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Motivation

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Why is it important to study the numerical stability of an algorithm?

It can be very sensitive to the rounding errors introduced by the floating-point number system and produce a result very far from what was expected.

Zoom in

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Limitation of finite-precision representation

Not all the data of problem can be exactly represented on a computer as a floating-point number

Rounding errors during the computation

Exact answer to the problem ≠ Result of the algorithm

The numerical stability of an algorithm expresses its behaviour with respect to the propagation of the errors that occur during the arithmetic operations executed by the computer.

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Errors introduced by rounded data

Errors introduced by floating-point operations

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Numerical stability of an algorithm

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How far is the computed solution from the exact one.

How much we should perturb the data to produce the computed solution.

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Barycentric interpolation

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Univariate case

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Our contribution

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THEOREM (Forward stability):

THEOREM (Backward stability):

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Bivariate case

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Existing methods for computing the mean value coordinates

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Potential stability problems

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New stable method

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Stability problems

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Numerical experiments - stability

(6)

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Numerical experiments - efficiency

(6)

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