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In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
Because the notation f n may refer to both iteration (composition) of the function f or exponentiation of the function f (the latter is commonly used in trigonometry), some mathematicians [citation needed] choose to use ∘ to denote the compositional meaning, writing f ∘n (x) for the n-th iterate of the function f(x), as in, for example, f ...
At any step in a Gauss-Seidel iteration, solve the first equation for in terms of , …,; then solve the second equation for in terms of just found and the remaining , …,; and continue to . Then, repeat iterations until convergence is achieved, or break if the divergence in the solutions start to diverge beyond a predefined level.
For example, the function f(x) = x 20 − 1 has a root at 1. Since f ′(1) ≠ 0 and f is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically. However, if initialized at 0.5, the first few iterates of Newton's method are approximately 26214, 24904, 23658, 22476, decreasing slowly, with only the 200th ...
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in.