American Institute of Mathematical Sciences

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

 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.

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, doi: 10.3934/mfc.2019007
References:
 [1] Z. Bohte, Numerične Metode, (Slevenian), Državna založba Slovenije, Ljubljana, 1978. [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. [3] D. A. Randall, An Introduction to Numerical Modeling of the Atmosphere, 2015, Chapter 6, Colorado State Univesity. [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. [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 [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. [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 [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.

show all references

References:
 [1] Z. Bohte, Numerične Metode, (Slevenian), Državna založba Slovenije, Ljubljana, 1978. [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. [3] D. A. Randall, An Introduction to Numerical Modeling of the Atmosphere, 2015, Chapter 6, Colorado State Univesity. [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. [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 [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. [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 [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.
 [1] Yang Wang. The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem. Communications on Pure & Applied Analysis, 2011, 10 (2) : 731-744. doi: 10.3934/cpaa.2011.10.731 [2] Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619 [3] Rana D. Parshad. Asymptotic behaviour of the Darcy-Boussinesq system at large Darcy-Prandtl number. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1441-1469. doi: 10.3934/dcds.2010.26.1441 [4] Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397 [5] Alexander Bobylev, Åsa Windfäll. Kinetic modeling of economic games with large number of participants. Kinetic & Related Models, 2011, 4 (1) : 169-185. doi: 10.3934/krm.2011.4.169 [6] Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 183-219. doi: 10.3934/naco.2016008 [7] Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288 [8] Jean-François Biasse. Subexponential time relations in the class group of large degree number fields. Advances in Mathematics of Communications, 2014, 8 (4) : 407-425. doi: 10.3934/amc.2014.8.407 [9] Freddy Dumortier. Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1465-1479. doi: 10.3934/dcds.2012.32.1465 [10] Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems & Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023 [11] Yang Wang, Yi-fu Feng. $\theta$ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019027 [12] Josef Diblík, Radoslav Chupáč, Miroslava Růžičková. Existence of unbounded solutions of a linear homogenous system of differential equations with two delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2447-2459. doi: 10.3934/dcdsb.2014.19.2447 [13] Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019033 [14] Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Dynamical behaviour of a large complex system. Communications on Pure & Applied Analysis, 2008, 7 (2) : 249-265. doi: 10.3934/cpaa.2008.7.249 [15] Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147 [16] Harald Fripertinger. The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications, 2011, 5 (2) : 407-416. doi: 10.3934/amc.2011.5.407 [17] Paolo Gidoni, Alessandro Margheri. Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 585-606. doi: 10.3934/dcds.2019024 [18] Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237 [19] Junmin Yang, Maoan Han. On the number of limit cycles of a cubic Near-Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 827-840. doi: 10.3934/dcds.2009.24.827 [20] Jishan Fan, Fucai Li, Gen Nakamura. Low Mach number limit of the full compressible Hall-MHD system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1731-1740. doi: 10.3934/cpaa.2017084

Impact Factor: