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The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
The stability of fixed points of a system of constant coefficient linear differential equations of first order can be analyzed using the eigenvalues of the corresponding matrix. An autonomous system ′ =, where x(t) ∈ R n and A is an n×n matrix with real entries, has a constant solution =
One often uses fixed-point iteration or (some modification of) the Newton–Raphson method to achieve this. It costs more time to solve this equation than explicit methods; this cost must be taken into consideration when one selects the method to use.
An algorithm for solving a linear evolutionary partial differential equation is stable if the total variation of the numerical solution at a fixed time remains bounded as the step size goes to zero. The Lax equivalence theorem states that an algorithm converges if it is consistent and stable (in this sense).
The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure. [4] This includes the whole left half of the complex plane, making it suitable for the solution of stiff equations. [5] In fact, the backward Euler method is even L-stable.
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.
Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points. In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
For r < 1, exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, appears between [0, 1] as a stable fixed point. When the parameter r = 1, the trajectory of the logistic map converges to 0 as before , but the convergence speed is slower at r = 1 .