Some Convalescent Methods for the Solution of Systems of ...article.acmath.org/pdf/10.11648.j.acm.20150403.23.pdf · Keywords: System of Equations, Matrix, Cholesky’s Method 1

Applied and Computational Mathematics2015; 4(3): 207-213

Published online June 8, 2015 (http://www.sciencepublishinggroup.com/j/acm)

doi: 10.11648/j.acm.20150403.23

ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online)

Some Convalescent of Linear Equations

M. Rafique, Sidra Ayub

Department of Mathematics, Faculty of Science,

Email address: [email protected] (M. Rafique), [email protected]

To cite this article: M. Rafique, Sidra Ayub. Some Convalescent Methods

Mathematics. Vol. 4, No. 3, 2015, pp. 207-213. doi:

Abstract: In a variety of problems in the

linear equations, Ax = b, comprising n linear

and x = [x1 x2 . . .xn]T, b

= [b1 b2 . . .bn]

T are

to solve such systems of equations, including

Crout’s method and Cholesky’s method, which

and upper triangular matrices respectively.

Turing[2]-[11]

in 1948. Here, in this paper we

above mentioned system Ax = b, with the least[4]

needs about 2n3/3 operations, while Doolittle’s

are required to evaluate n2

number of unknown

2n2/3 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 number

evaluate only (n–2)2 number of the said unknown

effort required for the purpose can substantially

Keywords: System of Equations, Matrix,

Cholesky’s Method

1. Introduction

There are numerous analytical and numerical

the solution of a linear system, Ax = b,

elimination method, and its modifications namely

In the case of Crout’s method this decomposition

Applied and Computational Mathematics

(http://www.sciencepublishinggroup.com/j/acm)

5613 (Online)

Methods for the Solution

Science, HITEC University, Taxila, Pakistan

[email protected] (S. Ayub)

Methods for the Solution of Systems of Linear Equations. Applied and Computational

doi: 10.11648/j.acm.20150403.23

the fields of physical sciences, engineering, economics,

linear equations in n unknowns x1, x2, …, xn, where A = [a

are the column vectors. There are many analytical as well

including Gauss elimination method, and its modification

which employ LU-decomposition method, where L = [

The LU-decomposition method was first introduced by

we have made an effort to modify the existing LU-decomposition

least possible endeavour. It may be seen that the Gauss

Doolittle’s and Crout’s methods require n2 operations. Accordingly,

unknown elements of the L and U matrices. Moreover,

requires evaluation of 2n2/3 number of unknown elements

Crout’s and Cholesky’s methods presented in this paper

the L and U matrices. Moreover, an innovative method

number of unknown elements of the L and U matrices.

unknown elements. Thus, by employing these methods,

substantially be reduced.

Matrix, Column Vector, Decomposition, Doolittle’s Method,

numerical methods for

b, including Gauss

namely Doolittle’s,

Crout's and Cholesky’s methods.

LU-decomposition of an nxn matrix

which is given as under,

=

decomposition is given as,

Solution of Systems

Applied and Computational

economics, etc., we are led to systems of

[aij] is an nxn coefficient matrix,

well as numerical methods[1}– [11]

modifications namely Doolittle’s method,

[iij] and u = [uij] are the lower

by the mathematician Alan M.

decomposition methods to solve the

Gauss elimination method[1], [2], [3],

Accordingly, in these methods we

Cholesky’s method[1]

requires

elements of the L and U matrices

paper require evaluation of only

is also presented in this paper

matrices. In this method we need to

methods, the computational time and

Method, Crout’s Method,

methods. In the Doolittle’s method

matrix A is of the form A = LU,

.

208 M. Rafique and Sidra Ayub: Some Convalescent Methods for the

It may be mentioned that both the above

For a symmetric and positive definite coefficient

under,

The above mentioned usual LU-decomposition

are well discussed in the books on numerical

In these methods the coefficient matrix A is

= LU, so that the linear system of equations

written as LUx = b. This equation may further

Ly = b, where, Ux = y. First we solve the former

y by forward substitution and then solve

backward substitution. Accordingly, we obtain

the given system of linear equations.

