Attractive Fixed Points
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 x1 in the basin of attraction of x, and let xn+1 = f(xn) for n ≥ 1, and the sequence {xn}n ≥ 1 will converge to the solution x. If the function f is continuously differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the fixed point. If this condition holds at the fixed point, then a sufficiently small neighborhood (basin of attraction) must exist.
Read more about this topic: Iterative Method
Famous quotes containing the words attractive, fixed and/or points:
“... when you do get a job everybody says, “Well, they wanted a black woman,” which necessarily puts you on a level where you have to prove yourself above being a woman and being black.... Now, I would say, in certain situations, it helped me simply because I was mildly attractive, not because I was black or a woman. That gets you more mileage than anything else.... God help you if you’re not an attractive woman.”
—Theresa Brown (b. 1957)
“... social evils are dangerously contagious. The fixed policy of persecution and injustice against a class of women who are weak and defenseless will be necessarily hurtful to the cause of all women.”
—Fannie Barrier Williams (1855–1944)
“The men who carry their points do not need to inquire of their constituents what they should say, but are themselves the country which they represent: nowhere are its emotions or opinions so instant and so true as in them; nowhere so pure from a selfish infusion.”
—Ralph Waldo Emerson (1803–1882)