interpolation and you a brief overview of some interpolation tools l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Interpolation and You: A Brief Overview of Some Interpolation Tools PowerPoint Presentation
Download Presentation
Interpolation and You: A Brief Overview of Some Interpolation Tools

Loading in 2 Seconds...

play fullscreen
1 / 22

Interpolation and You: A Brief Overview of Some Interpolation Tools - PowerPoint PPT Presentation


  • 321 Views
  • Uploaded on

Interpolation and You: A Brief Overview of Some Interpolation Tools. By: Mark Coose Joetta Swift Micah Weiss. What Problems Can Interpolation Solve?. Given a table of values, find a simple function that passes through the given points exactly

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Interpolation and You: A Brief Overview of Some Interpolation Tools' - Gabriel


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
interpolation and you a brief overview of some interpolation tools

Interpolation and You: A Brief Overview of Some Interpolation Tools

By:

Mark Coose

Joetta Swift

Micah Weiss

what problems can interpolation solve
What Problems Can Interpolation Solve?
  • Given a table of values, find a simple function that passes through the given points exactly
  • Given a table of experimental data, find a formula that approximates the data and possibly filters out errors
  • Given an arbitrary function f, find an approximation in the form of a simpler function g
how do we find an interpolating polynomial
How Do We Find An Interpolating Polynomial?
  • Lagrange form
  • Newton’s form
  • Cubic Splines
lagrange form of the interpolating polynomial
Lagrange Form of the Interpolating Polynomial
  • A linear combination of n+1 polynomials
  • Each term is a polynomial of degree n
consider the table
Consider the Table

Begin by finding n+1=3 cardinal functions li(x)

Or,

disadvantages of lagrange form
Disadvantages of Lagrange Form
  • Each cardinal polynomial term is degree n
  • To add one point we must recalculate each cardinal function from scratch
newton form of interpolating polynomial
Newton Form of Interpolating Polynomial
  • Data points can be added without recalculating existing sequence
  • Pn(xi)=f(xi) for i=0,1,…,n
  • The polynomial pn is of deg ≤ n
  • Pn(x) = a0 + a1(x-x0) + a2(x-x0)(x-x1) + … + an(x – xn-1)
adding a point
Adding a Point

The beauty of Newton’s form of the interpolating polynomial is that it allows us to add points and extend the polynomial without redoing previous calculations. The additional point only requires us to calculate the three values shown in red.

disadvantages of polynomial interpolation
Disadvantages of Polynomial Interpolation
  • The function can oscillate wildly
  • Each polynomial can be of degree n (computationally expensive for large n = number of points).
splines
Splines

Definition: A function is called a spline of degree k if

  • The domain of S is an interval[a,b]
  • There are points ti such that a=t0<t1<…<tn=b and S is a polynomial of degree at most k on each subinterval [ti, ti+1]
  • S, S’, S’’,…,S(k-1) are all continuous functions on [a,b]
splines15
Splines
  • Linear,Quadratic Splines
  • Cubic Splines
    • Cubic Splines are almost always used over polynomial interpolation because of its increased accuracy (smaller error) and the final curve is smoother than polynomial interpolation
cubic spline smoothness property
Cubic Spline Smoothness Property

If S is the natural cubic spline function that interpolates a twice-continuous differentiable function f at knots a=t0<t1<…<tn=b, then

what does this mean
What Does This Mean?
  • A function with large second derivatives is subject to wild oscillations. The Smoothness Property ensures that the spline S oscillates less than the function that it interpolates.
  • An example is shown, here.
proof of smoothness property
Proof of Smoothness Property
  • Let g(x)=f(x)-s(x)

Then, f”(x)=g”(x)+s”(x)

And,

apply integration by parts
Apply integration by parts
  • Let u=s’’(x) and dv=g’’(x)
    • Then,
  • Since, this is a natural cubic spline, s’’=0, so the first term is equal to zero.
    • Which leaves
integration by parts cont
Integration by Parts cont.
  • [break this up into sub integrals]

=

  • (s’’’ is a constant within this integral)

=

  • [g(t(i) + 1) – g(t(i))] = 0
    • because g(t(i) + 1) = g(t(i)) = 0
slide22

So,

  • But,
  • So,