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

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Presentation transcript:

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

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).

Intuitive approach

Higher Order interpolation Curve to be approximated

Linear interpolation Higher Order interpolation Curve to be approximated

Linear interpolation Higher Order interpolation Curve to be approximated

Linear interpolation Higher Order interpolation Curve to be approximated

Linear interpolation Quadratic interpolation Higher Order interpolation Curve to be approximated

Linear interpolation Quadratic interpolation Higher Order interpolation Curve to be approximated

Higher Order interpolation Interpolation with a polynomial of degree 2

Impose: Higher Order interpolation

Solve for the coefficients:

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

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

Case of a step function Curve to be approximated

Linear interpolation Case of a step function Curve to be approximated

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

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

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

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

Case of a step function Curve to be approximated

Case of a step function Curve to be approximated Linear

Case of a step function Curve to be approximated

Case of a step function Curve to be approximated

Case of a step function Curve to be approximated

Case of a step function Curve to be approximated

Quadratic interpolation Case of a step function Curve to be approximated