從公元前兩千多年到現在，方程組的求解陸續被人們討論著，隨著待求解的未知變數增加，問題的複雜度以及所耗的時間也越來越多，造成問題無法求解或是效率不高。然而，現今普遍用來運算的電腦科學軟體，例如MATLAB、OCTAVE等等，其效能及可計算的規模也不能完全解決在大尺度運算中會遇到的問題，若是線性方程組的解為數萬等級，則計算系統會超過負荷。
為了解決上述問題，本研究主要利用LU分解主元消去法，搭配了訊息傳遞介面(MPI)的高效能、大規模性以及可移植性等性質，先將方程組進行LU分解，再利用反向回代法求出線性方程組的解。除此之外，本研究所提出的方法可以支援跨核心以及跨機，若是硬體許可，無擴充限制的情況下，本方法所能求解的問題規模無上限。符合現今網路時代大數據求解之需求。而除了問題規模無限制之外，求解所需的執行時間，經比較之後，也已經超越了現今常用的運算軟體。因此本研究結合了不同運算方法，發展並建構出一套新的求解方式，提升了運算的效能以及效率。

The issue of solving equations has been discussed since two thousand years BC. Along with increasing unknown variables, the complexity of the problem and the solving time in-crease. However, scientific computation cannot be used to solve the problem completely, when it is so large-scale an operation that the loading of the system exceeds the limitation.
To solve the above problems, in this study, we use the Gaussian elimination with the Mes-sage Passing Interface(MPI), LU decompose equations firstly, and then use back-ward-substitution to obtain the solution of linear equations. In addition, the proposed method in this study supports large-scale parallelization across CPU cores and machine nodes, if the hardware resource allowed, the problems scale can be enlarged accordingly, meeting today's needs of big data in this Internet era. Moreover, we have compared the computing effectiveness and efficiency of our method with other computing software, showing better performance. Therefore, this study combines the different calculation methods, to develop and construct a new way of solving equations, enhancing the effec-tiveness and efficiency of operations.

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