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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 = ...
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in ...
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map. The inverse function theorem implies that a smooth map f : X → Y {\displaystyle f:X\to Y} is a local diffeomorphism if and only if the derivative D f x : T x X → T f ( x ) Y {\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y} is a linear ...
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
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.
Global optimization is distinguished from local optimization by its focus on finding the minimum or maximum over the given set, as opposed to finding local minima or maxima. Finding an arbitrary local minimum is relatively straightforward by using classical local optimization methods. Finding the global minimum of a function is far more ...