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:
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—Lois Wyse (20th century)
“I, who travel most often for my pleasure, do not direct myself so badly. If it looks ugly on the right, I take the left; if I find myself unfit to ride my horse, I stop.... Have I left something unseen behind me? I go back; it is still on my road. I trace no fixed line, either straight or crooked.”
—Michel de Montaigne (1533–1592)
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—Henry David Thoreau (1817–1862)