# GMRES for the Differentiation Operator

@article{Olver2009GMRESFT, title={GMRES for the Differentiation Operator}, author={Sheehan Olver}, journal={SIAM J. Numer. Anal.}, year={2009}, volume={47}, pages={3359-3373} }

We investigate using the gmres method with the differentiation operator. This operator is unbounded and thus does not fall into the framework of existing Krylov subspace theory. We establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane. These conditions are used to determine when gmres converges. This algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals, even when the kernel of…

## 26 Citations

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This work investigates the use of Krylov subspace methods to solve linear, oscillatory ODEs and demonstrates the effectiveness of this method by computing error and Mathieu functions.

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## References

SHOWING 1-10 OF 17 REFERENCES

### Moment-free numerical integration of highly oscillatory functions

- Mathematics
- 2006

The aim of this paper is to derive new methods for numerically approximating the integral of a highly oscillatory function. We begin with a review of the asymptotic and Filon-type methods developed…

### Efficient quadrature of highly oscillatory integrals using derivatives

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2005

In this paper, we explore quadrature methods for highly oscillatory integrals. Generalizing the method of stationary phase, we expand such integrals into asymptotic series in inverse powers of the…

### A method for numerical integration on an automatic computer

- Mathematics
- 1960

A new method for the numerical integration of a “well-behaved” function over a finite range of argument is described. It consists essentially of expanding the integrand in a series of Chebyshev…

### Moment-free numerical approximation of highly oscillatory integrals with stationary points

- MathematicsEuropean Journal of Applied Mathematics
- 2007

This article presents a method for the numerical quadrature of highly oscillatory integrals with stationary points. We begin with the derivation of a new asymptotic expansion, which has the property…

### GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

- Computer Science, Mathematics
- 1986

We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

### Restarted GMRES for Shifted Linear Systems

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 1998

This work develops a variant of the restarted GMRES method exhibiting the same advantage and investigates its convergence for positive real matrices in some detail and applies it to speed up "multiple masses" calculations arising in lattice gauge computations in quantum chromodynamics, one of the most time-consuming supercomputer applications.

### Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment

- Mathematics
- 1991

### An Extension of MATLAB to Continuous Functions and Operators

- Computer ScienceSIAM J. Sci. Comput.
- 2004

About eighty MATLAB functions from plot and sum to svd and cond have been overloaded so that one can work with "chebfun" objects using almost exactly the usual MATLAB syntax.

### Applied asymptotic analysis

- Mathematics
- 2006

Fundamentals: Themes of asymptotic analysis The nature of asymptotic approximations Asymptotic analysis of exponential integrals: Fundamental techniques for integrals Laplace's method for asymptotic…

### Convergence of Iterations for Linear Equations

- Mathematics
- 1993

1. Motivation, problem and notation.- 1.1 Motivation.- 1.2 Problem formulation.- 1.3 Usual tools.- 1.4 Notation for polynomial acceleration.- 1.5 Minimal error and minimal residual.- 1.6…