Decompositions of optimal averaged Gauss quadrature rules
Само за регистроване кориснике
2024
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Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating
matrix functionals of the form u
T
f (A)v with a large matrix A ∈ R
N×N by lowrank approximations that are obtained by applying a few steps of the symmetric or
nonsymmetric Lanczos processes to A; here u, v ∈ R
N
are vectors. The latter process
is used when the measure associated with the Gauss quadrature rule has support in
the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix,
whose entries determine the recursion coefficients of the monic orthogonal polynomials
associated with the measure, while the nonsymmetric Lanczos process determines a
nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of
sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results
of Peherstorfer, that optimal averaged Gauss quadrature rules, which a...re associated
with a nonnegative measure with support on the real axis, can be expressed as a
weighted sum of two quadrature rules. This decomposition allows faster evaluation of
optimal averaged Gauss quadrature rules than the previously available representation.
The present paper provides a new self-contained proof of this decomposition that
is based on linear algebra techniques. Moreover, these techniques are generalized to
determine a decomposition of the optimal averaged quadrature rules that are associated
with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also,
the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings
allow faster evaluation of optimal averaged Gauss quadrature rules than the previously
available representations. Computational aspects are discussed.
Извор:
Journal of Computational and Applied Mathematics, 2024, 438, Art. 115586Издавач:
- Elsevier
Финансирање / пројекти:
- Министарство науке, технолошког развоја и иновација Републике Србије, институционално финансирање - 200105 (Универзитет у Београду, Машински факултет) (RS-MESTD-inst-2020-200105)
Колекције
Институција/група
Mašinski fakultetTY - JOUR AU - Đukić, Dušan AU - Mutavdžić Đukić, Rada AU - Reichel, Lothar AU - Spalević, Miodrag PY - 2024 UR - https://machinery.mas.bg.ac.rs/handle/123456789/7068 AB - Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form u T f (A)v with a large matrix A ∈ R N×N by lowrank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u, v ∈ R N are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed. PB - Elsevier T2 - Journal of Computational and Applied Mathematics T1 - Decompositions of optimal averaged Gauss quadrature rules IS - Art. 115586 VL - 438 DO - 10.1016/j.cam.2023.115586 ER -
@article{ author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag", year = "2024", abstract = "Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form u T f (A)v with a large matrix A ∈ R N×N by lowrank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u, v ∈ R N are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.", publisher = "Elsevier", journal = "Journal of Computational and Applied Mathematics", title = "Decompositions of optimal averaged Gauss quadrature rules", number = "Art. 115586", volume = "438", doi = "10.1016/j.cam.2023.115586" }
Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2024). Decompositions of optimal averaged Gauss quadrature rules. in Journal of Computational and Applied Mathematics Elsevier., 438(Art. 115586). https://doi.org/10.1016/j.cam.2023.115586
Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Decompositions of optimal averaged Gauss quadrature rules. in Journal of Computational and Applied Mathematics. 2024;438(Art. 115586). doi:10.1016/j.cam.2023.115586 .
Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Decompositions of optimal averaged Gauss quadrature rules" in Journal of Computational and Applied Mathematics, 438, no. Art. 115586 (2024), https://doi.org/10.1016/j.cam.2023.115586 . .