# A Matrix Approach to the Newton Formula and Divided Differences

• J. M. Carnicer
• Y. Khiar
• J. M. Peña
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)

## Abstract

The Crout factorization of a Vandermonde matrix is related with the Newton polynomial interpolation formula expressed in terms of divided differences. Another triangular factorization, which can be related with the Newton formula in terms of finite differences, is provided by the Doolittle factorization. The influence of the order of the nodes on the conditioning of the corresponding linear system is analyzed, considering the three cases of increasing order, Leja order and increasing distances to the origin. The lower triangular systems for the computation of divided and finite differences are analyzed and the conditioning of the corresponding lower triangular matrices is studied. Numerical examples are included.

## Notes

### Acknowledgements

This work has been partially supported by the Spanish Research Grant MTM2015-65433, by Gobierno the Aragón and Fondo Social Europeo.

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