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The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
Example A: Find the truncation in calculating the first derivative of () = at = using a step size of = Solution: The first derivative of () = is ′ =, and at = ...
called the local Artin symbol, the local reciprocity map or the norm residue symbol. [4] [5] Let L⁄K be a Galois extension of global fields and C L stand for the idèle class group of L. The maps θ v for different places v of K can be assembled into a single global symbol map by multiplying the local
For example, = = =. The result 1 × 10 − 3 {\displaystyle 1\times 10^{-3}} is clearly representable, but there is not much faith in it. This is closely related to the phenomenon of catastrophic cancellation , in which the two numbers are known to be approximations.
For example, if the summands are uncorrelated random numbers with zero mean, the sum is a random walk, and the condition number will grow proportional to . On the other hand, for random inputs with nonzero mean the condition number asymptotes to a finite constant as n → ∞ {\displaystyle n\to \infty } .
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Verlet integration (French pronunciation:) is a numerical method used to integrate Newton's equations of motion. [1] It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics.