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# A Realization of the Simplex Method Based on Triangular Decompositions

• R. H. Bartels
• J. Stoer
• Ch. Zenger
Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)

## Abstract

Consider the following problem of linear programming
$${\text{Minimize }}{c_{\text{0}}} + {c_{ - m}} {x_{ - m}} + \cdots + {c_{ - 1}}{x_{ - 1}} + {c_1}{x_1} + \cdots + {c_n}{x_n}$$
(1.1.1a)
subject to
$${x_{ - i}} + \sum\limits_{k = 1}^n {a{a_{ik}}{x_k} = {b_i},{\text{ }}i{\text{ = 1,2,}} \ldots {\text{,}}m{\text{,}}}$$
(1.1.1b)
$${x_i} \geqq 0{\text{ for }}i \in {I^ + },{\text{ }}{x_i} = 0{\text{ for }}i \in {I^0},$$
(1.1.1c)
where I +, I 0, I ± are disjoint index sets with
$${I^ + } \cup {I^0} \cup {I^ \pm } = N: = \{ i| - m \leqq i \leqq - 1,1 \leqq i \leqq n\} .$$
The variable x i is called a nonnegative (zero, free) variable if i∈I + (i∈I 0, i∈I)±.

## Keywords

Basic Solution Free Variable Current Basis Exchange Step Triangular Decomposition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

1. 1.
Bartels, R. H.: A numerical investigation of the simplex method. Technical Report No. CS 104, 1968, Computer Science Department, Stanford University, California.Google Scholar
2. 2.
Bartels, R. H. Golub, G. H.: The simplex method of linear programming using LU decomposition . Comm. ACM. 12, 266 -268 (1969).
3. 3.
Dantzig, G. B.: Linear programming and extensions. Princeton: Princeton University Press 1963.
4. 4.
Wilkinson, J. H.: Rounding errors in algebraic processes. London: Her Majesty’s Stationery Office; Englewood Cliffs, N.Y.: Prentice Hall 1963. German edition: Rundungsfehler. Berlin-Heidelberg-New York: Springer 1969.
5. 5.
Wilkinson, J. H. The algebraic eigenvalue problem. London: Oxford University Press 1965.

## Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

## Authors and Affiliations

• R. H. Bartels
• J. Stoer
• Ch. Zenger

There are no affiliations available