Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems
This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say $$q(\tau ,A)$$ q ( τ , A ) , on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods ba...
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| Published in | Advances in computational mathematics Vol. 51; no. 2 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Springer Nature B.V
01.04.2025
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| Abstract | This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say $$q(\tau ,A)$$ q ( τ , A ) , on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of $$q(\tau ,w)$$ q ( τ , w ) have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms. |
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| AbstractList | This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say q(τ,A), on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of q(τ,w) have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms. This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say $$q(\tau ,A)$$ q ( τ , A ) , on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of $$q(\tau ,w)$$ q ( τ , w ) have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms. |
| ArticleNumber | 19 |
| Author | Gemignani, Luca Aceto, Lidia |
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| Cites_doi | 10.1137/0913071 10.1016/j.apnum.2023.05.026 10.1016/0898-1221(81)90036-5 10.1007/s11075-024-01946-1 10.2307/2005818 10.1155/2013/315748 10.3390/fractalfract5010015 10.1093/imanum/drp043 10.1002/nla.2141 10.1063/1.3047921 10.1137/0729014 10.2307/2035062 10.1007/s10496-007-0228-0 10.1016/j.na.2007.12.007 10.1007/BFb0072427 10.1007/s10444-021-09917-z |
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| References | L Gemignani (10231_CR11) 2023; 192 E Gallopoulos (10231_CR10) 1992; 13 10231_CR13 10231_CR17 10231_CR16 JN Lyness (10231_CR15) 1974; 28 10231_CR9 10231_CR8 Y Saad (10231_CR19) 1992; 29 P Boito (10231_CR4) 2018; 25 10231_CR6 10231_CR18 R Bellman (10231_CR3) 1966; 17 P Boito (10231_CR5) 2022; 48 10231_CR1 JE Kiefer (10231_CR12) 1981; 7 C Lanczos (10231_CR14) 1966 A Barkhudaryan (10231_CR2) 2007; 23 A Boucherif (10231_CR7) 2009; 70 |
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| SubjectTerms | Algorithms Boundary value problems Computation Fourier series Mathematics Numerical methods Polynomials Sparse matrices |
| Title | Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems |
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