A note on golden means, nonlinear matrix equations and structured doubling algorithms
The authors consider two matrix equations \[ BX^{-1}B-X-A=0, \quad XA^{-1}X+X-(B-A)=0, \] where \(0 < A \leq B\). They propose some structured doubling algorithms for computing solutions of these equations and so-called golden means \[ A\natural B = \tfrac{1}{2}[A + A\sharp(4B-3A)], \quad A\bar\natural B = \tfrac{1}{2}[-A + A\sharp(4B-3A)], \] where \(A\sharp B = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}\) [see the paper by \textit{Y.~Lim}, SIAM J. Matrix Anal. Appl. 29, No.~1, 54--66 (2006; Zbl 1171.15014)]. As illustration, some numerical examples are given.
matrix golden mean, structured doubling algorithm
scientific article; zbMATH DE number 5778016