swissgirl
  • swissgirl
Cubic Splines
Mathematics
jamiebookeater
  • jamiebookeater
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your free account and access expert answers to this
and thousands of other questions

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

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

swissgirl
  • swissgirl
Yes I figured it out \(S_0'(1)=S_1'(1)\) \(S_0''(1)=S_1''(1)\) \(S_1''(2)=0\) THHAANKKSSSS :)
anonymous
  • 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
  • swissgirl
Oh i forgot to mention that it is a natural spline lol
anonymous
  • 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
  • swissgirl
Gotcha Can i dirve you crazy with one more question?
anonymous
  • anonymous
Go for it. I'm trying to avoid my own homework as it is.
swissgirl
  • 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
  • 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
  • swissgirl
I think I am allowed to use a computer program but \i am wondering How can this be done by hand
swissgirl
  • swissgirl
Or is it wayyy too time consuming?
anonymous
  • 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
  • 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
  • swissgirl
ohhhh it didnt work out uughhhhh
swissgirl
  • swissgirl
whtvrrrr ok once i get the 16 diff values then what?
swissgirl
  • swissgirl
It seems like I am suppossed to integrate but not sure y we are integrating
anonymous
  • 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
  • swissgirl
ok I will write out all the equations then
swissgirl
  • 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
  • swissgirl
So there are my four equations
anonymous
  • anonymous
Have you tried plotting them? If those are correct, they should match f(x) = cos(pi*x) pretty well.
swissgirl
  • swissgirl
hmm let me try
swissgirl
  • swissgirl
ummm pretty much sooo
swissgirl
  • swissgirl
Its weird that one of the equations has nothing to do with the graph at all
anonymous
  • anonymous
Yeah, only looks like one or two of them are correct. You may want to double check your equations.
swissgirl
  • swissgirl
ohhh I found my mistake mixed up signs Its only suppossed to match from [0,1] and it does
swissgirl
  • swissgirl
Thats cooollllllllllllllllllll
swissgirl
  • swissgirl
its annoying to work with sooooo mannnyy decimals
swissgirl
  • swissgirl
So now i integrate?
anonymous
  • anonymous
Yes. Now you can integrate using those cubics (so, from 0.25 use S1, 0.25 to 0.5 use S2, etc.).
swissgirl
  • swissgirl
I am just wondering y we are integrating lol I am just not seeing the logic behind that?
anonymous
  • 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
  • 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
  • anonymous
You're welcome.
swissgirl
  • swissgirl
@DanielxAK hahah I have one more question It asks me to find f'(.5) How would I go abt it?
swissgirl
  • swissgirl
Do I add all the equations together and that will be my spline function?
swissgirl
  • swissgirl
Heyyyy :)
anonymous
  • anonymous
It sounds like it's just asking you to find the derivative of f(x) = cos(pi*x) at 0.5.
swissgirl
  • swissgirl
No it asks me to compare both so it cant mean that
anonymous
  • 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
  • swissgirl
Ohhh what i did for a previous question gottcchhhaaaaaaa
swissgirl
  • swissgirl
I am seriously driving u crazy here. Thanks I really appreciate your help
anonymous
  • anonymous
Ha. That's okay.

Looking for something else?

Not the answer you are looking for? Search for more explanations.