1 / 23

Computers in Civil Engineering 53:081 Spring 2003

Computers in Civil Engineering 53:081 Spring 2003. Lecture #14. Interpolation. Interpolation: Overview. Objective: estimate intermediate values between precise data points using simple functions Solutions Newton Polynomials Lagrange Polynomials Spline Interpolation. Interpolation.

beulah
Download Presentation

Computers in Civil Engineering 53:081 Spring 2003

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Computers in Civil Engineering53:081 Spring 2003 Lecture #14 Interpolation

  2. Interpolation: Overview • Objective: estimate intermediate values between precise data points using simple functions • Solutions • Newton Polynomials • Lagrange Polynomials • Spline Interpolation Interpolation Curve Fitting Curve goes through data points single value Curve need not go through data points multiple values

  3. Example High-precision data points

  4. Quad Cities Dresden LaSalle Braidwood

  5. Quad-Cities Nuke Station Diffuser Curve

  6. Examples of Simple Polynomials Fist-order (linear) Third-order (cubic) Second-order (quadratic)

  7. Newton’s Divided-Difference Interpolating Polynomials • General comments • Linear Interpolation • Quadratic Interpolation • General Form

  8. The notation: means the first order interpolatingpolynomial Linear Interpolation Formula By similar triangles: Rearrange:

  9. Example Problem: Estimate ln(2)(the true value is 0.69) Solution: We know that: at x = 1 ln(x) =0 at x = e ln(x) =1 (e=2.718...) Thus,

  10. Quadratic Interpolation General form: Equivalent form: (f2(x) means second-order interpolating polynomial) To solve for ,three points are needed:

  11. Set in (1) to find Substitute in (1) and evaluate at to find: Substitute in (1) and evaluate at to find: Quadratic Interpolation Note: this looks like a second derivative…

  12. Example Problem Estimate ln(2)(the true value is 0.69) Solution We know that: at x = x0 = 1 ln(x) =0 at x = x1 = e ln(x) =1 (e=2.718...) at x = x2= e2 ln(x) = 2

  13. How to Generalize This? It would get pretty tedious to do this for third, fourth, fifth, sixth, etc order polynominal We need a plan: Newton’s Interpolating Polynomials

  14. General form of Newton’s Interpolating Polynomials To solve for , n+1 points are needed: Solution What does this [ ] notation mean?

  15. Finite Divided Differences First finite divided difference: Second finite divided difference: nth finite divided difference:

  16. Finite Divided Differences Finite divided difference table, case n = 3:

  17. do i=0,n-1 fdd(i,1)=f(i) enddo do j=2,n do i=1,n-j+1 fdd(i,j)=(fdd(i+1,j-1)-fdd(i,j-1))/ & (x(i+j-1)-x(i)) enddo enddo Divided Differences Pseudo Code

  18. Example – ln(2) again

  19. Newton Interpolation Pseudo Code See the textbook!

  20. Features of Newton Divided-Differences to get Interpolating Polynomial • Data need not be equally spaced • Arrangement of data does not have to be ascending or descending, but it does influence error of interpolation • Best case is when the base points are close to the unknown value • Estimate of relative error: Error estimate for nth-order polynomial is the difference between the (n+1)th and nth-order prediction.

  21. Relative Error As a Function of Order Example 18.5 in text Determine ln(2) using the following table MATLAB function interp1 is very useful for this

  22. Midterm 2 Tuesday 15 April

More Related