A discrete Newton algorithm for minimizing a function of many variables

A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn > 39, and storage is proportional ton rather thann2.

[1]  C. Lanczos An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[2]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[3]  C. Lanczos Solution of Systems of Linear Equations by Minimized Iterations1 , 1952 .

[4]  A. Goldstein On Steepest Descent , 1965 .

[5]  S. Kaniel Estimates for Some Computational Techniques - in Linear Algebra , 1966 .

[6]  J. Daniel The Conjugate Gradient Method for Linear and Nonlinear Operator Equations , 1967 .

[7]  J. Bunch ON DIRECT METHODS FOR SOLVING SYMMETRIC SYSTEMS OF LINEAR EQUATIONS , 1969 .

[8]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[9]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[10]  R. Brent Some Efficient Algorithms for Solving Systems of Nonlinear Equations , 1973 .

[11]  Philip E. Gill,et al.  Newton-type methods for unconstrained and linearly constrained optimization , 1974, Math. Program..

[12]  M. Powell,et al.  On the Estimation of Sparse Jacobian Matrices , 1974 .

[13]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[14]  R. Dennemeyer,et al.  CGS algorithms for unconstrained minimization of functions , 1975 .

[15]  J. Bus Convergence of Newton-like methods for solving systems of nonlinear equations , 1976 .

[16]  O. Axelsson Solution of linear systems of equations: Iterative methods , 1977 .

[17]  R. Tapia,et al.  QDN: a variable storage algorithm for unconstrained optimization , 1978 .

[18]  M. R. Osborne An Efficient Weak Line Search with Guaranteed Termination. , 1978 .

[19]  R. Chandra Conjugate gradient methods for partial differential equations. , 1978 .

[20]  S. Kaniel,et al.  A Modified Newton's Method for Unconstrained Minimization , 1979 .

[21]  M. Bartholomew-Biggs A Matrix Modification Method for Calculating Approximate Solutions to Systems of Linear Equations , 1979 .

[22]  P. Gill,et al.  Conjugate-Gradient Methods for Large-Scale Nonlinear Optimization. , 1979 .

[23]  M. Powell,et al.  On the Estimation of Sparse Hessian Matrices , 1979 .

[24]  D. J. Bell,et al.  Numerical Methods for Unconstrained Optimization , 1979 .

[25]  Magnus R. Hestenes,et al.  Conjugate Direction Methods in Optimization , 1980 .

[26]  Trond Steihaug,et al.  Truncated-newtono algorithms for large-scale unconstrained optimization , 1983, Math. Program..