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A note on the efficient solution of matrix pencil systems

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Abstract

Algorithms for solving matrix pencil systems of linear equations, of the form (A+γB)x=c+γd, are developed and analysed. The techniques that are discussed are based on methods for the generalized eigenvalue problem and avoid refactoring a matrix when the scalar γ changes. Numerical results are presented which demonstrate the advantages of the new techniques.

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References

  1. I. Babuska,The finite element method with penalty, Math. Comp. 27 (1973), 221–228.

    Google Scholar 

  2. J. H. Bramble and J. A. Nitsche,A generalized Ritz least squares method for Dirichlet problems, SIAM Num. Anal. 10 (1973), 81–83.

    Google Scholar 

  3. J. H. Bramble and A. H. Schatz,Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions, Comm. Pure Appl. Math. XXIII (1970), 653–675.

    Google Scholar 

  4. S. Crandall and R. McCalley, Jr.,Numerical methods of analysis, in C. Harris and C. Crede,Shock and Vibration Handbook, v. 2, McGraw-Hill, New York, 1961.

    Google Scholar 

  5. C. R. Crawfod,Reduction of a band-symmetric generalized eigenvalue problem, Comm. ACM. 16 (1973), 41–44.

    Google Scholar 

  6. W. H. Enright,Improving the efficiency of matrix operations in the numerical solution of stiff ODEs, TOMS, to appear.

  7. P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders,Methods for modifying matrix factorizations, Math. Comp. 28 (1974), 505–535.

    Google Scholar 

  8. D. Goldfarb,Modification methods for inverting matrices and solving systems of linear equations, Math. Comp. 26 (1972), 829–852.

    Google Scholar 

  9. L. Kaufman,The LZ-algorithm to solve the generalized eigenvalue problem, SIAM Num. Anal. 11 (1974), 997–1025.

    Google Scholar 

  10. J. T. King and S. M. Serbin,Boundary flux estimates for elliptic problems by the perturbed variational method, Computing 16 (1976), 339–347.

    Google Scholar 

  11. J. T. King and S. M. Serbin,Computational experiments and techniques for the penalty method with extrapolation, Math. Comp., January, 1978.

  12. P. Lancaster,Lambda-matrices and vibrating systems, Pergamon Press, New York, 1966.

    Google Scholar 

  13. C. B. Moler and C. F. van Loan,Nineteen ways to compute the exponential of a matrix, Dept. of Comp. Sc. Tech. Rep. No. 76–283 (1976), Cornell University, Ithaca, New York.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, M5S 1A7, Toronto, Canada

    W. H. Enright & Steven M. Serbin

  2. Department of Mathematics, University of Tennessee, 37916, Knoxville, Tennessee, USA

    W. H. Enright & Steven M. Serbin

Authors
  1. W. H. Enright
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  2. Steven M. Serbin
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Additional information

The work of the first author was supported by the National Research Council of Canada and that of the second author by the National Science Foundation under Grant # MCS 76-24433.

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Enright, W.H., Serbin, S.M. A note on the efficient solution of matrix pencil systems. BIT 18, 276–281 (1978). https://doi.org/10.1007/BF01930897

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