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

Higham, Nicholas J. (1998) Factorizing complex symmetric matrices with positive definite real and imaginary parts. Mathematics of Computation, 67 (224). pp. 1591-1599. ISSN 1088-6842

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.
Item Type: Article Complex symmetric matrices, LU factorization, diagonal pivoting factorization, block $\mathrm{LDL^T}$ factorization, Bunch--Kaufman pivoting strategy, growth factor, band matrix, LINPACK, LAPACK MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Ms Lucy van Russelt 30 Jun 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/337