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Singular value decomposition for the Takagi factorization of symmetric matrices

Alexander M. Chebotarev
Organization: Laboratory ’’Mathematical Methods of Natural Sciences’’ at Higher School of Economics
Alexander E. Teretenkov
Organization: Dept. of Physics, Moscow State University
Journal / Anthology

Applied Mathematics and Computation
Year: 2014
Volume: 234
Page range: 380–384

We describe a simple implementation of the Takagi factorization of symmetric matrices A=U L UT with unitary U and diagonal L e T, e >= 0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of A. The method is based on an algebraically exact expression.
For parameterized family Ae =A+e R=UeLe UeT, e > 0 with distinct singular values, the unitary matrices Ue are discontinuous at the point e=0, if the singular values of A are multiple, but the composition UeLeUeT remains numerically stable and converges to A. The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available in Internet.

*Applied Mathematics