## Factorizing complex symmetric matrices with positive definite real and imaginary parts

HTML articles powered by AMS MathViewer

- by Nicholas J. Higham PDF
- Math. Comp.
**67**(1998), 1591-1599 Request permission

## Abstract:

Complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for LU factorization. This result adds to the classes of matrix for which it is known to be safe not to pivot in LU factorization. Block $\mathrm {LDL^T}$ factorization with the pivoting strategy of Bunch and Kaufman is also considered, and it is shown that for such matrices only $1\times 1$ pivots are used and the same growth factor bound of 2 holds, but that interchanges that destroy band structure may be made. The latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in LINPACK and LAPACK.## References

- E. Anderson, Z. Bai, C. H. Bischof, J. W. Demmel, J. J. Dongarra, J. J. Du Croz, A. Greenbaum, S. J. Hammarling, A. McKenney, S. Ostrouchov, and D. C. Sorensen,
*LAPACK users’ guide, Release 2.0*, second ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1995. - James R. Bunch and Linda Kaufman,
*Some stable methods for calculating inertia and solving symmetric linear systems*, Math. Comp.**31**(1977), no. 137, 163–179. MR**428694**, DOI 10.1090/S0025-5718-1977-0428694-0 - Jane K. Cullum and Ralph A. Willoughby,
*A $QL$ procedure for computing the eigenvalues of complex symmetric tridiagonal matrices*, SIAM J. Matrix Anal. Appl.**17**(1996), no. 1, 83–109. MR**1372924**, DOI 10.1137/S0895479894137639 - J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart,
*LINPACK users’ guide*, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1979. - Jack J. Dongarra, Iain S. Duff, Danny C. Sorensen, and Henk A. van der Vorst,
*Solving linear systems on vector and shared memory computers*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1991. MR**1084164** - Graeme Fairweather,
*A note on the efficient implementation of certain Padé methods for linear parabolic problems*, BIT**18**(1978), no. 1, 106–109. MR**488820**, DOI 10.1007/BF01947749 - Roland W. Freund,
*Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices*, SIAM J. Sci. Statist. Comput.**13**(1992), no. 1, 425–448. MR**1145195**, DOI 10.1137/0913023 - Philip E. Gill, Michael A. Saunders, and Joseph R. Shinnerl,
*On the stability of Cholesky factorization for symmetric quasidefinite systems*, SIAM J. Matrix Anal. Appl.**17**(1996), no. 1, 35–46. MR**1372921**, DOI 10.1137/S0895479893252623 - Gene H. Golub and Charles Van Loan,
*Unsymmetric positive definite linear systems*, Linear Algebra Appl.**28**(1979), 85–97. MR**549423**, DOI 10.1016/0024-3795(79)90122-8 - Emilie V. Haynsworth,
*Determination of the inertia of a partitioned Hermitian matrix*, Linear Algebra Appl.**1**(1968), no. 1, 73–81. MR**223392**, DOI 10.1016/0024-3795(68)90050-5 - Nicholas J. Higham,
*The Test Matrix Toolbox for Matlab (version 3.0)*, Numerical Analysis Report No. 276, Manchester Centre for Computational Mathematics, Manchester, England, September 1995. - Nicholas J. Higham,
*Accuracy and stability of numerical algorithms*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR**1368629** - Mark T. Jones and Merrell L. Patrick,
*Bunch–Kaufman factorization for real symmetric indefinite banded matrices*, SIAM J. Matrix Anal. Appl.**14**(1993), no. 2, 553–559. - Mark T. Jones and Merrell L. Patrick,
*Factoring symmetric indefinite matrices on high-performance architectures*, SIAM J. Matrix Anal. Appl.**15**(1994), no. 1, 273–283. MR**1257634**, DOI 10.1137/S089547989018008X - Roy Mathias,
*Matrices with positive definite Hermitian part: inequalities and linear systems*, SIAM J. Matrix Anal. Appl.**13**(1992), no. 2, 640–654. MR**1152771**, DOI 10.1137/0613038 - Giorgio Moro and Jack H. Freed,
*Calculation of ESR spectra and related Fokker-Planck forms by the use of the Lanczos algorithm*, J. Chem. Phys.**74**(1981), no. 7, 3757–3773. MR**610918**, DOI 10.1063/1.441604 - D. Schmitt, B. Steffen, and T. Weiland,
*2D and 3D computations of lossy eigenvalue problems*, IEEE Trans. Magnetics**30**(1994), no. 5, 3578–3581. - Steven M. Serbin,
*On factoring a class of complex symmetric matrices without pivoting*, Math. Comp.**35**(1980), no. 152, 1231–1234. MR**583500**, DOI 10.1090/S0025-5718-1980-0583500-9 - Robert J. Vanderbei,
*Symmetric quasidefinite matrices*, SIAM J. Optim.**5**(1995), no. 1, 100–113. MR**1315706**, DOI 10.1137/0805005 - H. A. Van Der Vorst and J. B. M. Melissen,
*A Petrov–Galerkin type method for solving $Ax=b$, where $A$ is symmetric complex*, IEEE Trans. Magnetics**26**(1990), no. 2, 706–708.

## Additional Information

**Nicholas J. Higham**- Affiliation: Department of Mathematics, University of Manchester, Manchester, M13 9PL, England
- Email: higham@ma.man.ac.uk
- Received by editor(s): December 8, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Math. Comp.
**67**(1998), 1591-1599 - MSC (1991): Primary 65F05
- DOI: https://doi.org/10.1090/S0025-5718-98-00978-8
- MathSciNet review: 1474652