Study on solution of nonlinear equations and development of their practical software

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In this research project, we considered iterative solver for nonlinear equations and their applications. We also develop practical software and apply them for various problems. We obtained factoring methods which can provide factors of close zeros with verification. In ordinary methods such as verified Newton's method, the verification for close zeros was a difficult problem. We proposed some algorithms to estimate parameters for verification. Software using MATLAB and INTLAB were developed, and efficiency of this method was considered by numerical experiments. We also apply our method for large-scale eigenvalue problems. This method is suitable for parallel computing in a grid computing environment. We implemented the algorithm with a hybrid of Grid RPC system and MPI. In the application of large molecular orbital computations, we obtained high performance with our method. An system environment to support development of mathematical software with graphical user interface was provided. On this system, we developed some mathematical algorithms.