Shapiro, A.; Botha, J. D. Dual algorithms for orthogonal Procrustes rotations. (English) Zbl 0664.65038 SIAM J. Matrix Anal. Appl. 9, No. 3, 378-383 (1988). Let \(A_ 1,...,A_ m\) be given \(n\times k\)-matrices. The problem is to find orthogonal \(k\times k\)-matrices \(Y_ 1,...,Y_ m\) which maximize \(g(Y_ 1,...,Y_ m):=\sum_{i<j}tr(Y_ i^ TA_ i^ TA_ jY_ j).\) This problem is related to a minimization problem for an upper bound for g. Under an eigenvalue condition, this upper bound is sharp in the sense that its minimization yields a global maximizer of g. The upper bound is minimized via a regularized Newton-type algorithm. Problems are caused by points where the upper bound is not differentiable. The authors also report about the numerical performance of their algorithm in detail. Reviewer: H.Engl Cited in 1 ReviewCited in 5 Documents MSC: 65F20 Numerical solutions to overdetermined systems, pseudoinverses Keywords:Procrustes rotations; orthogonal rotation; best least-squares fit; singular value decomposition; least upper bound; eigenvalues; nonsmooth optimization; minimization; regularized Newton-type algorithm PDFBibTeX XMLCite \textit{A. Shapiro} and \textit{J. D. Botha}, SIAM J. Matrix Anal. Appl. 9, No. 3, 378--383 (1988; Zbl 0664.65038) Full Text: DOI