Math 495B

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Math 495B

• Polynomial Interpolation
• Special case of a step function.

Frederic Gibou
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• 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).

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Intuitive approach
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Intuitive approach
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Intuitive approach
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Intuitive approach
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Higher Order interpolation
Curve to be approximated
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Higher Order interpolation
Curve to be approximated
Linear interpolation
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Higher Order interpolation
Curve to be approximated
Linear interpolation
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Higher Order interpolation
Curve to be approximated
Linear interpolation
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Higher Order interpolation
Curve to be approximated
Quadratic interpolation
Linear interpolation
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Higher Order interpolation
Curve to be approximated
Curve to be approximated
Linear interpolation
Quadratic interpolation
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Higher Order interpolation

• Interpolation with a polynomial of degree 2

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Higher Order interpolation

• Impose

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• Solve for the coefficients

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Higher Order interpolation
Moral Need 3 points to get a polynomial
interpolation of degree 2.
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• How to choose the points?
• Case of a smooth function.
• 3 consecutive points.

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