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A new quadrature scheme based on an Extended Lagrange Interpolation process.
- Source :
-
Applied Numerical Mathematics . Feb2018, Vol. 124, p57-75. 19p. - Publication Year :
- 2018
-
Abstract
- Let w ( x ) = e − x β x α , w ¯ ( x ) = x w ( x ) and let { p m ( w ) } m , { p m ( w ¯ ) } m be the corresponding sequences of orthonormal polynomials. Since the zeros of p m + 1 ( w ) interlace those of p m ( w ¯ ) , it makes sense to construct an interpolation process essentially based on the zeros of Q 2 m + 1 : = p m + 1 ( w ) p m ( w ¯ ) , which is called “Extended Lagrange Interpolation”. In this paper the convergence of this interpolation process is studied in suitable weighted L 1 spaces, in a general framework which completes the results given by the same authors in weighted L u p ( ( 0 , + ∞ ) ) , 1 ≤ p ≤ ∞ (see [31] , [28] ). As an application of the theoretical results, an extended product integration rule , based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type ∫ 0 + ∞ f ( x ) k ( x , y ) u ( x ) d x , u ( x ) = e − x β x γ ( 1 + x ) λ , γ > − 1 , λ ∈ R + , where the kernel k ( x , y ) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [23] . It is shown that the “compound quadrature sequence” represents an efficient proposal for saving 1/3 of the evaluations of the function f , under unchanged speed of convergence. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01689274
- Volume :
- 124
- Database :
- Academic Search Index
- Journal :
- Applied Numerical Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 126231029
- Full Text :
- https://doi.org/10.1016/j.apnum.2017.09.016