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Both minimize the 2-norm of the residual and do the same calculations in exact arithmetic when the matrix is symmetric. MINRES is a short-recurrence method with a constant memory requirement, whereas GMRES requires storing the whole Krylov space, so its memory requirement is roughly proportional to the number of iterations.
In this context, the cascading failure is known by the term cascade failure. A cascade failure can affect large groups of people and systems. The cause of a cascade failure is usually the overloading of a single, crucial router or node, which causes the node to go down, even briefly. It can also be caused by taking a node down for maintenance ...
Stability in this context means that a matrix norm of the matrix used in the iteration is at most unity, called (practical) Lax–Richtmyer stability. [2] Often a von Neumann stability analysis is substituted for convenience, although von Neumann stability only implies Lax–Richtmyer stability in certain cases. This theorem is due to Peter Lax.
Computing the square root of 2 (which is roughly 1.41421) is a well-posed problem.Many algorithms solve this problem by starting with an initial approximation x 0 to , for instance x 0 = 1.4, and then computing improved guesses x 1, x 2, etc.
Cascades in financial networks are situations in which the failure of one financial institution causes a cascading failure in another member of the financial network. In an extreme this can cause failure of the whole network in what is known as systemic failure.
The stability of numerical schemes can be investigated by performing von Neumann stability analysis. For time-dependent problems, stability guarantees that the numerical method produces a bounded solution whenever the solution of the exact differential equation is bounded.
Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to ...
In the former case, the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting. An equilibrium solution f e {\displaystyle f_{e}} to an autonomous system of first order ordinary differential equations is called: