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Model Problems and Preconditioning

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Abstract

Model problem ADI iteration is discussed for three distinct classes of problems. The first is discretized elliptic systems with separable coefficients so that difference equations may be split into two commuting matrices. The second is where the model ADI problem approximates the actual nonseparable problem and serves as a preconditioner. The third is an entirely different class of problems than initially considered. These are Lyapunov and Sylvester matrix equations in which commuting operations are inherent.

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Notes

  1. 1.

    Nowadays, a variety of names are attached to variants of the Lanczos recursion formulas derived by minimizing different functionals. Forty years ago Gabe Horvay (a GE mechanics expert and one of my associates at KAPL) introduced me to this new approach developed by his friend Lanczos and, influenced strongly by Gabe, I became accustomed to referring to all these schemes as “Lanczos algorithms.” Hence, the method of “conjugate gradients” is often referred to as “Lanczos’ method” in my early works.

  2. 2.

    My definition of positive definite in those days implied symmetry. More recently, the term has been used by some with a different definition so that it is now customary to impose symmetry and denote A as “SPD” for “symmetric and positive definite.” I still prefer the old definition in Wachspress,  1966 , but approve wholeheartedly of the use of SPD to resolve any doubt.

  3. 3.

    Al Schatz (Cornell) advised me when I was preparing work on this preconditioner for publication that he had considered a related approximation for solving finite element problems but I have not yet seen a published reference to this work. His effort was devoted more to approximating equations over nonrectangular grids by preconditioning equations over rectangular grids.

  4. 4.

    While at the University of Tennessee in Knoxville I interacted with Al Geist at Oak Ridge and awakened his interest in gaussian reduction to tridiagonal form. Our work stimulated renewed interest by several mathematicians with whom we communicated.

References

  • Bartels RH, Stewart GW (1972) Solution of the matrix equation AX + XB = C. Commun. ACM 15:820–826

    Article  Google Scholar 

  • Buzbee BL, Golub GH, Nielson CW (1970) On direct methods for solving Poisson’s equation. SIAM J Numer Anal 7:627–656

    Article  MathSciNet  MATH  Google Scholar 

  • Byers R (1983) Hamilton and symplectic algorithms for the algebraic Riccati equation. Ph.D. Thesis, Cornell University

    Google Scholar 

  • Cesari L (1937) Sulla risoluzione dei sistemi di equazioni lineari par aprossimazioni successive. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. VI S–25:422–428

    Google Scholar 

  • Concus P, Golub G (1973) Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAM J Numer Anal 20:1103–1120

    Article  MathSciNet  Google Scholar 

  • Dax A, Kaniel S (1981) The ELR method for computing the eigenvalues of a general matrix. SIAM J Numer Anal 18:597–605

    Article  MathSciNet  MATH  Google Scholar 

  • Diliberto SP, Strauss EG (1951) On the approximation of a function of several variables by a sum of functions of fewer variables. Pacific J Math 1:195–210

    Article  MathSciNet  MATH  Google Scholar 

  • D’Yakonov EG (1961) An iteration method for solving systems of linear difference equations. Soviet Math (AMS translation) 2:647

    Google Scholar 

  • Engeli M, Ginsburg T, Rutishauser H, Stiefel E (1959) Refined iterative methods for computation of the solution and the eigenvalues of self-adjoint boundary value problems. Mitteilungen aus dem Institut für angewandte Mathematik, vol 8. Birkhäuser, Basel

    Book  Google Scholar 

  • Geist GA (1989) Reduction of a general matrix to tridiagonal form. ORNL/TM-10991

    Google Scholar 

  • Geist GA, Lu A, Wachspress EL (1989) Stabilized Gaussian reduction of an arbitrary matrix to tridiagonal form. ORNL/TM-11089

    Google Scholar 

  • Golub GH, O’Leary DP (1987) Some history of the conjugate gradient and Lanczos algorithms. UMIACS-TR-87-20, CS-TR-1859, University of Maryland Institute for Advanced Computer Studies and Computer Science Department

    Google Scholar 

  • Golub GH, Nash S, Van Loan C (1979) A Hessenberg-Shur method for solution of the problem AX + XB = C. IEEE Trans Automat Contr AC24:909–913

