# A Flexible Inner-Outer Preconditioned GMRES Algorithm

@article{Saad1993AFI, title={A Flexible Inner-Outer Preconditioned GMRES Algorithm}, author={Yousef Saad}, journal={SIAM J. Sci. Comput.}, year={1993}, volume={14}, pages={461-469} }

A variant of the GMRES algorithm is presented that allows changes in the preconditioning at every step. There are many possible applications of the new algorithm, some of which are briefly discussed. In particular, a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner. For example, the standard GMRES algorithm itself can be used as a preconditioner, as can CGNR (or CGNE), the conjugate gradient method applied to the normal equations. Howeverâ€¦Â

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## References

SHOWING 1-7 OF 7 REFERENCES

### GMRESR: a family of nested GMRES methods

- Computer ScienceNumer. Linear Algebra Appl.
- 1994

Recently Eirola and Nevanlinna have proposed an iterativ<: solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggestâ€¦

### Hybrid Krylov Methods for Nonlinear Systems of Equations

- Computer ScienceSIAM J. Sci. Comput.
- 1990

To improve the global convergence properties of these basic algorithms, hybrid methods based on Powell's dogleg strategy are proposed, as well as linesearch backtracking procedures.

### An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix

- Computer Science
- 1977

A particular class of regular splittings of not necessarily symmetric M-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide aâ€¦

### 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...

### Iterative Solution of Indefinite Symmetric Linear Systems by Methods Using Orthogonal Polynomials over Two Disjoint Intervals

- Mathematics, Computer Science
- 1983

A new technique based upon the least squares polynomial in the set S is proposed, i.e. thePolynomial $t_k $ which minimizes $||1 - \lambda t_k (\lambda )||_w $, where $|| \cdot ||_w$ is an $L_2 $ norm with respect to a weight function defined on S.

### Massively parallel preconditioned Krylov subspace methods

- Massively parallel preconditioned Krylov subspace methods
- 1991

### A mixed CEBE/CC preconditionning for nite element computations

- A mixed CEBE/CC preconditionning for nite element computations
- 1991