本論文主要是對以多項式近似正弦函數之演算法做最佳化之調整，在一定準確度的要求下，可針對任ㄧ特定數值處理器選擇最合適的多項式項數與相關係數值。多項式係數的最佳化採用限制峰值最小平方法進行，其中最大峰值誤差和平方誤差之比重可根據需求予以適當選擇。當多項式係數之準確度受限時，各係數值可依其重要性依序使用限制峰值最小平方法予以求解，可大幅降低函數近似誤差。當考慮多項式近似演算法之計算誤差及使用霍納演算法之情況下，各係數值仍可經由一局部搜尋之步驟求得較佳的函數近似結果。
在本論文中利用MATLAB軟體以模擬方式成功驗證本論文所提出之方法於16位元定點處理器、32位元定點處理器、單精確度浮點處理器與雙精確度浮點處理器並探討相關特性。此一多項式近似之最佳化技術可延伸使用於各類型數值處理器及其他適用多項式近似之函數上。

In this thesis, we have explored the optimization of polynomial approximation algorithm for sinusoidal function such that a polynomial function of suitable degree and the corresponding coefficients can be determined for a specific numerical processor to satisfied a predetermined error tolerance. The peak-constrained least-square (PCLS) algorithm is employed to compute the polynomial coefficients iteratively, where the relative weighting factor between the peak error and least-square error in the sinusoidal function approximation can be judiciously chosen. When the finite-precision effects of the polynomial coefficients are considered, the PCLS algorithm can be applied iteratively to find an appropriate set of polynomial coefficients in accordance to the order of significance of each coefficient. If the polynomial approximation is calculated with the Horner algorithm and the computation errors are to be considered, the polynomial coefficients can be further refined through the use of a local minimum search algorithm developed for the integer programming problem.
The techniques proposed in this thesis have been verified with Matlab software simulation for 16-bit and 32-bit fixed-point processors, single-precision floating-point processor, double-precision floating- point processor, and the pertinent convergence properties and error character- istics of these techniques have been investigated. It is clearly seen that these polynomial approximation techniques can be applicable to various numerical processor and to a number of common functions suitable for polynomial approximation.

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