當使用套裝數值運算軟體(MATLAB)解多項式時，若多項式有重根，則可以發現到MATLAB求出之重根的準確率並不高。為了解決此問題，本論文首先先建立一個假想的線性離散系統模型，將待分解的多項式視為假想轉移函數的分母，並將待求多項式之微分視為假想轉移函數之分子，透過假想系統的控制輸入來刺激產生輸出，接著藉由觀察系統的響應，以達求解多項式根之目的。由於線性離散系統的輸出是由各極點的模式響應所加成，而各模式響應又是對應之極點對離散時間的乘方，因此隨著乘方越高，範數(norm)較小的根將會衰退，而整個線性離散系統輸出的特性最後將只受到具有最大範數的根影響，其它根對線性離散系統輸出的影響則漸漸消失，這樣便可找到此系統具有最大範數的極點，也就是待求多項式的根。接著，由於對離散時間的乘方之數值非常大，而導致截斷誤差也隨之而被放大，一般的套裝數值運算軟體並沒有辦法解決此問題，因此本論文發展出超高精確度程式來解決數值困難的問題，超高精確度程式提供更高的精確度和更大的運算空間，因此對於MATLAB求解重根之準確率以及誤差之問題，可以加以修正。

Solving of polynomial roots using Matlab often suffers from high numerical error when the roots of the polynomial are repeated. Errors in the solutions escalate when the repentance in the roots increase. In this work, polynomial roots are separated and then identified according to their norms. This task is accomplished by using the polynomial of interest as the denominator of the input/output transfer function of a fictitious discrete linear system. Because the unit pulse response of a discrete linear system consists of the sum of modal response which vary in the power of discrete time of the norm of root of the respective mode, the root that is of largest norm gradually becomes the only mode that contributes to the output responses and therefore can be easily identified. In order to free the results from the effect of repeating roots, the derivative of the polynomial of interest is used as the numerator of fictitious transfer function. In this way, any repeating poles will be explicitly cancelled by the same roots in the numerator. In this work, the numerical difficulty of such a computation is resolved by developing an ultra-precision (UP) arithmetic package and by conducting computations using the UP package, and therefore will not affect the solution.

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