Abstract
The aim of this work is to propose a fast and reliable algorithm for computing
integrals of the type
$$\int_{-\infty}^{\infty} f(x) e^{\scriptstyle -x^2 -\frac{\scriptstyle 1}{\scriptstyle x^2}} dx,$$
where $f(x)$ is a sufficiently smooth function, in floating point arithmetic.
The algorithm is based on a product integration rule, whose rate of convergence
depends only on the regularity of $f$, since the coefficients of the rule are ``exactly'' computed by means of suitable recurrence relations here derived.
We prove stability and convergence in the space of locally continuous functions on $\RR$ equipped with weighted uniform norm.
By extensive numerical tests, the accuracy of the proposed product rule is compared with that of the Gauss--Hermite quadrature formula w.r.t. the function $f(x) e^{-\frac{\scriptstyle 1}{\scriptstyle x^2}}$. The numerical results confirm the effectiveness of the method, supporting the proven theoretical estimates.
Anno
2023
Autori IAC
Tipo pubblicazione
Altri Autori
Teresa Laudadio, Nicola Mastronardi, Donatella Occorsio
Editore
North-Holland
Rivista
Applied numerical mathematics