Proof for interpolating polynomial and error approximation

[bremenfallturm]

TL;DR

I am looking to find a source that proves (or gives insight) that the error of an interpolating polynomial at a point $x$ can be approximated by the formula $E \approx \left| p_{n+1}(x)-p_n(x)\right|$ given $n+2$ datapoints.

Hello!

I took an introductory course in numerical analysis earlier this year. I feel like I did not get insight into all the material, especially the material about error approximation.

The lecture notes I saved from the course state that the (absolute) error of an interpolating polynomial over a set of datapoints can be approximated by fitting a polynomial of one degree higher. That's how I have always calculated the error - but I cannot find a source that provides a proof (or insight!) to this fact. I've looked through various sources online.

That is, what I'm looking for a proof of (or where I can find the proof):

The error of an interpolating polynomial at a point $x$ can be approximated by the formula $E \approx \left| p_{n+1}(x)-p_n(x)\right|$ (given $n+2$ datapoints)

I am (literally) in a cottage right now so I don't have access to pop by a library and browse through all the numerical analysis book, so thought I'd head online and ask where I can find the proof.

 

[fresh_42]

Well, let's see if my standard recommendation works: google "interpolation polynomials + error + pdf".

Besides some interesting short papers with formulas and a PowerPoint presentation, I found
https://faculty.sites.iastate.edu/jia/files/inline-files/interpolate.pdf

You can check by yourself whether there are more suitable results for your purposes. The keyword "error" restricted the search results significantly. I only added it because you asked particularly about it. I would have searched more generally for "interpolation polynomials + pdf" to find complete lecture notes on the subject and then looked inside them about the considerations of error margins. The "+ pdf" part is important for finding university servers, not private homepages.

 

[FactChecker]

error of an interpolating polynomial over a set of datapoints can be approximated by fitting a polynomial of one degree higher. That's how I have always calculated the error - but I cannot find a source that provides a proof (or insight!) to this fact. I've looked through various sources online.

I don't think that is true. Consider the points from $y=A\sin(x),\ x_i = \pi i,\ i=0,m$. An interpolating polynomial $p_{m+1} \equiv 0$ would have errors up to $|A|$.

 

[fresh_42]

I don't think that is true. Consider the points from $y=A\sin(x),\ x_i = \pi i,\ i=0,m$. An interpolating polynomial $p_{m+1} \equiv 0$ would have errors up to $|A|$.

The link I quoted has a precise formula.

 

[FactChecker]

The link I quoted has a precise formula.

Right. The formula depends on the existence and bounds on the (n+1)st derivative. Maybe the OP omitted details of the question.

 

[bremenfallturm]

Well, let's see if my standard recommendation works: google "interpolation polynomials + error + pdf".

Haha, I wouldn't have asked here before trying to Google myself. But apparently, I need to practice Googling, it looks like the PDF you linked contains what I want - so thank you!
I'll read through the proof and come back here if I have any questions!

The formula depends on the existence and bounds on the (n+1)st derivative. Maybe the OP omitted details of the question.

Yes, I was being informal in my OP. Sorry about that :)

 

[fresh_42]

Haha, I wouldn't have asked here before trying to Google myself. But apparently, I need to practice Googling,

The "... + pdf" is the essential trick! Lecture notes on university servers are usually .PDF, some boring notes on someone's private homepage are just in .HTML. If you're unlucky you catch a .PPT. The first part of the keyword can be adjusted by words like "introduction", "lecture notes" or with words in your native language. That regularly led me to university servers and profound information.