20.500.12556/RUL-175108 Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding It is well known that a family of $n \times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers. commuting matrices joint eigenvalues Rayleigh quotient randomized numerical linear algebra multiparameter eigenvalue problems polynomial systems komutirajoče matrike skupne lastne vrednosti Rayleighov kvocient verjetnostna numerična linearna algebra večparametrični problemi lastnih vrednosti polinomski sistemi true false true Angleški jezik Slovenski jezik Članek v reviji 2025-10-16 10:41:49 2025-10-16 10:41:55 2025-11-18 13:36:19 0000-00-00 00:00:00 2025 0 0 Str. 861–892 iss. 3 Vol. 100 Nov. 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.6 1017-1398 10.1007/s11075-024-01971-0 215046403 s11075-024-01971-0_(1).pdf s11075-024-01971-0_(1).pdf 1 ABE4307B337C79A3B5324E1572579258 282fe7988f397c174e7383b1f04e02d0e1f54b445424f8c9ba2a8cd0f6afacdb c38b49b3-aa6c-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=220181 https://link.springer.com/article/10.1007/s11075-024-01971-0 1 b5accc6b-aa6b-11f0-9328-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=220180 Fakulteta za matematiko in fiziko 0 0 0