# Steepest descent, method of

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A special instance of the method of descent (cf. Descent, method of), when the direction of descent is chosen as the direction opposite to . The formulas of the method of steepest descent are where the parameters are determined by the condition that the function has maximum decrease at each step. If is twice continuously-differentiable and its matrix of second derivatives satisfies the inequality for any , with constants , then (see , ) the sequence converges to a solution of the problem of minimizing , the convergence rate being that of a geometric progression with quotient .

The method of steepest descent has been widely applied to the solution of systems of linear algebraic equations with a Hermitian or positive-definite matrix . In the real symmetric case, the problem of solving this system is equivalent to finding a vector in an -dimensional vector space minimizing the functional (*)

Applied to (*), the method of steepest descent takes the form where the value of is determined by minimization of the functional (*), according to the formula If the spectrum of lies in an interval , , on the real axis, then the sequence converges to the solution at the rate of a geometric progression with quotient .

The method of steepest descent can be applied to solve an operator equation with a self-adjoint positive-definite bounded operator . If does not satisfy these conditions, the problem can be symmetrized, reduced to the problem and then one can apply the method of steepest descent (see also Minimal discrepancy method).

How to Cite This Entry:
Steepest descent, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steepest_descent,_method_of&oldid=13529
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article