Search results
Results from the WOW.Com Content Network
This page was last edited on 13 February 2025, at 01:06 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.
The Taylor series of f converges uniformly to the zero function T f (x) = 0, which is analytic with all coefficients equal to zero. The function f is unequal to this Taylor series, and hence non-analytic. For any order k ∈ N and radius r > 0 there exists M k,r > 0 satisfying the remainder bound above.
That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions ...
Suppose we have a continuous differential equation ′ = (,), =, and we wish to compute an approximation of the true solution () at discrete time steps ,, …,.For simplicity, assume the time steps are equally spaced:
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 (+) + (() +).
The step size is =. The same illustration for = The midpoint method converges faster than the Euler method, as .. Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs).
A computation using Taylor series shows that a linear multistep method is consistent if and only if = = = = + =. All the methods mentioned above are consistent (Hairer, Nørsett & Wanner 1993, §III.2).
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.