# A Lanczos method for approximating composite functions

@article{Constantine2011ALM,
title={A Lanczos method for approximating composite functions},
author={Paul G. Constantine and Eric T. Phipps},
journal={Appl. Math. Comput.},
year={2011},
volume={218},
pages={11751-11762}
}
• Published 1 October 2011
• Mathematics
• Appl. Math. Comput.
11 Citations

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## References

SHOWING 1-10 OF 28 REFERENCES

• Computer Science
SIAM J. Sci. Comput.
• 2005
A high-order stochastic collocation approach is proposed, which takes advantage of an assumption of smoothness of the solution in random space to achieve fast convergence and requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods.
• Mathematics
Numerische Mathematik
• 2007
It is shown that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates.
• Computer Science, Mathematics
SIAM J. Matrix Anal. Appl.
• 1992
It is demonstrated that finite precision Lanczos and conjugate gradient computations for solving a symmetric positive definite linear system $Ax = b$ or computing the eigenvalues of A behave very
• Computer Science, Mathematics
Acta Numerica
• 2006
A tribute is paid to those who have made an understanding of the Lanczos and conjugate gradient algorithms possible through their pioneering work, and to review recent solutions of several open problems that have also contributed to knowledge of the subject.
Partial reorthogonalization (PRO) is proposed as a new method for maintain- ing semiorthogonality among the Lanczos vectors and is applied to the solution of large symmetric systems of linear equations and results obtained compare favorably with the conjugate gradient method.
• Mathematics
• 2009
This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient
• A. Greenbaum
• Computer Science
Frontiers in applied mathematics
• 1997
This paper presents a brief overview of the State of the Art Notation Review of Relevant Linear Algebra and some of the algorithms used in this review, as well as some basic ideas of Domain Decomposition Methods.