Search results
Results from the WOW.Com Content Network
A series acceleration method is a sequence transformation that transforms the convergent sequences of partial sums of a series into more quickly convergent sequences of partial sums of an accelerated series with the same limit.
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
Anderson acceleration is a method to accelerate the convergence of the fixed-point sequence. [2] Define the residual () = (), and denote = and = (where corresponds to the sequence of iterates from the previous paragraph).
For example, to calculate the autocorrelation of the real signal sequence = (,,) (i.e. =, =, =, and = for all other values of i) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: +
As a vector, jerk j can be expressed as the first time derivative of acceleration, second time derivative of velocity, and third time derivative of position: = = = ()Where:
Illustration of gradient descent on a series of level sets. Gradient descent is based on the observation that if the multi-variable function is defined and differentiable in a neighborhood of a point , then () decreases fastest if one goes from in the direction of the negative gradient of at , ().
A Zeno machine is a Turing machine that can take an infinite number of steps, and then continue take more steps. This can be thought of as a supertask where units of time are taken to perform the -th step; thus, the first step takes 0.5 units of time, the second takes 0.25, the third 0.125 and so on, so that after one unit of time, a countably infinite number of steps will have been performed.