Generalized averaged Gauss quadrature rules for the approximation of matrix functionals
Abstract
The need to compute expressions of the form , where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and then evaluate the reduced problem. This paper describes a new method to determine error estimates for computed quantities and shows how to achieve higher accuracy than available methods for essentially the same computational effort. Our methods are based on recently proposed generalized averaged Gauss quadrature formulas.
Keywords:
Matrix functional / Gauss quadrature / Averaged Gauss rulesSource:
Bit Numerical Mathematics, 2016, 56, 3, 1045-1067Publisher:
- Springer, Dordrecht
Funding / projects:
- Methods of Numerical and Nonlinear Analysis with Applications (RS-MESTD-Basic Research (BR or ON)-174002)
- NSF [DMS-1115385]
DOI: 10.1007/s10543-015-0592-7
ISSN: 0006-3835
WoS: 000382137200012
Scopus: 2-s2.0-84949763422
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Reichel, Lothar AU - Spalević, Miodrag AU - Tang, Tunan PY - 2016 UR - https://machinery.mas.bg.ac.rs/handle/123456789/2489 AB - The need to compute expressions of the form , where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and then evaluate the reduced problem. This paper describes a new method to determine error estimates for computed quantities and shows how to achieve higher accuracy than available methods for essentially the same computational effort. Our methods are based on recently proposed generalized averaged Gauss quadrature formulas. PB - Springer, Dordrecht T2 - Bit Numerical Mathematics T1 - Generalized averaged Gauss quadrature rules for the approximation of matrix functionals EP - 1067 IS - 3 SP - 1045 VL - 56 DO - 10.1007/s10543-015-0592-7 ER -
@article{ author = "Reichel, Lothar and Spalević, Miodrag and Tang, Tunan", year = "2016", abstract = "The need to compute expressions of the form , where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and then evaluate the reduced problem. This paper describes a new method to determine error estimates for computed quantities and shows how to achieve higher accuracy than available methods for essentially the same computational effort. Our methods are based on recently proposed generalized averaged Gauss quadrature formulas.", publisher = "Springer, Dordrecht", journal = "Bit Numerical Mathematics", title = "Generalized averaged Gauss quadrature rules for the approximation of matrix functionals", pages = "1067-1045", number = "3", volume = "56", doi = "10.1007/s10543-015-0592-7" }
Reichel, L., Spalević, M.,& Tang, T.. (2016). Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. in Bit Numerical Mathematics Springer, Dordrecht., 56(3), 1045-1067. https://doi.org/10.1007/s10543-015-0592-7
Reichel L, Spalević M, Tang T. Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. in Bit Numerical Mathematics. 2016;56(3):1045-1067. doi:10.1007/s10543-015-0592-7 .
Reichel, Lothar, Spalević, Miodrag, Tang, Tunan, "Generalized averaged Gauss quadrature rules for the approximation of matrix functionals" in Bit Numerical Mathematics, 56, no. 3 (2016):1045-1067, https://doi.org/10.1007/s10543-015-0592-7 . .