Singular value decomposition for the Takagi factorization of symmetric matrices

We describe a simple implementation of the Takagi factorization of symmetric matrices A = U Lambda U-T with unitary U and diagonal Lambda >= 0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of Lambda. The method is based on an algebraically exact expression. For parameterized family A(epsilon) = A + epsilon R = U-epsilon Lambda U-epsilon(epsilon)T, epsilon >= 0 with distinct singular values, the unitary matrices U-epsilon are discontinuous at the point epsilon = 0, if the singular values of A are multiple, but the composition U-epsilon Lambda U-epsilon(epsilon)T 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 on Internet. (c) 2014 Elsevier Inc. All rights reserved.