在此論文中，我們主要探討的是有關於加權多項式迴歸模型D之最適設計的問題，其中加權函數限定為大於零的函數，在[m_0+a,m_0-a]內去建構此最適設計。我們發現在此區間內，如果加權函數一次微分除以加權函數自己本身為有理函數且a趨近於零時，則建構此D之最適設計的問題可以轉為解微分方程的問題，在解微分方程過程中，利用矩陣在代數上的相關知識，以泰勒展開式去逼近矩陣中的未知參數，而在泰勒展開式中的係數部分，我們提供了一個遞迴演算法來估計它們，因此，從這個線性系統中，我們可以估算出那些以D之最適設計的實驗點為零根的多項式之係數。

This paper is concerned with the problem of computing theapproximate D-optimal design for polynomial regression with weight function w(x)>0 on the design interval I=[m_0-a,m_0+a]. It is shown that if w'(x)/w(x) is a rational function on I and a is close to zero, then the problem of constructing D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be approximated by a Taylor expansion. We provide a recursive algorithm to compute Taylor expansion of these constants. Moreover, the D-optimal
interior support points are the zeros of a polynomial which has coefficients that can be computed from a linear system.

Contens:
Abstract..................................................ii
1. Introduction...........................................1
2. The differential equation..............................2
3. Taylor expansion.......................................5
4. Examples...............................................7
References................................................12
Appendix..................................................14

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