    Google Scholar 

  • Hare DE, Tang WP (1989) Toward a stable tridiagonalization algorithm for general matrices. University of Waterloo technical report CS-89-03, 1989

    Google Scholar 

  • Howell GW (1994) Toward an efficient and stable determination of spectra of a general matrix and a more efficient solution of the Lyapunov equation. First world congress of nonlinear analysis, 1994

    Google Scholar 

  • Hestenes M, Stieffel E (1952) Methods of conjugate gradients for solving linear systems. J Res NBS 49(6):409–436

    MATH  Google Scholar 

  • Howell G, Diaa N (2005) Algorithm 841: BHESS: Gaussian reduction to a similar banded Hessenberg form, ACM transactions on Mathematical Software vol. 31 (1)

    Google Scholar 

  • Hurwitz H (1984) Infinitesimal scaling: A new procedure for modeling exterior field problems. IEEE Trans 20:1918–1923

    Google Scholar 

  • Istace MP, Thiran JP (1993) On the third and fourth Zolotarev problems in the complex plane. Math Comput

    Google Scholar 

  • Opfer G, Schober G (1984) Richardson’s iteration for nonsymmetric matrices. Linear Algebra Appl 58:343–361

    Article  MathSciNet  MATH  Google Scholar 

  • Peaceman DW, Rachford HH Jr (1955) The numerical solution of parabolic and elliptic differential equation. J SIAM 3:28–41

    MathSciNet  MATH  Google Scholar 

  • Smith RA (1968) Matrix equation XA + XB = C. SIAM J Appl Math 16(1):198–201

    Article  MathSciNet  MATH  Google Scholar 

  • Starke G (1989) Rational minimization problems in connection with optimal ADI-parameters for complex domains. Ph.D. thesis, Institute für Praktische Mathematik, Universität Karlsruhe, FRG

    Google Scholar 

  • Tang WP (1988) A stabilized algorithm for tridiagonalization of an unsymmetric matrix. University of Waterloo Technical Report CS-88-14

    Google Scholar 

  • Varga RS (1962) Matrix iterative analysis. Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Wachspress EL (1957) CURE: A generalized two-space-dimension multigroup coding for the IBM-704. Knolls Atomic Power Report KAPL-1724, AEC R&D Physics and Mathematics TID-4500, 13th edn. Knolls Atomic Power Lab, Schenectady

    Google Scholar 

  • Wachspress EL (1963) Extended application of alternating-direction-implicit iteration model problem theory. J Soc Indust Appl Math 11:994–1016

    Article  MathSciNet  Google Scholar 

  • Wachspress EL (1966) Iterative solution of elliptic systems. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Wachspress EL (1984) Generalized ADI preconditioning. Comput Math Appl 10(6):457–461

    Article  MathSciNet  MATH  Google Scholar 

  • Wachspress EL (1988a) ADI solution of the Lyapunov equation. MSI Workshop on practical iterative methods for large-scale computations, Minneapolis, MN, 1988

    Google Scholar 

  • Wachspress EL (1988b) Iterative solution of the Lyapunov matrix equation. Appl Math Lett 1: 87–90

    Article  MathSciNet  MATH  Google Scholar 

  • Wachspress EL (1991) Optimum alternating direction implicit iteration parameters for rectangular regions in the complex plane. Center for Mathematical Analysis of the Australian National University Report CMA-R33-90, 1991

    Google Scholar 

  • Watkins D (1988) Use of the LR algorithm to tridiagonalize a general matrix. SIAM annual meeting, Minneapolis, MN, 1988

    Google Scholar 

  • Wilkinson JH (1965) The algebraic eigenvalue problem. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Young DM, Wheeler MF (1964) Alternating direction methods for solving partial difference equations. Nonlinear problems in engineering. Academic, New York, pp 220–246

    Google Scholar 

Download references

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Authors and Affiliations

  1. East Windsor, New Jersey, USA

    Eugene Wachspress

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  1. Eugene Wachspress
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Wachspress, E. (2013). Model Problems and Preconditioning. In: The ADI Model Problem. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5122-8_3

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  • DOI: https://doi.org/10.1007/978-1-4614-5122-8_3

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