Abstract
We examine a Romberg-like quadrature in which the number of subintervals taken grows only linearly, not exponentially. Such a scheme is known to be numerically unstable. We show, however, that this instability is only mild, and we obtain a bound on the size of the error, caused by extrapolation of the initial trapezoidal sums.
A similar formula is obtained for the standard deviation of the error when a statistical model is used. Numerical examples show that the bound is not too pessimistic. Quantitatively, not more than two-fifths of a digit of accuracy get lost per extrapolation step, and this economical choice of points can thus be used practically.
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Laurie, D.P. Propagation of initial rounding error in Romberg-like quadrature. BIT 15, 277–282 (1975). https://doi.org/10.1007/BF01933660
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DOI: https://doi.org/10.1007/BF01933660