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Math 495B Polynomial Interpolation Special case of a step function. Frederic Gibou.

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Presentation on theme: "Math 495B Polynomial Interpolation Special case of a step function. Frederic Gibou."— Presentation transcript:

1 Math 495B Polynomial Interpolation Special case of a step function. Frederic Gibou

2 What is polynomial interpolation? -Linear interpolation (An intuitive approach) -Higher order polynomial interpolation (second order) How to choose the nodes. -Case of a smooth function (Straight forward) -Case of a step function (Decision to choose the nodes).

3 Intuitive approach

4

5

6

7 Higher Order interpolation Curve to be approximated

8 Linear interpolation Higher Order interpolation Curve to be approximated

9 Linear interpolation Higher Order interpolation Curve to be approximated

10 Linear interpolation Higher Order interpolation Curve to be approximated

11 Linear interpolation Quadratic interpolation Higher Order interpolation Curve to be approximated

12 Linear interpolation Quadratic interpolation Higher Order interpolation Curve to be approximated

13 Higher Order interpolation Interpolation with a polynomial of degree 2

14 Impose: Higher Order interpolation

15 Solve for the coefficients:

16 Moral : Need 3 points to get a polynomial interpolation of degree 2. Higher Order interpolation

17 §How to choose the points? Case of a smooth function. –3 consecutive points.

18 Case of a step function Curve to be approximated

19 Linear interpolation Case of a step function Curve to be approximated

20 Linear interpolation Quadratic interpolation Case of a step function Curve to be approximated

21 Linear interpolation Quadratic interpolation Case of a step function Curve to be approximated

22 Linear interpolation Quadratic interpolation Case of a step function Curve to be approximated

23 Linear interpolation Quadratic interpolation Gibbs phenomenon Case of a step function Curve to be approximated

24 Case of a step function Curve to be approximated

25 Case of a step function Curve to be approximated Linear

26 Case of a step function Curve to be approximated

27 Case of a step function Curve to be approximated

28 Case of a step function Curve to be approximated

29 Case of a step function Curve to be approximated

30 Quadratic interpolation Case of a step function Curve to be approximated

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