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:
“The men—the undergraduates of Yale and Princeton are cleaner, healthier, better-looking, better dressed, wealthier and more attractive than any undergraduate body in the country.”
—F. Scott Fitzgerald (1896–1940)
“Traditionally, marriage involved a kind of bartering, rather than mutual inter-dependence or role sharing. Husbands financially and economically supported wives, while wives emotionally, psychologically and socially supported husbands. He brought home the bacon, she cooked it. He fixed the plumbing, she the psyche.”
—Bettina Arndt (20th century)
“It’s my feeling that God lends you your children until they’re about eighteen years old. If you haven’t made your points with them by then, it’s too late.”
—Betty Ford (b. 1918)