Newton Interpolation and Divided Differences

In this video, we introduce the Newton Interpolation method and Divided Differences. We start with the general concept, then the recurrence relation and the divided difference table. Finally, we run through a quick example in order to understand how the method is used in practice.

Chapters
0:00 - Introduction
05:40 - The Recurrence Relation
10:56 - The Divide Difference Table
12:48 - Example

Errata and Clarifications:
- at 9:00, the numerator should read (x_k - x_0)qn(x_k) + (x_n+1 - x_k)pn(x_k)

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In-video References:

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Thank you so much, I hope you find the content useful.

[amansartaz]

This is my first video of yours and you got my subs ! Nice explaination.

[tzwtxcs]

Thanks for high-level explantion, examples!

[avyakthaachar.]

Amazing explanation and proof, just what I needed, thank you so much

[rishadislam]

Thanks for this great video! Would really appreciate if you made new videos on forward, backward and centererd difference based off of divided differnce formula.

[duartecastro]

Great explanation! Thanks for the help

[dvir-ross]

Great video! This is high level but understandable. Love it.

[avocadolover]

thank you so much sir, well comprehensive

[armanboss]

sir you are lifesaver for my upcoming semesters

[twomoonsinthesky_]

Good explanation, thanks to you I was able to attain a better understanding of the Newton Interpolation and the idea of divided differences. I still don't understand the correlation between divided differences and writing them as integrals, which is what I need for proving the Hermite relation; but I understand this is not part of this video.

[oraviram]

Before I forget: small error at 9:15, in the second line, it's p_n that should be multiplied by x_n+1, and q_n is multiplied by x_0. The signs should stay as they are. This doesn't really matter since q_n and p_n agree at x_k, but it may still be a good idea to add it to the errata.

Anyway, the main reason I came here is to say thanks and that this was a great video. My university does a really bad job with numerical analysis. Most courses here are taught with the university's own books, and they are very high quality. First year courses like Calculus 1 and Linear Algebra 1 set very high standards for rigor, both using precise definitions and giving complete proofs of everything. Last semester I took set theory which basically completed the hole in my knowledge by giving an axiomatic foundation of math.
Now I am taking numerical analysis and it's like all the rigor is thrown into the trash. We are using Burden's book and it's hard to even call it a math book. Definitions are not consistent, it contradicts itself, the proofs are sloppy and sometimes incorrect, statements of theorems often require extra conditions which are not mentioned, and theorems are applied even when not all the conditions are met. The worst thing, though, is their favorite proof method: do a few examples and notice the pattern, hope it applies forever.

Sorry this became a rant about the book, but I just want to make it clear how this video saved me.
The way divided differences were introduced in our book was...stupid: we already studied interpolation polynomials (with that they did an alright job), so they went like "let's try to present them in this form". They calculated the first two coefficients, introduced divided differences using the recurrence relation, and said "as you may guess, the nth coefficient is f[x0, ..., xn]". No proof no nothing, they just moved on like nothing.
It's fine if they did that if the proof was trivial, but like, it's far from it???
Even on Wikipedia I could barely get much help, but then your video introduced it in like the most perfect way it could. Defining the divided differences as the coefficients of the interpolation polynomials in terms of the newton polynomial basis makes more sense than an out-of-the-blue recurrence relation (and it's also easy to make rigorous using linear algebra). After doing that you just gave a simple proof which makes you understand why the recurrence relation makes sense.
I just don't get why it was omitted from the course I am taking. It's far from trivial, and it's not that complicated, but our book just decided to brush it under the rug and hope we take its word for it.

Alright that's it, sorry for the rant, and thanks for putting out this video.

[mohamadkhanafer]

thank you, much better than my teacher's 4hours useless lecture.

[pauselab]

there are even cooler things about newton polynomials. they are in some sense the discrete counterparts to taylor series and have relations with the calculus of discrete difference where you just have to adjust the divided difference a little.

[kalinkochnev]

At 10:54, I am having a hard time following where x^n+1 comes from

[QT-ytdb]

At 6:34, I seemed to have lost of what is the purpose of proof. First, what's the definition of f[x0, x1, ..., xn+1]? seems the proof is to prove the definition. But a definiton is just definition and it needs not to be proved, unless there is another definition of f[x0, x1, ..., xn+1] in the first place.

[pydblyn]

how can we get the equation at 9:16? seems like it has to be Xk (Qn (Xk)+Pn (Xk)) - (X0Qn(Xk)+Xn+1Pn(Xk))

[casperspook]

Great content! Considering extrapolation rather than interpolation, would this newton polynomial work better, worse or identically to a taylor polynomial of the same order whose coefficients are determined by finite difference schemes?

[kalkhasse]

Thanks for awesome videos!
Does this method give the same polynomials as Lagrange interpolation? And if so, when would you use one method over the other?

[azzabadawy]

When can i using only Newton's method?

[mikewood]

Hey, Wonderful Video! Really I was struggling to understand this and now its so simple. Can you make a video on Radial Basis function interpolation or Kernel Interpolation? I am really working on some stuffs and I need it. Also What is the difference between Lagrange's and Newton Interpolation? When to use which? Also why can't we use Taylor series for polynomial approximation? Thank you!

[linco]

For a second I thought the music in the background was L’s music from death note. Tbh I could put that behind any math video lol