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:
“Military glory—the attractive rainbow that rises in showers of blood.”
—Abraham Lincoln (1809–1865)
“The aim of every artist is to arrest motion, which is life, by artificial means and hold it fixed so that a hundred years later, when a stranger looks at it, it moves again since it is life. Since man is mortal, the only immortality possible for him is to leave something behind him that is immortal since it will always move. This is the artist’s way of scribbling “Kilroy was here” on the wall of the final and irrevocable oblivion through which he must someday pass.”
—William Faulkner (1897–1962)
“PLAIN SUPERFICIALITY is the character of a speech, in which any two points being taken, the speaker is found to lie wholly with regard to those two points.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)