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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.

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Computers in Civil Engineering 53:081 Spring 2003

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  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

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