Optimally exact algorithm for solution of a certain numerical integration problem

Optimal minimax algorithm for integrating fast oscillatory functions in two dimensions

“…; M p 2 1; and p ¼ 1; 2: For the given problem the hereditary error is zero. By assuming that the calculations are performed in a floating-point regime with round-off the results of arithmetic operations using the standard rule up to t binary digits in normalised mantissae of numbers, we can derive the round-off errors for all main arithmetic EC 21,8 operations participating in computing Rðv k 1 ; v k 2 Þ: For example, of fl(•) is the result of computing x 1^x2 on a computer with the above characteristics, we have that flðx 1^x2 Þ ¼ ðx 1^x2 Þð1 þ 1 1 Þ where j1 1 j # 2 2t : Furthermore, we have (Zadiraka and Abatov, 1991;Zadiraka and Melnikova, 1993)…”

“…In the context of numerical integration of fast oscillatory functions, one of the first works on applications of spline‐based approximations was due to Einarson (1968). It was also shown that spline‐based approximations in numerical integration can lead to optimal‐by‐accuracy and optimal‐by‐order quadrature formulae which were obtained for some important functional classes (Berezovskii and Ivanov, 1977; Melnik and Melnik, 1998, 1999; Zadiraka and Abatov, 1991; Zadiraka and Kasenov, 1986). However, the results in this direction are typically limited to the one‐dimensional case.…”

“…For example, of fl(·) is the result of computing x 1 ± x 2 on a computer with the above characteristics, we have that fl( x 1 ± x 2 )=( x 1 ± x 2 )(1+ ε 1 ) where | ε 1 |≤2 − τ . Furthermore, we have (Zadiraka and Abatov, 1991; Zadiraka and Melnikova, 1993) thatEquation 30where the bounds for constants ξ i , i =1,…, n and ε 1 can be given as follows: (1−2 − τ ) n −1 ≤1+ ε 1 ≤(1+2 − τ ) n −1 with the same bounds for ξ 1 , andEquation 31If we assume that ( n +2− i )2 − τ ≤0.1 we can conclude thatEquation 32Applying these estimates to equation (4.7), we derive the following a priori estimate of round‐off error for computing R¯ ( ω k 1 , ω k 2 )Equation 33The estimate of the round‐off error in computing R¯¯ ( ω k 1 , ω k 2 ) can be obtained analogously.…”