Cubic Splines

- swissgirl

Cubic Splines

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- swissgirl

@DanielxAK Can you help me through this problem. It seems quite simple

- swissgirl

\(S_0(x)=1+2x-x^3 \to 0 \leq x \leq 1\)
\(S_1(x)=2+b(x-1)+c(x-1)^2+d(x-3)^3 \to 1 \leq x \leq 2\)

- anonymous

Sorry for the late reply. I was at class and then attended a seminar. Anyway:
You know your three points. 0, 1 and 2. You have been given some of your coefficients ( specifically five of them) and need to find the other 3. So, you need 3 equations.
Can you tell me which 3 equations you need to use? You're almost given one of them to start.

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- swissgirl

Yes I figured it out
\(S_0'(1)=S_1'(1)\)
\(S_0''(1)=S_1''(1)\)
\(S_1''(2)=0\)
THHAANKKSSSS :)

- anonymous

Um. Be careful there. S1''(2) = 0 implies that you have a natural spline. You don't know that. So, that doesn't work. But, you're almost there. Looking at the first two equations you gave me (which are correct), can you find a very similar equation that follows the same pattern as those two?

- swissgirl

Oh i forgot to mention that it is a natural spline lol

- anonymous

Ah, okay. Well, in case you didn't know, you should still be able to solve it by using the fact that:
S0(1) = S1(1), as they have the same y value.

- swissgirl

Gotcha Can i dirve you crazy with one more question?

- anonymous

Go for it. I'm trying to avoid my own homework as it is.

- swissgirl

hahaha I do the same. I sit on here answering questions and im like thinking y the hell am I on here when I gotta do my own

- swissgirl

Construct a natural cubic spline to approximate f(x)=Cos(pi*x) by using the values given by f(x) at x=0, .25,.5, .75 and1
Integrate the spline over [0,1]

- swissgirl

I think I am allowed to use a computer program but \i am wondering How can this be done by hand

- swissgirl

Or is it wayyy too time consuming?

- anonymous

Well, you have 5 points. So, you'll end up having 16 coefficients to solve. You're better off solving the equations using a computer program. So, your points will be the x defined above and your five y values will be f(x) for each x. Then, just go back to making cubics on those intervals with unknown coefficients and do all the steps for defining all 16 equations.

- swissgirl

A(I) B(I) C(I) D(I)
1.00000000 -0.7573593129 0. -6.627416998
0.7071067812 -2. -4.970562748 6.627416998
6.123233996*10^-17 -3.242640687 4.440892099*10^-16 6.627416998
-0.7071067812 -2. 4.970562748 -6.627416998

- swissgirl

ohhhh it didnt work out uughhhhh

- swissgirl

whtvrrrr ok once i get the 16 diff values then what?

- swissgirl

It seems like I am suppossed to integrate but not sure y we are integrating

- anonymous

Don't worry about integrating right now. You want to find your coefficients first so you can find all 4 cubics. Then, after you know your cubics, you integrate them on their intervals. But, not right now. Worry about finding your coefficients using your equations.

- swissgirl

ok I will write out all the equations then

- swissgirl

\(S_0(x)=1-.7573593129x-6.627416998x^3\)
\(S_1(x)=.707106812-2(x-.25)-4.970562748(x-.25)^2+6.627416998(x-.25)^3\)
\(S_2(x)=-3.242640687(x-.5)+6.627416998(x-.5)^3\)
\(S_3(x)=-.7071067812-2(x-.75)+4.970562748(x-.75)^2-6.627416998(x-.75)^3\)

- swissgirl

So there are my four equations

- anonymous

Have you tried plotting them? If those are correct, they should match f(x) = cos(pi*x) pretty well.

- swissgirl

hmm let me try

- swissgirl

ummm pretty much sooo

- swissgirl

Its weird that one of the equations has nothing to do with the graph at all

- anonymous

Yeah, only looks like one or two of them are correct. You may want to double check your equations.

- swissgirl

ohhh I found my mistake mixed up signs
Its only suppossed to match from [0,1] and it does

- swissgirl

Thats cooollllllllllllllllllll

- swissgirl

its annoying to work with sooooo mannnyy decimals

- swissgirl

So now i integrate?

- anonymous

Yes. Now you can integrate using those cubics (so, from 0.25 use S1, 0.25 to 0.5 use S2, etc.).

- swissgirl

I am just wondering y we are integrating lol
I am just not seeing the logic behind that?

- anonymous

The idea is you're able to approximate the integral of any function in the real numbers by building a cubic spline and integrating the pieces of the spline.

- swissgirl

ohhhhh I get it
Ya that makes sense
Like I am taking this course online and its def getting annoying wish I had a teacher to explain everything clearly
THANKS YOU ARE GREATTTTTTTT

- anonymous

You're welcome.

- swissgirl

@DanielxAK hahah I have one more question
It asks me to find f'(.5)
How would I go abt it?

- swissgirl

Do I add all the equations together and that will be my spline function?

- swissgirl

Heyyyy :)

- anonymous

It sounds like it's just asking you to find the derivative of f(x) = cos(pi*x) at 0.5.

- swissgirl

No it asks me to compare both so it cant mean that

- anonymous

Compare both?
Ah okay. So, yes. Find the derivative at 0.5 of the original function. Then, find the derivative of 0.5 using the cubic defined at 0.5 (there should be two of them, since it was one of your points you used when building your piecewise function).
The values of both should be pretty close.

- swissgirl

Ohhh what i did for a previous question gottcchhhaaaaaaa

- swissgirl

I am seriously driving u crazy here. Thanks I really appreciate your help

- anonymous

Ha. That's okay.

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