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A simpler GMRES method for computing oscillatory integral is presented. Theoretical analysis shows that this method is mathematically equivalent to the GMRES method proposed by Olver (2009). Moreover, the simpler GMRES does not require upper Hessenberg matrix factorization, which leads to much simpler program and requires less work. Numerical experiments are conducted to illustrate the performance of the new method and show that in some cases the simpler GMRES method could achieve higher accuracy than GMRES.

In this paper we consider iterative methods for computing high oscillatory integral

Recently, Olver introduced GMRES and shifted GMRES methods for oscillatory integrals in [

In this paper, we deal with a different approach. We look for an orthogonal basis

The paper is organized as follows. In Section

Given a differentiation operator

Specially, for an integral

For convenience, assume that the initial guess of approximation

Given

Compute the

For

Compute

For

End For

Compute

If

return

Due to the orthogonality property

Define

It is well known that equivalence between the simpler GMRES [

Suppose that

We prove the result by induction. For

Now, we assume that the theorem is true for

Theorem

Besides asymptotic order, S-GMRES shares some other properties which shifted GMRES possesses such as reuse of the Arnoldi process required by S-GMRES to compute

In this section, we give some numerical examples to illustrate the efficiency of the proposed methods. When use S-GMRES or GMRES methods, we encounter inner (or semi-inner product)

In all the following examples, we plot the absolute error

We consider

It is easy to see from Figure

Example

In our second experiment, we consider

Figure

Example

In our third experiment, we consider

Generally, for

Figure

Example

In this paper we have proposed a minimum residual method mathematically equivalent to the GMRES method for computing oscillatory integrals with irregular oscillations. The proposed S-GMRES method shares some properties as those possessed by GMRES, such as guaranteed to converge for a large class of analytic functions. But S-GMRES method does not need the factorization of an upper Hessenberg matrix. Numerical experiments show that in some cases S-GMRES methods could achieve higher accuracy than GMRES; also S-GMRES method possesses the fact that the higher the frequency the more the accuracy of the approximations.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the anonymous referees for their constructive comments and helpful suggestions to improve this paper greatly. This work is supported by the Scientific Research Foundation of Education Bureau of Hunan Province under Grant 14C0495.