Abstract

By multiplying a truncated aberration series by an appropriate, low-order polynomial containing a given number of terms, it is possible to produce a new power series for which a significantly larger number of the high-order coefficients are markedly smaller than those of the original power series. This indicates that, to a reasonable approximation, the higher-order coefficients form simple patterns that can be expected to extend into the unknown coefficients. This suggests the possibility of estimating unknown coefficients in order to reduce the error incurred by simply ignoring them. This is effectively realized when the original power series is replaced with the quotient of the new power series (which is expected to have a smaller truncation error) and the polynomial, yielding what is, for most purposes, a rational function. In this way it is possible to gain significantly greater advantage from the higher-order coefficients (say, beyond the first few orders) than was previously obtained. These rational functions are generalizations of the well-known Padé approximants for power series in one variable. Some numerical examples are presented that demonstrate the remarkable effectiveness of this method.

© 1986 Optical Society of America

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