Polynomial interpolation with Lagrange polynomials, Numerical Methods #9
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What are Lagrange polynomials and why are Lagrange polynomials used for polynomial interpolation and not the Vandermonde matrix? In this video, physicist Dietmar Haase explains why the Lagrange representation is used for polynomial interpolation using the Lagrange basis polynomials. An interpolation polynomial is generally not determined from the solution of the linear equation system using the Vandermonde matrix because, among other things, this would be too complex with a large number of support points. The most important properties of the Lagrange basis polynomials are derived and it is shown how an interpolation polynomial can be represented as a linear combination of the Lagrange basis polynomials. Using a selected example, it is demonstrated how the Lagrange representation of an interpolation polynomial can be found for a table of measured values that describes the temperature curve on a summer day over the course of the day. It turns out that the interpolation for an argument that lies within the support points is approximated relatively well. On the other hand, it turns out that a prediction, i.e. an extrapolation, for an argument outside the support point interval is approximated very poorly. Website: https://www.ingmathe.de Youtube channel: / ingmathede Online calculator: https://www.wolframalpha.com/
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