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The purpose of this paper is to focus on the performance of three typical nonlinear least-squares estimation algorithms in atmospheric density model calibration.

The error of Jacchia-Roberts atmospheric density model is expressed as an objective function about temperature parameters. The estimation of parameter corrections is a typical nonlinear least-squares problem. Three algorithms for nonlinear least-squares problems, Gauss–Newton (G-N), damped Gauss–Newton (damped G-N) and Levenberg–Marquardt (L-M) algorithms, are adopted to estimate temperature parameter corrections of Jacchia-Roberts for model calibration.

The results show that G-N algorithm is not convergent at some sampling points. The main reason is the nonlinear relationship between Jacchia-Roberts and its temperature parameters. Damped G-N and L-M algorithms are both convergent at all sampling points. G-N, damped G-N and L-M algorithms reduce the root mean square error of Jacchia-Roberts from 20.4% to 9.3%, 9.4% and 9.4%, respectively. The average iterations of G-N, damped G-N and L-M algorithms are 3.0, 2.8 and 2.9, respectively.

This study is expected to provide a guidance for the selection of nonlinear least-squares estimation methods in atmospheric density model calibration.

The study analyses the performance of three typical nonlinear least-squares estimation methods in the calibration of atmospheric density model. The non-convergent phenomenon of G-N algorithm is discovered and explained. Damped G-N and L-M algorithms are more suitable for the nonlinear least-squares problems in model calibration than G-N algorithm and the first two algorithms have slightly fewer iterations.

The purpose of this paper is to investigate the conditions under which working with an incumbent downstream competitor could be a beneficial strategy for upstream firms as the case of the relationship between banks and third-party payment providers.

Using a game model, this study considers a market with two upstream firms (banks) and two downstream firms (third-party payment providers). One downstream firm is an incumbent that poses a competitive threat to the upstream market, and the other downstream firm does not.

The results show that the optimal decision for banks depends on the number of loyal users the incumbent third-party payment providers and banks have. When the bank has more loyal users than the competitive third-party provider to a certain level, it would terminate cooperation with the provider; otherwise, the bank would maintain cooperation. This is true whether the duopoly banks are symmetrical or asymmetrical.

This study makes contribution to the theory of co-opetition lies in the fact that it examines a special case of competition and cooperation between vertical enterprises in the bank context. This study investigates how the upstream firms do when threatened by a downstream firm while the upstream firms have other options. This study also contributes to bank marketing theory through providing explanations for some of the incomprehensible cooperation in China's payment market, which is characterized by consumer loyalty. This study extends previous new-entry competition for banks by differentiating between incumbent and new-entry downstream firms.

The peer review history for this article is available at: https://publons.com/publon/10.1108/IJBM-11-2019-0414