Volume 24, Issue 6
A Simplicial Algorithm for Computing an Integer Zero Point of a Mapping with the Direction Preserving Property
DOI:

J. Comp. Math., 24 (2006), pp. 711-718

Published online: 2006-12

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• Abstract

A mapping $f:Z^n\rightarrow R^n$ is said to possess the direction preserving property if $f_i(x)\gt 0$ implies $f_i(y)\geq 0$ for any integer points $x$ and $y$ with $\|x-y\|_{\infty}\leq 1$. In this paper, a simplicial algorithm is developed for computing an integer zero point of a mapping with the direction preserving property. We assume that there is an integer point $x^0$ with $c\leq x^0\leq d$ satisfying that $\max_{1\leq i\leq n}(x_i-x^0_i)f_i(x)\ge0$ for any integer point $x$ with $f(x)\neq 0$ on the boundary of $H=\{x\in R^n\;|\;c-e\leq x\leq d+e\}$, where $c$ and $d$ are two finite integer points with $c\leq d$ and $e=(1,1,\cdots,1)^{\top}\in R^n$. This assumption is implied by one of two conditions for the existence of an integer zero point of a mapping with the preserving property in van der Laan et al. (2004). Under this assumption, starting at $x^0$, the algorithm follows a finite simplicial path and terminates at an integer zero point of the mapping. This result has applications in general economic equilibrium models with indivisible commodities.

• Keywords

Integer Zero Point Direction Preserving Simplicial Algorithm Triangulation Existence

@Article{JCM-24-711, author = {}, title = {A Simplicial Algorithm for Computing an Integer Zero Point of a Mapping with the Direction Preserving Property}, journal = {Journal of Computational Mathematics}, year = {2006}, volume = {24}, number = {6}, pages = {711--718}, abstract = { A mapping $f:Z^n\rightarrow R^n$ is said to possess the direction preserving property if $f_i(x)\gt 0$ implies $f_i(y)\geq 0$ for any integer points $x$ and $y$ with $\|x-y\|_{\infty}\leq 1$. In this paper, a simplicial algorithm is developed for computing an integer zero point of a mapping with the direction preserving property. We assume that there is an integer point $x^0$ with $c\leq x^0\leq d$ satisfying that $\max_{1\leq i\leq n}(x_i-x^0_i)f_i(x)\ge0$ for any integer point $x$ with $f(x)\neq 0$ on the boundary of $H=\{x\in R^n\;|\;c-e\leq x\leq d+e\}$, where $c$ and $d$ are two finite integer points with $c\leq d$ and $e=(1,1,\cdots,1)^{\top}\in R^n$. This assumption is implied by one of two conditions for the existence of an integer zero point of a mapping with the preserving property in van der Laan et al. (2004). Under this assumption, starting at $x^0$, the algorithm follows a finite simplicial path and terminates at an integer zero point of the mapping. This result has applications in general economic equilibrium models with indivisible commodities. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8785.html} }
TY - JOUR T1 - A Simplicial Algorithm for Computing an Integer Zero Point of a Mapping with the Direction Preserving Property JO - Journal of Computational Mathematics VL - 6 SP - 711 EP - 718 PY - 2006 DA - 2006/12 SN - 24 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8785.html KW - Integer Zero Point KW - Direction Preserving KW - Simplicial Algorithm KW - Triangulation KW - Existence AB - A mapping $f:Z^n\rightarrow R^n$ is said to possess the direction preserving property if $f_i(x)\gt 0$ implies $f_i(y)\geq 0$ for any integer points $x$ and $y$ with $\|x-y\|_{\infty}\leq 1$. In this paper, a simplicial algorithm is developed for computing an integer zero point of a mapping with the direction preserving property. We assume that there is an integer point $x^0$ with $c\leq x^0\leq d$ satisfying that $\max_{1\leq i\leq n}(x_i-x^0_i)f_i(x)\ge0$ for any integer point $x$ with $f(x)\neq 0$ on the boundary of $H=\{x\in R^n\;|\;c-e\leq x\leq d+e\}$, where $c$ and $d$ are two finite integer points with $c\leq d$ and $e=(1,1,\cdots,1)^{\top}\in R^n$. This assumption is implied by one of two conditions for the existence of an integer zero point of a mapping with the preserving property in van der Laan et al. (2004). Under this assumption, starting at $x^0$, the algorithm follows a finite simplicial path and terminates at an integer zero point of the mapping. This result has applications in general economic equilibrium models with indivisible commodities.