Steepest descent, method of
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
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).
|||L.V. Kantorovich, "On the method of steepest descent" Dokl. Akad. Nauk SSSR , 56 : 3 (1947) pp. 233–236 (In Russian)|
|||L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)|
|||D.K. Faddeev, V.N. Faddeeva, "Computational methods of linear algebra" , Freeman (1963) (Translated from Russian)|
|||B.N. Pshenichnyi, Yu.M. Danilin, "Numerical methods in extremal problems" , MIR (1978) (Translated from Russian)|
|||N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)|
|[a1]||G.H. Golub, C.F. van Loan, "Matrix computations" , Johns Hopkins Univ. Press (1989)|
Steepest descent, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Steepest_descent,_method_of&oldid=13529