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May  2019, 2(2): 83-93. doi: 10.3934/mfc.2019007

"Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns

Aleksa Srdanov 1,, , Radiša Stefanović 2, , Aleksandra Janković 2, and  Dragan Milovanović 2,

1. 

Stevana Mokranjca 6, 12000 Požarevac, Serbia
2. 

Visoka Tehnička škola strukovnih studija, Njegoseva 2, 12000 Požarevac, Serbia

* Corresponding author: Aleksa Srdanov

Received  February 2019 Revised  March 2019 Published  May 2019

Solving linear systems with a relatively large number of equationsand unknowns can be achieved using an approximate method to obtain a solution with specified accuracy within numerical mathematics. Obtaining theexact solution using the computer today is only possible within the frameworkof symbolic mathematics. It is possible to define an algorithm that does notsolve the system of equations in the usual mathematical way, but still findsits exact solution in the exact number of steps already defined. The methodconsists of simple computations that are not cumulative. At the same time,the number of operations is acceptable even for a relatively large number ofequations and unknowns. In addition, the algorithm allows the process to startfrom an arbitrary initial n-tuple and always leads to the exact solution if itexists.

Keywords: Linear system equations, extremely large number of unknowns, hyperplanes.
Mathematics Subject Classification: Primary: 15A06; Secondary: 65F99.
Citation: Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007
References:
[1]

Z. Bohte, Numerične Metode, (Slevenian), Državna založba Slovenije, Ljubljana, 1978.  Google Scholar

[2]

N. Higham, Accuracy and Stability of Numerical Algorithms, 2002. 2nd ed. Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. doi: 10.1137/1.9780898718027.  Google Scholar

[3]

D. A. Randall, An Introduction to Numerical Modeling of the Atmosphere, 2015, Chapter 6, Colorado State Univesity. Google Scholar

[4]

A. Srdanov, The universal formulas for the number of partitions, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), Art. 40, 17 pp. doi: 10.1007/s12044-018-0418-z.  Google Scholar

[5]

A. Srdanov and R. Stefanović, How to solve a system of linear equations with extremely many unknown, (Serbian), 16th International Symposium INFOTEH-Jahorina, 16 (2017), 593–596, Available from: https://www.infoteh.rs.ba/zbornik/2017/radovi/RSS-2/RSS-2-12.pdf Google Scholar

[6]

A. Srdanov, R. Stefanović, N. Ratković Kovačević, A. Jovanović and D. Milovanović, The method of external spiral for solving large system of linear equations, Military Technical Courier, 66 (2018), 391-414.  doi: 10.5937/vojtehg66-14625.  Google Scholar

[7]

A. Srdanov, R. Stefanović, N. Ratković Kovačević, A. Jovanović, D. Milovanović and D. Marjanović, Method of trihedrals for finding the exact solution of a linear system with a large number of unknowns, (Serbian), 17th International Symposium INFOTEH-Jahorina, (2018), 388–392, Available from: https://www.infoteh.rs.ba/zbornik/2018/radovi/RSS-3/RSS-3-1.pdf Google Scholar

[8]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Texts in Applied Mathematics, (Third edition. Texts in Applied Mathematics, 12. Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

show all references

References:
[1]

Z. Bohte, Numerične Metode, (Slevenian), Državna založba Slovenije, Ljubljana, 1978.  Google Scholar

[2]

N. Higham, Accuracy and Stability of Numerical Algorithms, 2002. 2nd ed. Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. doi: 10.1137/1.9780898718027.  Google Scholar

[3]

D. A. Randall, An Introduction to Numerical Modeling of the Atmosphere, 2015, Chapter 6, Colorado State Univesity. Google Scholar

[4]

A. Srdanov, The universal formulas for the number of partitions, Proc. Indian Acad. Sci. Math. Sci., 128 (2018), Art. 40, 17 pp. doi: 10.1007/s12044-018-0418-z.  Google Scholar

[5]

A. Srdanov and R. Stefanović, How to solve a system of linear equations with extremely many unknown, (Serbian), 16th International Symposium INFOTEH-Jahorina, 16 (2017), 593–596, Available from: https://www.infoteh.rs.ba/zbornik/2017/radovi/RSS-2/RSS-2-12.pdf Google Scholar

[6]

A. Srdanov, R. Stefanović, N. Ratković Kovačević, A. Jovanović and D. Milovanović, The method of external spiral for solving large system of linear equations, Military Technical Courier, 66 (2018), 391-414.  doi: 10.5937/vojtehg66-14625.  Google Scholar

[7]

A. Srdanov, R. Stefanović, N. Ratković Kovačević, A. Jovanović, D. Milovanović and D. Marjanović, Method of trihedrals for finding the exact solution of a linear system with a large number of unknowns, (Serbian), 17th International Symposium INFOTEH-Jahorina, (2018), 388–392, Available from: https://www.infoteh.rs.ba/zbornik/2018/radovi/RSS-3/RSS-3-1.pdf Google Scholar

[8]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Texts in Applied Mathematics, (Third edition. Texts in Applied Mathematics, 12. Springer-Verlag, New York, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

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