A Study on Numerical Stability of Finite Difference Formulae for Numerical Differentiation and Integration

The numerical differentiation based on the interpol ating polynomial is basically an unstable process and one cannot expect good accu racy even when the original data are known to be accurate. We analyze the stability of c omputation of derivatives through polynomial interpolation at given point numerically and prove it has poor stability when closer to the interpolating nodes however it has a quite good stability between interpolating nodes. The numerical integration by use of lower order f ormulas such as trapezoidal rule and Simpson rule gives accuracy of results than use of higher order Newton-Cotes formulae. In this paper, we also analyze the reason for the poor stability of higher order Newton-Cotes formulae. Numerical examples are given to study roundfoff error analysis of numerical differentiation and integration.