UTC)

See [1]. -- Pberndt (talk) 14:02, 12 July 2009 (UTC)

Ouch. Without going (further) into original research, can you try this?

def min(a,b)
  if a<b then a else b end
end

def hermite(x, y)
  base = Array.new(y.length)
  q = Array.new(y.length)
  base[0] = 0
  q[0] = y[0]
  1.upto(y.length-1) do |j|
    base[j] = (x[j] == x[j-1]) ? base[j-1] : j   # point to corresponding y value
    q[j] = y[base[j]]                                         # fill with y value
  end

  1.upto(y.length-1) do |i|
    qq = q[i-1]           # in the loop below qq = q[j-1] from previous iteration
    i.upto(y.length-1) do |j|
      if x[j-i] == x[j] then
        qq, q[j] = q[j], y[min(j,base[j]+i)]   # do not go beyond i-th derivative
      else
        qq, q[j] = q[j], (q[j]-qq) / (x[j]-x[j-i])
      end
    end
  end
  [x,q]
end

def poly(x, hermite)
  xi, q = hermite
  y = q[0]
  c = 1
  1.upto(q.length-1) do |i|
    c *= x - xi[i-1]
    y += q[i]*c
  end
  y
end

I had placed the program in the article in good faith as a helper to decypher the tableau, in case anyone who wanted to improve the article, but maybe it's better left here (and the code above communicates the intention much better anyway). Balabiot (talk) 16:50, 18 August 2009 (UTC)

Works for me. You could place your code in the "Algorithm Implementation" wikibook and put a link into the article. --Pberndt (talk) 13:38, 31 October 2009 (UTC)