## swissgirl Group Title Cubic Splines one year ago one year ago

1. swissgirl Group Title

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

2. swissgirl Group Title

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

3. DanielxAK Group Title

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.

4. swissgirl Group Title

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

5. DanielxAK Group Title

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?

6. swissgirl Group Title

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

7. DanielxAK Group Title

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.

8. swissgirl Group Title

Gotcha Can i dirve you crazy with one more question?

9. DanielxAK Group Title

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

10. swissgirl Group Title

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

11. swissgirl Group Title

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]

12. swissgirl Group Title

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

13. swissgirl Group Title

Or is it wayyy too time consuming?

14. DanielxAK Group Title

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.

15. swissgirl Group Title

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

16. swissgirl Group Title

ohhhh it didnt work out uughhhhh

17. swissgirl Group Title

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

18. swissgirl Group Title

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

19. DanielxAK Group Title

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.

20. swissgirl Group Title

ok I will write out all the equations then

21. swissgirl Group Title

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

22. swissgirl Group Title

So there are my four equations

23. DanielxAK Group Title

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

24. swissgirl Group Title

hmm let me try

25. swissgirl Group Title

ummm pretty much sooo

26. swissgirl Group Title

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

27. DanielxAK Group Title

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

28. swissgirl Group Title

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

29. swissgirl Group Title

Thats cooollllllllllllllllllll

30. swissgirl Group Title

its annoying to work with sooooo mannnyy decimals

31. swissgirl Group Title

So now i integrate?

32. DanielxAK Group Title

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

33. swissgirl Group Title

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

34. DanielxAK Group Title

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.

35. swissgirl Group Title

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

36. DanielxAK Group Title

You're welcome.

37. swissgirl Group Title

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

38. swissgirl Group Title

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

39. swissgirl Group Title

Heyyyy :)

40. DanielxAK Group Title

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

41. swissgirl Group Title

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

42. DanielxAK Group Title

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.

43. swissgirl Group Title

Ohhh what i did for a previous question gottcchhhaaaaaaa

44. swissgirl Group Title

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

45. DanielxAK Group Title

Ha. That's okay.