Application of Linear Algebra on Polynomial Root Estimation

• Main Authors: Yung-ChuChuang, 莊詠筑
• Other Authors: Jenq-Tzong H. Chan
• Format: Others
• Language: zh-TW
• Published: 2016
• Online Access: http://ndltd.ncl.edu.tw/handle/78207187880572138637

碩士 === 國立成功大學 === 航空太空工程學系 === 104 === 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.