Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Generation of conjugate directions for unconstrained minimization without derivatives

Author: Larry Nazareth
Journal: Math. Comp. 30 (1976), 115-131
MSC: Primary 65K05
MathSciNet review: 0398100
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze a technique for unconstrained minimization without derivatives. This stems from two theorems proved by M. J. D. Powell. A particular version, which we consider in detail, is related to the Jacobi process for finding the eigensystem of a symmetric matrix, and the two processes, although different, help to illuminate one another. We study convergence of the search directions to mutual conjugacy, cases when cycling occurs and identify a broad class of 'cyclic patterns' for which convergence to mutual conjugacy is proven.

References [Enhancements On Off] (What's this?)

  • [1] M. J. D. POWELL, "An efficient method for finding the minimum of a function of several variables without calculating derivatives," Comput. J., v. 7, 1964, pp. 155-162. MR 32 #4828. MR 0187376 (32:4828)
  • [2] W. I. ZANGWILL, "Minimizing a function without calculating derivatives," Comput. J., v. 10, 1967, pp. 293-296. MR 38 #2930. MR 0234614 (38:2930)
  • [3] R. P. BRENT, Algorithms for Minimization Without Derivatives, Prentice-Hall Ser. in Automatic Computation, Prentice-Hall, Englewood Cliffs, N. J., 1973. MR 49 #4251. MR 0339493 (49:4251)
  • [4] N. W. BRODLIE, A New Method for Unconstrained Minimization without Evaluating Derivatives, IBM Report UKSC-0019-0019, IBM, Peterlee, United Kingdom, 1972.
  • [5] M. J. D. POWELL, Unconstrained Minimization Algorithms Without Computation of Derivatives, UK. A. E. A. Harwell Report HL72/1713, 1972.
  • [6] J. L. NAZARETH, Part I: Unified Approach to Unconstrained Minimization. Part II: Generation of Conjugate Directions for Unconstrained Minimization Without Derivatives, Dept. of Computer Science, Univ. of California, Berkeley, Rep. No. 23, 1973.
  • [7] ELDON R. HANSEN, "On cyclic Jacobi methods," J. Soc. Indust. Appl. Math., v. 11, 1963, pp. 448-459. MR 27 #6381. MR 0156458 (27:6381)
  • [8] J. L. NAZARETH, On the Convergence of the Cyclic Jacobi Process, Argonne National Lab. Tech. Memo, ANL-AMD, 1974.
  • [9] J. H. WILKINSON, "Note on the quadratic convergence of the cyclic Jacobi process," Numer. Math., v. 4, 1962, pp. 296-300. MR 26 #4473. MR 0146954 (26:4473)
  • [10] J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. MR 0184422 (32:1894)
  • [11] G. E. FORSYTHE & P. HENRICI, "The cyclic Jacobi method for computing the principal values of a complex matrix," Trans. Amer. Math. Soc., v. 94, 1960, pp. 1-23. MR 22 #710. MR 0109825 (22:710)
  • [12] PETER HENRICI, "On the speed of convergence of cyclic and quasicyclic Jacobi methods for computing eigenvalues of Hermitian matrices," J. Soc. Indust. Appl. Math., v. 6, 1958, pp. 144-162. MR 20 #2084. MR 0095582 (20:2084)
  • [13] H. P. M. VAN KEMPEN, "On the convergence of the classical Jacobi method for real symmetric matrices with non-distinct eigenvalues," Numer. Math., v. 9, 1966, pp. 11-18. MR 34 #2163. MR 0202290 (34:2163)
  • [14] J. N. FRANKLIN, Matrix Theory, Prentice-Hall, Englewood Cliffs, N. J., 1968. MR 38 #5798. MR 0237517 (38:5798)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65K05

Retrieve articles in all journals with MSC: 65K05

Additional Information

Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society