A Stable Version of the Modified Algorithm for Error Minimization in Combined Numerical Integration

The present study derives a stable version of “A Modified Algorithm for Reduction of Error in Combined Numerical Integration”. It is discovered that the earlier proposed scheme "A Modified Algorithm for Error Reduction in Combined Numerical Integration" exhibits accuracy fluctuations when the number of slits, , is increased .  Starting with the number of slits  and increasing the count of sub-intervals, the error increases spontaneously. This spontaneous spike in error is resolved by considering a better combination of quadrature rules. To this, the notable result of this study is the identification of an optimal choice for quadrature formulae that could minimizes error fluctuations in combined numerical integration regardless the number of slits ( ). With this revised choice, the error remains relatively stable and predictable even as the count of sub-intervals is increased.