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The first Dahlquist barrier states that a zero-stable and linear q-step multistep method cannot attain an order of convergence greater than q + 1 if q is odd and greater than q + 2 if q is even. If the method is also explicit, then it cannot attain an order greater than q ( Hairer, Nørsett & Wanner 1993 , Thm III.3.5).
A linear multistep method is zero-stable if all roots of the characteristic equation that arises on applying the method to ′ = have magnitude less than or equal to unity, and that all roots with unit magnitude are simple. [2]
For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors. Linear multistep methods that satisfy the condition of zero-stability have the same relation between local and global errors as one-step methods.
Explicit multistep methods can never be A-stable, just like explicit Runge–Kutta methods. Implicit multistep methods can only be A-stable if their order is at most 2. The latter result is known as the second Dahlquist barrier; it restricts the usefulness of linear multistep methods for stiff equations. An example of a second-order A-stable ...
General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include multistage Runge–Kutta methods that use intermediate collocation points , as well as linear multistep methods that save a finite time history of the solution.
In 1974, Baddeley and Hitch [5] introduced and made popular the multicomponent model of working memory.This theory proposes a central executive that, among other things, is responsible for directing attention to relevant information, suppressing irrelevant information and inappropriate actions, and for coordinating cognitive processes when more than one task must be done at the same time.
Germund Dahlquist (16 January 1925 – 8 February 2005) was a Swedish mathematician known primarily for his early contributions to the theory of numerical analysis as applied to differential equations.
More specifically, the signal-detection model, which assumes that memory strength is a graded phenomenon (not a discrete, probabilistic phenomenon) predicts that the ROC will be curvilinear, and because every recognition memory ROC analyzed between 1958 and 1997 was curvilinear, the high-threshold model was abandoned in favor of signal ...