Abstract
對任一矩陣 A,A 的每個元素非負,我們希望對 A 的每個元素做最微量的 調整,使調整後的 A,稱之為 C,使 C 的每個元素非負,而且使 C 的 2x2 階子矩陣,其行列式值皆大於或等於零,也就是所謂二階全正(Totally Positive of Order 2)的概念。在這□,所謂的最微量的調整,是以歐幾里 得向量空間的範數(Euclidean 2-Norm),來加以衡量的,更希望藉此程序, 二階全正配適度分析(最佳化 C,在 C 是二階全正的條件下,使得 E 具有 最小範數,其中 E = C - A )來明確的定義所謂的近似二階全正矩陣的概 念。最後我們提議一個量來作為 $ TP_2 $ 的相似性測度,並討論其相關 的幾何意義。 We are interested in the following optimization problem: Given a contigency table A, we search for the minimal pertubations of A, named E, such that C=A+E is totally positive of order 2. In this paper, we have showen that systematical reduction can be done by considering when C is strictly positive, and when the saddle-point of the Lagrange function exists, the optimization problem is equivalent to the problem of solving the non-linear system in terms of the (m-1)(n-1) Kuhn-Tucker multipliers. Geomatrical properties of the global minima has been explored. Solutions for small matrix is optained and analyzed by 3000 times of simulation studies. From this we hypothsis that the local minima C is unique when C is strictly positive, and establish MDUB (Monotone in Determinant and Upper Bounded) criterion for initial value of the Kuhn-Tucker multipliers in order to give a starting point when solving the non-linear system of equations.