Some Convalescent Methods for the Solution of Systems of Linear Equations

M. Rafique; Sidra Ayub

doi:doi:10.11648/j.acm.20150403.23

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Some Convalescent Methods for the Solution of Systems of Linear Equations

M. Rafique, Sidra Ayub

Published in Applied and Computational Mathematics (Volume 4, Issue 3)

Received: 16 May 2015 Accepted: 26 May 2015 Published: 9 June 2015

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Abstract

In a variety of problems in the fields of physical sciences, engineering, economics, etc., we are led to systems of linear equations, Ax = b, comprising n linear equations in n unknowns x1, x2, …, xn, where A = [aij] is an nxn coefficient matrix, and x = [x1 x2 . . .xn]T, b = [b1 b2 . . .bn]T are the column vectors. There are many analytical as well as numerical methods[1}– [11] to solve such systems of equations, including Gauss elimination method, and its modifications namely Doolittle’s method, Crout’s method and Cholesky’s method, which employ LU-decomposition method, where L = [iij] and u = [uij] are the lower and upper triangular matrices respectively. The LU-decomposition method was first introduced by the mathematician Alan M. Turing[2]-[11] in 1948. Here, in this paper we have made an effort to modify the existing LU-decomposition methods to solve the above mentioned system Ax = b, with the least possible endeavour. It may be seen that the Gauss elimination method[1], [2], [3], [4] needs about 2n3/3 operations, while Doolittle’s and Crout’s methods require n2 operations. Accordingly, in these methods we are required to evaluate n2 number of unknown elements of the L and U matrices. Moreover, Cholesky’s method[1] requires 2n2/3 operations. Accordingly this method requires evaluation of 2n2/3 number of unknown elements of the L and U matrices But, in contrast, the improved Doolittle’s, Crout’s and Cholesky’s methods presented in this paper require evaluation of only (n–1)2 number of unknown elements of the L and U matrices. Moreover, an innovative method is also presented in this paper which requires evaluation of even less number of unknown elements of the L and U matrices. In this method we need to evaluate only (n–2)2 number of the said unknown elements. Thus, by employing these methods, the computational time and effort required for the purpose can substantially be reduced.

Published in	Applied and Computational Mathematics (Volume 4, Issue 3)
DOI	10.11648/j.acm.20150403.23
Page(s)	207-213
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

System of Equations, Matrix, Column Vector, Decomposition, Doolittle’s Method, Crout’s Method, Cholesky’s Method

References

[1]	E. Kreyszig: Advanced Engineering Mathematics, John Wiley, (2011)
[2]	A. M. Turing: Rounding-Off Errors in Matrix Processes”, The Quarterly Journal of Mechanics and Applied Mathematics 1: 287-308.doi:10.1093/qjmam/1.1.287 (1948)
[3]	Myrick H. Doolittle:. Method employed in the solution of normal equations and the adjustment of a triangulation, U.S. Coast and Geodetic Survey Report, Appendix 8, Paper No. 3, blz.. 115–120.( 1878)
[4]	David Poole: Linear algebra: A Modern Introduction (2nd Edition), Thomson Brooks/Cole ISSN 0-534-99845-3 (2006)
[5]	James R. Bunch, and John Hopkins: Triangular factorization and inversion by fast matrix multiplication, Mathematics of Computation, 28, 231-236, doi:10.2307/2005828, ISSN 0025-5718
[6]	J H Wilkinson: The Algebraic Eigenvalue Problem, Oxford University Press (1988)
[7]	B. Faires, Numerical Analysis, PWS Pub., Boston, (1993).
[8]	G.H.Golub and C.F.Van Loan, Matrix Computations, John Hopkins, Baltimore, (1989).
[9]	J.H. Mathews and K. D. Fink: Numerical Methods Using MATLAB (4th Edition)
[10]	L. V. Fausett: Applied Numerical Analysis Using MATLAB, Pearson, (2009)
[11]	S.C.Chapra and R.P.Canale, Numerical Methods for Engineers, Mc-Graw-Hill, New York,(1990).
[12]	Petre Teodorescu, Nicolae-Doru Stanescu, Nicolae Pandrea: Numerical Analysis with Applications in Mechanics and Engineering, John Wiley (2013).

