Applied and Computational Ma 2015; 4(3): 207-213 Published online June 8, 2015 (http://www.scien doi: 10.11648/j.acm.20150403.23 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (On Some Convalescent of Linear Equations M. Rafique, Sidra Ayub Department of Mathematics, Faculty of Science, Email address: [email protected] (M. Rafique), sidra. To cite this article: M. Rafique, Sidra Ayub. Some Convalescent Me Mathematics. Vol. 4, No. 3, 2015, pp. 207-213. d Abstract: In a variety of problems in the linear equations, Ax = b, comprising n linea and x = [x 1 x 2 . . .x n ] T , b = [b 1 b 2 . . .b n ] T are to solve such systems of equations, includ Crout’s method and Cholesky’s method, w and upper triangular matrices respectively. Turing [2]-[11] in 1948. Here, in this paper we above mentioned system Ax = b, with the l [4] needs about 2n 3 /3 operations, while Doo are required to evaluate n 2 number of unk 2n 2 /3 operations. Accordingly this method But, in contrast, the improved Doolittle’s, (n–1) 2 number of unknown elements of the which requires evaluation of even less nu evaluate only (n–2) 2 number of the said un effort required for the purpose can substanti Keywords: System of Equations, Matrix Cholesky’s Method 1. Introduction There are numerous analytical and nume the solution of a linear system, Ax = b, elimination method, and its modifications n In the case of Crout’s method this decom athematics ncepublishinggroup.com/j/acm) nline) Methods for the Solution , HITEC University, Taxila, Pakistan [email protected](S. Ayub) ethods for the Solution of Systems of Linear Equations. App doi: 10.11648/j.acm.20150403.23 e fields of physical sciences, engineering, economics, ar equations in n unknowns x 1 , x 2 , …, x n , where A = [a e the column vectors. There are many analytical as wel ding Gauss elimination method, and its modification which employ LU-decomposition method, where L = [ The LU-decomposition method was first introduced b e have made an effort to modify the existing LU-decom least possible endeavour. It may be seen that the Gauss olittle’s and Crout’s methods require n 2 operations. Acc known elements of the L and U matrices. Moreover, requires evaluation of 2n 2 /3 number of unknown elem Crout’s and Cholesky’s methods presented in this pa e L and U matrices. Moreover, an innovative method umber of unknown elements of the L and U matrice nknown elements. Thus, by employing these method ially be reduced. x, Column Vector, Decomposition, Doolittle’s Method, merical methods for , including Gauss namely Doolittle’s, Crout's and Cholesky’s metho LU-decomposition of an nxn m which is given as under, = mposition is given as, n of Systems plied and Computational , etc., we are led to systems of a ij ] is an nxn coefficient matrix, ll as numerical methods [1}– [11] ns namely Doolittle’s method, [i ij ] and u = [u ij ] are the lower by the mathematician Alan M. mposition methods to solve the ss elimination method [1], [2], [3], cordingly, in these methods we Cholesky’s method [1] requires ments of the L and U matrices aper require evaluation of only is also presented in this paper es. In this method we need to ds, the computational time and , Crout’s Method, ods. In the Doolittle’s method matrix A is of the form A = LU, .
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Applied and Computational Mathematics2015; 4(3): 207-213
Published online June 8, 2015 (http://www.sciencepublishinggroup.com/j/acm)
212 M. Rafique and Sidra Ayub: Some Convalescent Methods for the
unity, i.e., uii =1, for all i.
(v) The remaining elements in the second
Thus, schematically, LU-decomposition of the coefficient matrix A = LU, is given as under,
=
After having accomplished the LU-decomposition
described above, we solve the equation Ly
solve the equation Ux = b for x, to obtain
given system of linear equations. Here
examples to elaborate the method discussed
5.1. Solution of a System of Three Equations
Unknowns
We consider the system of equations,
5x1 + x2 + 2x3 = 19
x1 + 4x2 – 2x3 = – 2
2x1 + 3x2 + 8x3 = 39.
We divide the first equation by 5 and the
2, throughout as required by the method elaborated
get,
=
Next, the coefficient matrix is decomposed
above. In this decomposition only (n – 2)2 =
(3–2)2 = 1 number of unknown, namely
be evaluated which comes to 4. Accordingly,
required decomposition A = LU, as,
After having carried out this decomposition,
system Ly = b for y to get y = [15 –5 4.33
solving the system Ux = y, for x we obtain
given system of equations as, x = [2 –3 4 –1]
By following the method discussed above,
linear system having any number of equations
number of unknowns.
6. Conclusion
The solution of a system of linear equations
M. Rafique and Sidra Ayub: Some Convalescent Methods for the Solution of Systems of Linear Equations
second row of matrix U are transcribed as: u2j
decomposition of the coefficient matrix A = LU, is given as under,
x
decomposition as
= b for y and next
the solution to the
we consider few
discussed above.
Equations in Three
the third equation by
elaborated above to
decomposed as described
=
namely l33 is required to
Accordingly, we obtain the
=
Further, solving Ly = b, we
Thereafter, solving Ux = y, we
required solution to the given systems
5.2. Solution of a System of Four
Unknowns
We consider the system of equations,
4x1 – 4x2 + 12x
–x1 + 5x2 – 5x
3x1 – 5x2 + 19x
2x1 – 2x2 + 3x
We divide the pivot equation
equation by –1, divide the third
divide the fourth equation by
transform the given system of equations
[ 15 35 31.33 0.5]. Here we need
(4–2)2 = 4 unknown number
matrices, which are obtained as:
and u34 = – 0.33. Thus we get A
= x
decomposition, we solve the
4.33 1]T. Thereafter,
the solution to the
1]T.
above, we can solve a
equations with the same
equations by means of
LU-decomposition of the coefficient
method that can be employed
numerically. It may be seen that
the usual Doolittle's, and Crout's
of a total of n2 number of unknown
matrices, and in the case of usual
required to evaluate 2n2/3 number
these matrices. However, the
and Cholesky’s methods need
number of elements of these matrices
LU-Decomposition method requires
Solution of Systems of Linear Equations
u2j = �� !�" ���#$"�
for j ≥ 3.
.
we obtain, y = [3.8 – 1.3 4]T.
get, x = [ 2 1 4]T, which is the
systems of equations.
Four Equations in Four
equations,
12x3 + 8x4 = 60
5x3 – 2x4 = – 35
19x3 + 3x4 = 94
3x3 + 21x4 = 1
equation by 4, multiply the second
third equation by 3, and also
by 2, as discussed above, and
equations as Ax = b, where b =
need to evaluate only (n –2)2 =
of elements of the L and U
as: l33 = 3, l43 = – 1.50, l44 = 8,
A = LU as under,
coefficient matrix is a plausible
employed analytically as well as
that for an nxn coefficient matrix,
Crout's methods require evaluation
unknown elements of the L and U
usual Cholesky’s method we are
number of unknown elements of
improved Doolittle’s, Crout’s
need evaluation of only (n –1)2
matrices, while the Innovative
requires evaluation of only (n –
Applied and Computational Mathematics 2015; 4(3): 207-213 213
2)2 number of unknown elements of the L and U matrices.
This difference becomes significant for systems of large
number of linear equations. As such a considerable amount of
computational time and energy can be saved by employing
the methods presented in this paper.
References
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[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)
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