6.2.2-Numerical Integration: Romberg Integration and Richardson's Extrapolation

These videos were created to accompany a university course, Numerical Methods for Engineers, taught Spring 2013. The text used in the course was "Numerical Methods for Engineers, 6th ed." by Steven Chapra and Raymond Canale.

[@mechwarreir2]

Oh my god you explained this soooo much better than my boring engineer teacher. He took 2 hours of taylor series crap and missed the whole point. Props, and thanks!

[@davidboozer2410]

At about 4:20, it should be pointed out he DOUBLED the number of function evaluations. The way he said it in the video sounded like he only needed 2 more evaluations to go from O(h2) to O(h4). It's a true statement provided you've only taken 2 evaluations to start with, but it's better to be more general when teaching.

[@SunFilmsProduction]

4:49 General Romberg formula should have I(j+1),(k-1) not I(j+1),(k+1) as written in the video

[@chrainy]

not clear w.r.t I_{j,k}. You did not explain what I_{j,k} is so it is difficult to understand the table.

[@tomiadventures]

Yes, as Jacob mentioned in a comment below, it should be k-1 instead of k+1 (around 4:30-5:30 time mark).

[@qsdfcvgyjmkl]

At 5:10, the first integration of the general equation is supposed to be I_(j+1, k-1), not I_(j+1, k+1)

[@SEBA149]

thanks for the lessons, it's helping me a lot. What do j and k stand for in the general Romberg algorithm?

[@umutsim]

Thanks for the video. I have one comment on the audio quality. I think you should lower the treble of your recordings, it's a little bit disturbing if you watch the video with a headphone.

[@rafaelortega1376]

Thank you for the video. I think you should explain in depth the meaning of the subindexes. I figured it out after a few minutes and the comments confirmed my suspicions but I think that would improve dramatically the understanding of the video. Anyhow, thank you very much.

[@AcademiaCS1]

It was awesome!! Thanks a lot

[@Exorcisto1]

Thank you so much you that was really helpful

[@jafarmat22]

i think your general formula is wrong...... At 4:29 you formula says something like this : I(1,2) = [ 4 I(2,1) - I(1,1) ] / 3 , there is no I(1,2) , maybe you mean I(2,2) , because we're moving forward in the algorithm :)

[@kvyi]

Thanks for the comment. I just went back over this, and there is an error, but it's not what you might think. The formula I(1,2) = [ 4 * I(2,1) - I(1,1) ] / 3 is actually correct. As I mentioned in my previous comment, it's important to keep in mind what we mean by j and k. I'm not sure why you think there is no I(1,2). I(1,2) refers to the integral approximation accurate to O(h^4) for one interval segment. This is equal to a weighted combination of two separate O(h^2) approximations (each using the trapezoid rule): one assuming a single trapezoid for the entire region [ I(1,1) ] and another assuming two trapezoids for the region [ I(2,1) ].
The error in the general formula is on the second index of the first I in the numerator. It should be k-1 not k+1. The general formula is thus:
I(j,k) =~ [ 4^(k-1)*I(j+1,k-1) - I(j,k-1) ] / [ 4^(k-1) - 1]. This is given correctly on p. 636 of Chapra & Canale.

[@kikokimo2]

thought the same as @trollLoki. Thought they were just indexes or sthing.. Thx for explanation

[@kikokimo2]

@jacob 1st question please: 1) Do you mean, I(1,1) is like using a single trapezoid rule on the whole region, I(2,1) is like using a "composite" trapezoid on the region, and I(1,2) is like using two (normal / single) trapezoid-rules on the whole region? 2nd question: 2) What is more accurate, 2 "normal" (single) trapezoid-rule on 1 region, or a "composite" trapezoid rule? Thanks in advance!

[@palashkhanra7816]

5:58 , sir what is the name of that book ?

[@anthroporraistes_]

"Numerical Methods for Engineers, 6th ed." by Steven Chapra and Raymond Canale.

[@danielgarnando8254]

gokil