Ibm Mainframe Algebra Library Development And Gr (?) Bner-based Algorithm

Most Computer Algebra Systems put great emphasis on requirement for computer hardware. Usually, it needs a great amount of memory and time to perform symbolic computation. However, the precise algebraic manipulation is always at the price of time and space. At present, there hasn't any prevailing algebra system integrated into IBM mainframe system. At the advantage of strong capacity of mainframe to perform scientific computation, the development of algebraic function set under mainframe system can resolve many algebraic manipulations efficiently, which put great demands on time and space, such as the multiplication and division of large integers, the reducing of polynomial group and generating the Gr(o|¨)bner basis of Ideal etc.. Buchbeger put forward the improved Gr(o|¨)bnerRefined algorithm which mainly adopted the standard presenting theory. Although it is perfect theoretically, it is unpractical due to the degree of middle term increasing too fast in the process of computation. The improved algorithm applying the degree-dropping-reducing method can eliminate such phenomenon. Furthermore, it demands less memory, which is a great advantage in large scale algebraic manipulation.Basing on the central idea of improved Gr(o|¨)bner basis algorithm, that is, degree-dropping-reducing the S-polynomial, this paper introduces the presentation and arithmetic of large integer in the IBM mainframe system. Breaking through the regular idea of storing polynomial in the machine using array or link-list, it develops a new way to achieve that by storing the polynomial as a whole unit in the machine and implements the arithmetic of polynomial in the IBM mainframe system. In the process of reducing polynomial, in order to keep the number of middle term from increasing too fast and the degree of polynomial being too high, it adopts a mechanism called dynamic order relation that is, modifying the order relation immediately after performing a reducing. It introduces the concept and related theorems of S-Varieties. At first, it is adopted to categorize the S-polynomial by relating terms, and then, every relating terms set is processed according to the degree-dropping-reducing algorithm. Thus, it completes the improvement of old algorithm. The author gets acquainted with the interacting integrated environment SDSF of IBM mainframe system and the process of developing C program under it. After that, the author implements several functions to perform the arithmetic of large integer and polynomial. They are INTEGER, POLYADD, POLYMUL and POLYQUO. At last, the author describes the degree-dropping-reducing algorithm in Maple language and the improved Gr?bner basis algorithm in Maple language and related pseudo code.