04016066

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04016066 Mikhalevich, V.S. Redkovskij, N.N. Numerical method of minimization on the set of positive-definite matrices. Sov. Phys., Dokl. 31, 777-779 (1986); translation from Dokl. Akad. Nauk SSSR 290, 809-812 (1986). 1986 Consultants Bureau, New York EN Newton method convergence acceleration linear convergence smooth functions nonnegative-definite matrices positive-definite tridiagonal matrices local minimum Zbl 0598.65045 The problem of seeking the minimum of smooth functions f(A) on the set of nonnegative-definite matrices $\{$ $A\}$ is considered. The set $\{$ $A\}$ is parametrized i.e. a mapping A(x) is constructed such that for any X from some Euclidean space the matrix A(x) is symmetric and nonnegative-definite. The original problem is then reduced to minimization of the function f[A(x)] over X. In their previous paper [ibid. 283, 1081-1085 (1985; Zbl 0598.65045)] the authors defined the mapping A(x) using diagonal matrices with nonnegative elements. Here A(x) is constructed by parametrizing positive-definite tridiagonal matrices. The authors suggest to reduce any symmetric positive-definite matrix A to tridiagonal form $A\sb T$ by some orthogonal transformation U, $A=U\sp*A\sb TU$. The search for a local minimum of a function f on $\{$ $A\}$ is equivalent to the minimization of the function $f[A(x)]=f(U\sp*A\sb TU)$ over variables which parametrize the matrices U, $A\sb T$. A linearly convergent algorithm for the minimization of f[A(x)] based on a gradient scheme is presented. Newton's procedure for the minimization of f[A(x)] which enables to accelerate the rate of convergence is also described. V.S.Izhutkin