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AN ADAPTIVE NONLINEAR LEAST-SQUARES ALGORITHMS

IP.com Disclosure Number: IPCOM000148240D
Original Publication Date: 1979-Jul-31
Included in the Prior Art Database: 2007-Mar-29
Document File: 24 page(s) / 2M

Publishing Venue

Software Patent Institute

Related People

Dennis Jr., John E.: AUTHOR [+4]

Abstract

AN ADAPTIVE ~~JONLXNEAR LEAST-SQUARES~ .-- /---, --'.* *Cornell University. Research supported in part by NSF Grant MCS76-00324 *NBER Computer Research Center Research supported in part by NSF Grant MCS76-00324NSF Grant MCS78-09525United States Army Contract DAAG29-75-C-0024 tMassachusetts Institute of Technology. Research supported in part by NSF Grants SOC76-14311 and MC:S76-00324 ALGORITHM John E. Dennis, Jr.*, David M. Gay**, and Roy E. Welscht TR77-321 Revised July 1979 ABSTRACT NL2SOL is a modular program for solving nonlinear least- squares problems that incorporates a number of novel features. It maintains a secant approximation S to the second-orderpart of the least-squares Hessian and adaptively decides when to use this approximation. S is "sized" before updating,something which is similar t o Oren-Luenberger scaling. The step choice algorithm is based on minimizing a local quadratic modelof the sum of squares function const:rained t o an elliptical trust region centered a t the current approximate minimizer. This is accomplished using ideas discussed by Xor6, together with a special module for assessing the quality of the step thus computed. These and other ideas behind ML2SOL stre discussed and its evolution and current implementation are also described briefly.

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AN ADAPTIVE ~~JONLXNEAR LEAST-SQUARES~ .-- /---, --'.*

'

*Cornell University. Research supported in part by NSF Grant MCS76-00324

* *NBER Computer Research Center

Research supported in part by NSF Grant MCS76-00324
NSF Grant MCS78-09525
United States Army Contract

DAAG29-75-C-0024

tMassachusetts Institute of

Technology. Research supported in part by NSF Grants SOC76-14311 and MC:S76-00324

ALGORITHM

John E. Dennis, Jr.*, David M. Gay**, and Roy E. Welscht

TR77-321

Revised July 1979

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ABSTRACT

   NL2SOL is a modular program for solving nonlinear least- squares problems that incorporates a number of novel features.

It maintains a secant approximation S to the second-order
part of the least-squares Hessian and adaptively decides when to use this approximation. S is "sized" before updating,
something which is similar t o Oren-Luenberger scaling. The step
choice algorithm is based on minimizing a local quadratic model
of the sum of squares function const:rained t o an elliptical trust region centered a t the current approximate minimizer. This is accomplished using ideas discussed by Xor6, together with a special module for assessing the quality of the step thus computed. These and other ideas behind ML2SOL stre discussed and its evolution and current implementation are also described briefly.

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MI ADAPTIVE NOllLINEAR LEAST-SQUARES ALGORITHM

bv 2. The llonlincsr Least-Squaras Problem

                                                                There are good reasons for numerical analysts to study this
J.E. Dennis, Jr., David H. Gay, Roy E. Welsch problem. In the first place, it is a computation of prisary i=- portance i n statistical data analysis and hence in the social
1. Introduction

   This project began in order to meet a need a t the Computer Research Center of the llational Bureau of Economic Research for
a nonlinear least-squares algorithm which, in the large residual case, uould be more reliable than the Gauss-Newton or Levenberg-
P!arquardt nethod [Dennis, 19771 and more efficient than the secant or variable metric algorithms [Dennis 6 Mor6, 19771 such as the DaviCon-Fletcher-bkell, which 6-e inteded for g e n d function mirlrP;ization.

We have developed a satisfactory computer program called

XLZSOL based on ideas in [Dennis 5 Weisch, 19761 and o u ~ primary

parpose here i s to report the details and to give some test results. Ca the othsr hand, we learned so much during the development which

seess likely to be applicabiie in the development of other algoriths

that we have chosen to expand our exposition t o include tlds experiexe.

sciences, a8 well as in the more traditional areas within the physical sciences. Th...