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:
“One of the most attractive things about the flowers is their beautiful reserve.”
—Henry David Thoreau (1817–1862)
“Genius detects through the fly, through the caterpillar, through the grub, through the egg, the constant individual; through countless individuals the fixed species; through many species the genus; through all genera the steadfast type; through all the kingdoms of organized life the eternal unity. Nature is a mutable cloud which is always and never the same.”
—Ralph Waldo Emerson (1803–1882)
“We only part to meet again.
Change, as ye list, ye winds: my heart shall be
The faithful compass that still points to thee.”
—John Gay (1685–1732)