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Finding a partial solution to a linear system of equations in positive integers. (English) Zbl 0649.65033
An algorithm is given to determine the set of positive integer solutions of the system $$AX=b$$ where $$A\in M_{m,n}({\mathbb{Z}})$$ and $$b\in {\mathbb{Z}}^ n$$. Denoting $$X_{| B}\in N^ B$$ as $$X_ i=(X_{| B})_ i$$ for every $$i\in B\subset \{1,...,n)$$ and $$x\in N^ n$$; $$S=\{X\in N^ n| AX=b\},\quad \sup p A=\{1\leq i\leq n;\quad \exists X\in N^ n,\quad X_ i>0;\quad AX=0\};\quad B=\{1\leq i\leq n;\quad i\not\in \sup p A\},$$ two-problems are solved: (A) “compute B and $$u\in Q^ m$$ such that $$A^ tu\geq 0$$ and $$(A^ tu)_ i>0$$ iff $$i\not\in \sup p A''$$. (B) “If $$B=\emptyset$$, is $$S=\emptyset ?$$ Else for $$y\in R'$$ determine $$X\in N^ n$$ such that $$X_{| B}=Y$$ and $$AX=b''$$ where $$R=S_{| B}=\{X_ B,\quad X\in S\}$$ and R’ is the set of $$X\in N^ B$$ satisfying $$u^ tAX=u^ tb.$$
Reviewer: P.Stavre

##### MSC:
 65K05 Numerical mathematical programming methods 65F05 Direct numerical methods for linear systems and matrix inversion 90C10 Integer programming
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