在區間[a, b]上具有權重為 之d次多項式回歸模型，我們提出一個代數方式建構(d+1)點之D最適設計。假如 是一個有理函數，而且最佳設計點是否包含邊界點a和b為已知，那麼建構最少設計點(d+1)點之D最適設計的問題可以被轉換成一個微分方程式的問題。此方程式可以用矩陣、向量和有限個輔助未知參數表示之，而這些未知參數可以從一個聯立多項式方程組解得。
此外，(d+1)點D最適設計的內部設計點是某一個多項式之零根，此多項式的係數可以從一個線性方程組求得。在許多情況下，(d+1)點D最適設計剛好也是連續D最適設計。

We propose an algebraic construction of $(d+1)$-point $D$-optimal
designs for $d$th degree polynomial regression with weight
function $omega(x)ge 0$ on the interval $[a,b]$. Suppose that
$omega'(x)/omega(x)$ is a rational function and the information
of whether the optimal support contains the boundary points $a$
and $b$ is available. Then the problem of constructing
$(d+1)$-point $D$-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 solved from a system of polynomial equations in those
constants. Moreover, the $(d+1)$-point $D$-optimal interior
support points are the zeros of a certain polynomial which the
coefficients can be computed from a linear system. In most cases
the $(d+1)$-point $D$-optimal designs are also the approximate
$D$-optimal designs.

1.Introduction............................1
2.The differential equation...............2
3.Algebraic method........................4
4.Examples................................7
Figure 1..................................9
Figure 2.................................10
Appendix.................................11
References...............................13

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