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The partial Schur factorization can be used to represent several eigenpairsof a matrix in a numerically robust way. Different adaptions of the Arnoldimethod are often used to compute partial Schur factorizations. We propose herea technique to compute a partial Schur factorization of a nonlinear eigenvalueproblem (NEP). The technique is inspired by the algorithm in [8], now calledthe infinite Arnoldi method. The infinite Arnoldi method is a method designedfor NEPs, and can be interpreted as Arnoldi's method applied to a linearinfinite-dimensional operator, whose reciprocal eigenvalues are the solutionsto the NEP. As a first result we show that the invariant pairs of the operatorare equivalent to invariant pairs of the NEP. We characterize the structure ofthe invariant pairs of the operator and show how one can carry out amodification of the infinite Arnoldi method by respecting the structure. Thisalso allows us to naturally add the feature known as locking. We nest thisalgorithm with an outer iteration, where the infinite Arnoldi method for aparticular type of structured functions is appropriately restarted. Therestarting exploits the structure and is inspired by the well-known implicitlyrestarted Arnoldi method for standard eigenvalue problems. The final algorithmis applied to examples from a benchmark collection, showing that bothprocessing time and memory consumption can be considerably reduced with therestarting technique.

Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method