| With the progress of science and technology in today's society,computers have found application in more and more families and companys. Apparently the automation in every industry is imevitable. Nonetheless it has been a grand task to perform and a long way to go for software programmars in developing the software products of its kind due to the fact that most of them at the moment are of low safety and less maintainance whereas a constantly increasing cost and relatively low efficiency. Under the multi-sponsorships at the governmental level.professor xue jinyun introduced his own unique method, "PAR",which has been proved to be a simple and effective solution to the problem above mentioned.In his program,pratical developing formal method and the resarch of its tools.sponsored by the Chinese nature&science fund project, "The research of Applied Formal Develop Methods and Their Tools",an auto program transforming system has been successfully worked out to shift any abstract programs described by Apia into another one recognized by most of the popular languages and operated directly by the target languge system.The interface of the transforming sysytem is convenient.PAR method is honored as the program design of mathematician.This method embodies rich mathematic philosophy.is suitable for solving numeric methd.In addition.In PAR method,there are many powerful tools and appropriate developing environments,which add new blood to the present program designing methodology and therefore enable the mathematics researchers to get away from studying program design language.PAR method is a advance of the current numeric method.The application of PAR in numeric mthod is a tendency.This program,aiming to the application of PAR method in numerical methods. Briefly,here is what we have been doing to the aims above: Studying the main factors which set back the development of today's software industry.analyzing and comparing the existing formal methods.With regard to the deficiencies of the current formal method.The paper points out that PAR method is an ideal formal method and introduces it; Demonstating the effectiveness of PAR method in arithmetic computing,studying the contribution to the numerical methods by PAR method; Analyzing and comparing the current ways of numeric methods, studying the superiority of Loop Invariants and numerical methods and developing a lot of algorithms on numerical methods,a fast algorithm in the diagonalization of Hankel matrices and a fast algorithm for computing the products of Schur-cohn matrices.hi this paper ,we have sought references from many widely accepted ideas as formal method, numeric methods,and the characteristics of the structure in which data store. At the same time ,we made many creative novelties in many aspects.PAR method is a unified algorithms design method,suitable for developing numeric problems,and is natural in the whole process of developing numeric method. Only with PAR method ,mathematic researchers can be absorbed in studying numeric problems, and what follows will be done with transferring systemjapplying the generic system of PAR method in numeric mthods. The system of PAR method is embodied with the idea of generic design,bring up many abstract data types.such as set ,etc,which make the operation in set (and the alike)easy.Besides,there exists recurrent relationship in many matrices fast algorithm,complicated as the operation on present data structure,it turns out to be fairly direct and simple with the abstract data type-list in PAR method for matrices storage,thus,the storage problem is solved perfectly.Developing Loop Invariants is required for the developing process with PAR method. It's the first time to describe Loop hi variants in terms of set theory in order to get a correct understanding of Loop Invariants .In the course of whole study, developing a great many of algorithms on numerical methods as well as a fast algorithm in the diagonalization of Hankel matrices and a algorithm for computing the products of Schur-cohn matrices.
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