5.2.7-Curve Fitting: Spline Interpolation

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.

[@silverKirilljedi]

This video was recorded 9 years ago, but here am I in 2022, happily watching these awesome videos that Jacob recorded and planning to apply it in my work 😄

[@nurmuhammadsumon2729]

Now 2023 😊

[@dieminestinkt]

ty. The only one on here who could explain it nice and simple

[@ImanMatar]

I don't know if it has a sequel, but considering this video alone, it doesn't have an in-depth analysis for those having a test in numerical computing in few hours. It's just a nice clean overview on what to expect. Thank you for your effort

[@kvyi]

+Iman Matar Thanks for the comment. It's part of a series of videos, which are organized into a playlist, but also organized by number. This is the seventh video in the second section of the fifth part of the video series...which is why it's labeled 5.2.7. I hope that helps.

[@spkearney16]

+Iman Matar funny... i have a test in numerical computing in a few hours

[@snake1625b]

the best explained video on this topic.

[@bettertony]

Hi Jacob, Thanks for sharing. There is one question. As for the natural spline, my understanding is that the second derivatives on starting and ending points are zeros. For a quadratic function, it seems to be not possible. We know that quadratic function only has one stationary point. In your video, you got conclusion that "a1=0", then it is going to be a linear function.

[@kvyi]

Hi Wenliang, Thanks for the question. You are exactly right. If we constrain the second derivative to be zero at the endpoint, it will mean that the first segment is a linear function. An alternative is to let the second derivatives be equal at the first or last interior knot (typically at the first, x1). This is the not-a-knot condition. Since these already share a point, and the first derivatives are equal, when we add the condition that the second derivatives are equal, this implies that the two functions are the same. For a cubic spline, the not-a-knot condition says to let the third derivatives at the first and last interior knots to be equal. Contrast this with the natural spline condition, which (as you correctly pointed out) says to let the second derivatives be equal to zero at the endpoints. Similar to this situation, the natural spline condition means that for a cubic spline, the first and last segments are quadratic, not cubic.

[@kvyi]

+Everett You Correct. I am not showing that condition. Perhaps it would be better to do so.

[@chunchen3450]

Thanks for the video. Nice hand on explanation on quadratic splines

[@shivakumarnatrajan]

This is awesome, could you make another video in b spline surface fitting?

[@Bonehand]

Nicely explained, thanks!

[@tag_of_frank]

Looks like more conditions than unknowns. If there are 2 intervals then there are 2 equations with 3 unknowns each, a,b, and c. = 6 unknowns 2 equations. The equations are equal at the interior points => 6 unknowns, 3 equations. They pass through endpoints => endpoints are known, that is 2 equations, 1 for each endpoint. 6 unknowns, 5 equations. First derivatives are equal at the knots => 6 equations, 6 unknowns a1 = 0 => 6 equations, 5 unknowns.

[@anasuyadr5]

Thank u for nice explanation. could u pl explain how to ensemble average the cubic splined graphs to get a final pattern

[@super-panda]

great video, thank you!

[@MarkElDeiry]

Did I miss it? Is the quadratic spline identical to the sine wave?

[@jonahbranch5625]

no, just very very close

[@eggtimer2]

Anyone got a derivation of the spline in tension?

[@M43STR0_dnb]

How did I end up here

[@SuprafluidHD]

Wir haben das grade.

[@mswoonc]

nice

[@Joshua-dc4hu]

For this video, you seem unprepared at points and were vague at points. For improvement, you could make a script so you know what you are doing.

[@kvyi]

Thank you for the feedback. That's a good idea.

[@yashshah8264]

Jacob Bishop None the less, its very informative and gets the point across. Thanks for uploading this.