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swissgirl
 2 years ago
Cubic Splines
swissgirl
 2 years ago
Cubic Splines

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swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0@DanielxAK Can you help me through this problem. It seems quite simple

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0\(S_0(x)=1+2xx^3 \to 0 \leq x \leq 1\) \(S_1(x)=2+b(x1)+c(x1)^2+d(x3)^3 \to 1 \leq x \leq 2\)

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Sorry 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.

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0Yes I figured it out \(S_0'(1)=S_1'(1)\) \(S_0''(1)=S_1''(1)\) \(S_1''(2)=0\) THHAANKKSSSS :)

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Um. 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
 2 years ago
Best ResponseYou've already chosen the best response.0Oh i forgot to mention that it is a natural spline lol

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Ah, 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
 2 years ago
Best ResponseYou've already chosen the best response.0Gotcha Can i dirve you crazy with one more question?

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Go for it. I'm trying to avoid my own homework as it is.

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0hahaha 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
 2 years ago
Best ResponseYou've already chosen the best response.0Construct 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
 2 years ago
Best ResponseYou've already chosen the best response.0I think I am allowed to use a computer program but \i am wondering How can this be done by hand

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0Or is it wayyy too time consuming?

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Well, 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
 2 years ago
Best ResponseYou've already chosen the best response.0A(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
 2 years ago
Best ResponseYou've already chosen the best response.0ohhhh it didnt work out uughhhhh

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0whtvrrrr ok once i get the 16 diff values then what?

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0It seems like I am suppossed to integrate but not sure y we are integrating

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Don'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
 2 years ago
Best ResponseYou've already chosen the best response.0ok I will write out all the equations then

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0\(S_0(x)=1.7573593129x6.627416998x^3\) \(S_1(x)=.7071068122(x.25)4.970562748(x.25)^2+6.627416998(x.25)^3\) \(S_2(x)=3.242640687(x.5)+6.627416998(x.5)^3\) \(S_3(x)=.70710678122(x.75)+4.970562748(x.75)^26.627416998(x.75)^3\)

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0So there are my four equations

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Have you tried plotting them? If those are correct, they should match f(x) = cos(pi*x) pretty well.

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0ummm pretty much sooo

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0Its weird that one of the equations has nothing to do with the graph at all

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Yeah, only looks like one or two of them are correct. You may want to double check your equations.

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0ohhh I found my mistake mixed up signs Its only suppossed to match from [0,1] and it does

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0Thats cooollllllllllllllllllll

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0its annoying to work with sooooo mannnyy decimals

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Yes. Now you can integrate using those cubics (so, from 0.25 use S1, 0.25 to 0.5 use S2, etc.).

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0I am just wondering y we are integrating lol I am just not seeing the logic behind that?

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1The 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
 2 years ago
Best ResponseYou've already chosen the best response.0ohhhhh 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

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0@DanielxAK hahah I have one more question It asks me to find f'(.5) How would I go abt it?

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0Do I add all the equations together and that will be my spline function?

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1It sounds like it's just asking you to find the derivative of f(x) = cos(pi*x) at 0.5.

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0No it asks me to compare both so it cant mean that

DanielxAK
 2 years ago
Best ResponseYou've already chosen the best response.1Compare 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
 2 years ago
Best ResponseYou've already chosen the best response.0Ohhh what i did for a previous question gottcchhhaaaaaaa

swissgirl
 2 years ago
Best ResponseYou've already chosen the best response.0I am seriously driving u crazy here. Thanks I really appreciate your help
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