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 intellect,—that is miraculous! Who has it, has the talisman: his skin and bones, though they were of the color of night, are transparent, and the everlasting stars shine through, with attractive beams.”
—Ralph Waldo Emerson (1803–1882)
“It is not merely the likeness which is precious ... but the association and the sense of nearness involved in the thing ... the fact of the very shadow of the person lying there fixed forever! It is the very sanctification of portraits I think—and it is not at all monstrous in me to say ... that I would rather have such a memorial of one I dearly loved, than the noblest Artist’s work ever produced.”
—Elizabeth Barrett Browning (1806–1861)
“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)