On the Stability of Filon–Clenshaw–Curtis Rules

Numerical stability of the Filon–Clenshaw–Curtis rules is considered, when applied to oscillatory integrals with the linear oscillator. The following results are proved: (1) the coefficients of the (N + 1)-point rule, for any N 2, never lie in a right sector of the complex plane; (2) the coefficients of the 2-point rule lie in a right sector onlywhen k ∈ [dπ − 3π/4, dπ − π/4), for any integer d 0 large enough; and (3) the coefficients of the 3-point rule lie in a right sector only when k ∈ (dπ − π/2, dπ − π/4), for any integer d 0 large enough. These results imply that the condition numbers associated with the 2-point and the 3-point rules are bounded by π/2 when k satisfies the aforementioned conditions. Then, we extend the stability intervals for k and show that in the following cases, the FCC rules can be applied in a stable manner: (1) the 2-point rule with k far enough from dπ for any integer d 0; (2) the 3-point rule with k ∈ [dπ − π/2, dπ) far enough from dπ; and (3) the 4-point rule with k ∈ [dπ − π/2, dπ) far enough from both dπ − π/2 and dπ.