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M. Rafique, Sidra Ayub. (2015). Some Convalescent Methods for the Solution of Systems of Linear Equations. Applied and Computational Mathematics, 4(3), 207-213. https://doi.org/10.11648/j.acm.20150403.23

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M. Rafique; Sidra Ayub. Some Convalescent Methods for the Solution of Systems of Linear Equations. Appl. Comput. Math. 2015, 4(3), 207-213. doi: 10.11648/j.acm.20150403.23

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AMA Style

M. Rafique, Sidra Ayub. Some Convalescent Methods for the Solution of Systems of Linear Equations. Appl Comput Math. 2015;4(3):207-213. doi: 10.11648/j.acm.20150403.23

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@article{10.11648/j.acm.20150403.23,
  author = {M. Rafique and Sidra Ayub},
  title = {Some Convalescent Methods for the Solution of Systems of Linear Equations},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3},
  pages = {207-213},
  doi = {10.11648/j.acm.20150403.23},
  url = {https://doi.org/10.11648/j.acm.20150403.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.23},
  abstract = {In a variety of problems in the fields of physical sciences, engineering, economics, etc., we are led to systems of linear equations, Ax = b, comprising n linear equations in n unknowns x1, x2, …, xn, where A = [aij] is an nxn coefficient matrix, and x = [x1 x2 . . .xn]T, b = [b1 b2 . . .bn]T are the column vectors. There are many analytical as well as numerical methods[1}– [11] to solve such systems of equations, including Gauss elimination method, and its modifications namely Doolittle’s method, Crout’s method and Cholesky’s method, which employ LU-decomposition method, where L = [iij] and u = [uij] are the lower and upper triangular matrices respectively. The LU-decomposition method was first introduced by the mathematician Alan M. Turing[2]-[11] in 1948. Here, in this paper we have made an effort to modify the existing LU-decomposition methods to solve the above mentioned system Ax = b, with the least possible endeavour. It may be seen that the Gauss elimination method[1], [2], [3], [4] needs about 2n3/3 operations, while Doolittle’s and Crout’s methods require n2 operations. Accordingly, in these methods we are required to evaluate n2 number of unknown elements of the L and U matrices. Moreover, Cholesky’s method[1] requires 2n2/3 operations. Accordingly this method requires evaluation of 2n2/3 number of unknown elements of the L and U matrices But, in contrast, the improved Doolittle’s, Crout’s and Cholesky’s methods presented in this paper require evaluation of only (n–1)2 number of unknown elements of the L and U matrices. Moreover, an innovative method is also presented in this paper which requires evaluation of even less number of unknown elements of the L and U matrices. In this method we need to evaluate only (n–2)2 number of the said unknown elements. Thus, by employing these methods, the computational time and effort required for the purpose can substantially be reduced.},
 year = {2015}
}

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TY  - JOUR
T1  - Some Convalescent Methods for the Solution of Systems of Linear Equations
AU  - M. Rafique
AU  - Sidra Ayub
Y1  - 2015/06/09
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.23
DO  - 10.11648/j.acm.20150403.23
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JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 207
EP  - 213
PB  - Science Publishing Group
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AB  - In a variety of problems in the fields of physical sciences, engineering, economics, etc., we are led to systems of linear equations, Ax = b, comprising n linear equations in n unknowns x1, x2, …, xn, where A = [aij] is an nxn coefficient matrix, and x = [x1 x2 . . .xn]T, b = [b1 b2 . . .bn]T are the column vectors. There are many analytical as well as numerical methods[1}– [11] to solve such systems of equations, including Gauss elimination method, and its modifications namely Doolittle’s method, Crout’s method and Cholesky’s method, which employ LU-decomposition method, where L = [iij] and u = [uij] are the lower and upper triangular matrices respectively. The LU-decomposition method was first introduced by the mathematician Alan M. Turing[2]-[11] in 1948. Here, in this paper we have made an effort to modify the existing LU-decomposition methods to solve the above mentioned system Ax = b, with the least possible endeavour. It may be seen that the Gauss elimination method[1], [2], [3], [4] needs about 2n3/3 operations, while Doolittle’s and Crout’s methods require n2 operations. Accordingly, in these methods we are required to evaluate n2 number of unknown elements of the L and U matrices. Moreover, Cholesky’s method[1] requires 2n2/3 operations. Accordingly this method requires evaluation of 2n2/3 number of unknown elements of the L and U matrices But, in contrast, the improved Doolittle’s, Crout’s and Cholesky’s methods presented in this paper require evaluation of only (n–1)2 number of unknown elements of the L and U matrices. Moreover, an innovative method is also presented in this paper which requires evaluation of even less number of unknown elements of the L and U matrices. In this method we need to evaluate only (n–2)2 number of the said unknown elements. Thus, by employing these methods, the computational time and effort required for the purpose can substantially be reduced.
VL  - 4
IS  - 3
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Author Information

M. Rafique

Department of Mathematics, Faculty of Science, HITEC University, Taxila, Pakistan
Sidra Ayub

Department of Mathematics, Faculty of Science, HITEC University, Taxila, Pakistan

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APA Style

M. Rafique, Sidra Ayub. (2015). Some Convalescent Methods for the Solution of Systems of Linear Equations. Applied and Computational Mathematics, 4(3), 207-213. https://doi.org/10.11648/j.acm.20150403.23

