Relaxed fixed point iterations for matrix equations arising in Markov chain modeling

Fixed Point Iterations

Recall that a vector norm on R is a mapping ‖·‖ : R → R satisfying the following conditions: • ‖x‖ > 0 for x 6= 0. • ‖λx‖ = |λ|‖x‖ for x ∈ R and λ ∈ R. • ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ R. Since the space Rn×n of all matrices is also a vector space, it is also possible to consider norms there. In contrast to usual vectors, it is, however, also possible to multiply matrices (that is, the matric...

Anderson Acceleration for Fixed-Point Iterations

This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Machinery, 12 (1965), 547-560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anders...

Fixed Point Iterations for Real Functions

On Fixed-point Iterations for the Solution of Control Equations in Power Systems Transients

This paper contributes towards the establishment of a formal analysis method of control system equations solved through fixed-point iterations. The success of fixed-point iterations relies on contraction properties of the function to be iterated. A convergence criterion is presented and accuracy is not sacrificed over gain in computational time. The presented algorithms are illustrated in EMTP-...

analysis of ruin probability for insurance companies using markov chain

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