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:
“I confess, the motto of the Globe newspaper is so attractive to me, that I can seldom find much appetite to read what is below it in its columns, “The world is governed too much.””
—Ralph Waldo Emerson (1803–1882)
“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)
“There are good points about all such wars. People forget self. The virtues of magnanimity, courage, patriotism, etc., etc., are called into life. People are more generous, more sympathetic, better, than when engaged in the more selfish pursuits of peace.”
—Rutherford Birchard Hayes (1822–1893)