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ACS Style

M. Rafique; Sidra Ayub. Some Convalescent Methods for the Solution of Systems of Linear Equations. Appl. Comput. Math. 2015, 4(3), 207-213. doi: 10.11648/j.acm.20150403.23

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AMA Style

M. Rafique, Sidra Ayub. Some Convalescent Methods for the Solution of Systems of Linear Equations. Appl Comput Math. 2015;4(3):207-213. doi: 10.11648/j.acm.20150403.23

Copy | Download

@article{10.11648/j.acm.20150403.23,
  author = {M. Rafique and Sidra Ayub},
  title = {Some Convalescent Methods for the Solution of Systems of Linear Equations},
  journal = {Applied and Computational Mathematics},
  volume = {4},
  number = {3},
  pages = {207-213},
  doi = {10.11648/j.acm.20150403.23},
  url = {https://doi.org/10.11648/j.acm.20150403.23},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150403.23},
  abstract = {In a variety of problems in the fields of physical sciences, engineering, economics, etc., we are led to systems of linear equations, Ax = b, comprising n linear equations in n unknowns x1, x2, …, xn, where A = [aij] is an nxn coefficient matrix, and x = [x1 x2 . . .xn]T, b = [b1 b2 . . .bn]T are the column vectors. There are many analytical as well as numerical methods[1}– [11] to solve such systems of equations, including Gauss elimination method, and its modifications namely Doolittle’s method, Crout’s method and Cholesky’s method, which employ LU-decomposition method, where L = [iij] and u = [uij] are the lower and upper triangular matrices respectively. The LU-decomposition method was first introduced by the mathematician Alan M. Turing[2]-[11] in 1948. Here, in this paper we have made an effort to modify the existing LU-decomposition methods to solve the above mentioned system Ax = b, with the least possible endeavour. It may be seen that the Gauss elimination method[1], [2], [3], [4] needs about 2n3/3 operations, while Doolittle’s and Crout’s methods require n2 operations. Accordingly, in these methods we are required to evaluate n2 number of unknown elements of the L and U matrices. Moreover, Cholesky’s method[1] requires 2n2/3 operations. Accordingly this method requires evaluation of 2n2/3 number of unknown elements of the L and U matrices But, in contrast, the improved Doolittle’s, Crout’s and Cholesky’s methods presented in this paper require evaluation of only (n–1)2 number of unknown elements of the L and U matrices. Moreover, an innovative method is also presented in this paper which requires evaluation of even less number of unknown elements of the L and U matrices. In this method we need to evaluate only (n–2)2 number of the said unknown elements. Thus, by employing these methods, the computational time and effort required for the purpose can substantially be reduced.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Some Convalescent Methods for the Solution of Systems of Linear Equations
AU  - M. Rafique
AU  - Sidra Ayub
Y1  - 2015/06/09
PY  - 2015
N1  - https://doi.org/10.11648/j.acm.20150403.23
DO  - 10.11648/j.acm.20150403.23
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 207
EP  - 213
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20150403.23
AB  - In a variety of problems in the fields of physical sciences, engineering, economics, etc., we are led to systems of linear equations, Ax = b, comprising n linear equations in n unknowns x1, x2, …, xn, where A = [aij] is an nxn coefficient matrix, and x = [x1 x2 . . .xn]T, b = [b1 b2 . . .bn]T are the column vectors. There are many analytical as well as numerical methods[1}– [11] to solve such systems of equations, including Gauss elimination method, and its modifications namely Doolittle’s method, Crout’s method and Cholesky’s method, which employ LU-decomposition method, where L = [iij] and u = [uij] are the lower and upper triangular matrices respectively. The LU-decomposition method was first introduced by the mathematician Alan M. Turing[2]-[11] in 1948. Here, in this paper we have made an effort to modify the existing LU-decomposition methods to solve the above mentioned system Ax = b, with the least possible endeavour. It may be seen that the Gauss elimination method[1], [2], [3], [4] needs about 2n3/3 operations, while Doolittle’s and Crout’s methods require n2 operations. Accordingly, in these methods we are required to evaluate n2 number of unknown elements of the L and U matrices. Moreover, Cholesky’s method[1] requires 2n2/3 operations. Accordingly this method requires evaluation of 2n2/3 number of unknown elements of the L and U matrices But, in contrast, the improved Doolittle’s, Crout’s and Cholesky’s methods presented in this paper require evaluation of only (n–1)2 number of unknown elements of the L and U matrices. Moreover, an innovative method is also presented in this paper which requires evaluation of even less number of unknown elements of the L and U matrices. In this method we need to evaluate only (n–2)2 number of the said unknown elements. Thus, by employing these methods, the computational time and effort required for the purpose can substantially be reduced.
VL  - 4
IS  - 3
ER  -

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