68,159 research outputs found
Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method
The partial Schur factorization can be used to represent several eigenpairs
of a matrix in a numerically robust way. Different adaptions of the Arnoldi
method are often used to compute partial Schur factorizations. We propose here
a technique to compute a partial Schur factorization of a nonlinear eigenvalue
problem (NEP). The technique is inspired by the algorithm in [8], now called
the infinite Arnoldi method. The infinite Arnoldi method is a method designed
for NEPs, and can be interpreted as Arnoldi's method applied to a linear
infinite-dimensional operator, whose reciprocal eigenvalues are the solutions
to the NEP. As a first result we show that the invariant pairs of the operator
are equivalent to invariant pairs of the NEP. We characterize the structure of
the invariant pairs of the operator and show how one can carry out a
modification of the infinite Arnoldi method by respecting the structure. This
also allows us to naturally add the feature known as locking. We nest this
algorithm with an outer iteration, where the infinite Arnoldi method for a
particular type of structured functions is appropriately restarted. The
restarting exploits the structure and is inspired by the well-known implicitly
restarted Arnoldi method for standard eigenvalue problems. The final algorithm
is applied to examples from a benchmark collection, showing that both
processing time and memory consumption can be considerably reduced with the
restarting technique
Policy iteration for perfect information stochastic mean payoff games with bounded first return times is strongly polynomial
Recent results of Ye and Hansen, Miltersen and Zwick show that policy
iteration for one or two player (perfect information) zero-sum stochastic
games, restricted to instances with a fixed discount rate, is strongly
polynomial. We show that policy iteration for mean-payoff zero-sum stochastic
games is also strongly polynomial when restricted to instances with bounded
first mean return time to a given state. The proof is based on methods of
nonlinear Perron-Frobenius theory, allowing us to reduce the mean-payoff
problem to a discounted problem with state dependent discount rate. Our
analysis also shows that policy iteration remains strongly polynomial for
discounted problems in which the discount rate can be state dependent (and even
negative) at certain states, provided that the spectral radii of the
nonnegative matrices associated to all strategies are bounded from above by a
fixed constant strictly less than 1.Comment: 17 page
Random Unitaries Give Quantum Expanders
We show that randomly choosing the matrices in a completely positive map from
the unitary group gives a quantum expander. We consider Hermitian and
non-Hermitian cases, and we provide asymptotically tight bounds in the
Hermitian case on the typical value of the second largest eigenvalue. The key
idea is the use of Schwinger-Dyson equations from lattice gauge theory to
efficiently compute averages over the unitary group.Comment: 14 pages, 1 figur
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