For numerical stability, pivoting is very

the coefficient matrix is diagonally dominant

and positive definite. Therefore, in all the methods

in this paper, pivoting has been implemented

out further steps.

2. Improved Doolittle’s Method

The usual Doolittle’s method, mentioned

devised by the American mathematician

Doolittle (1830-1913)[1], [3], [7]–[12]

, which

Here all aij are known while lij and uij are

are to be evaluated. For this purpose we

hand side matrices L and U, and equate

elements on both sides, i.e., set aij

correspondence enables us to evaluate

elements lij and uij. Here we see that for

coefficient matrix A, it is required to evaluate

number of unknown elements lij, and uij

M. Rafique and Sidra Ayub: Some Convalescent Methods for the Solution of Systems of Linear Equations

=

��

�� 0 0 … 0�� 0 … 0�� … 0… … … … … … … … … …… … … … … … … … … … .… … … … … … … … … … . .�� … ��

��

��

1 u� u� 0 1 u 0 0 1 … … … … … … …… … … … … … …… … … … … … …

0 0 0

quoted decompositions lead to the same solution of any

coefficient matrix, the LU-decomposition by the Cholesky’s

=

decomposition methods

umerical linear algebra.

is decomposed as A

equations Ax = b can be

further be written as,

former equation for

the latter for x by

obtain the solution to

very essential unless

dominant or symmetric

methods discussed

implemented before carrying

Method

mentioned above, was

mathematician Myrick Hascall

appeared in U.S.

Coast and Geodetic Survey in

method, we consider a linear

unknowns, Ax = b. To start with

numerical stability. Thereafter,

is transcribed in such a way that

have the pivot element as unity,

if required, we may have to select

divide it throughout by the coefficient

and make it the pivot equation

main considerations for the decomposition

matrix A, as A = LU, are as under:

(i) The first column of the matrix

as that of matrix A, i.e., li1 = ai1,

(ii) The first row of the matrix

same as that of the matrix A, i.e.,

(iii) The diagonal elements of

unity, i.e., lii = 1 for all i.

Accordingly, we decompose

as,

=

are unknown which

multiply the right

the corresponding

= (lu)ij . This

all the unknown

for an nxn order

evaluate only (n–1)2

j of the L and U

matrices respectively. It may be

the above mentioned scheme

computational effort is substantially

above, after having carried

decomposition, we solve Ly =

of Ux = y for x, to get the required

system of equations. Here we

illustrate the above mentioned method.

Solution of Systems of Linear Equations

… u�� … u� … u�

… … … …… … … … .… … … … . .

0 … 1 ��

any linear system of equations.

Cholesky’s method is carried out as

1878. Here, in our improved

linear system of n equations in n

with pivoting is done to ensure

the system of linear equations

that the pivot equation is made to

unity, i.e., a11 = 1. For this purpose,

select the requisite equation and

coefficient of x1 for the needful,

equation. Following these steps, the

decomposition of the coefficient

under:

matrix L = [lij] is kept the same

, for all i.

matrix U = [uij] is also kept the

i.e., u1j = a1j, for all j.

of the matrix L= [lij] are kept as

the coefficient matrix A = LU

be appreciated that by following

scheme of LU-decomposition, the

substantially reduced. As described

out the above mentioned

b for y, followed by solution

required solution to the given

we discuss few examples to

method.

Applied and Computational Mathematics 2015;

2.1. Solution of a System of Three Equations

Unknowns

We consider the following system of equations,

5x1 + x2 + 2x3 = 19

6x1 + 24x2 – 12x3 = –12

2x1 + 3x2 + 8x3 = 39

We interchange 1st and 2

nd equations and

pivot equation by 6 so as to have a11 = 1,

method explained above. Then we decompose

matrix as A = LU, according to the scheme

Here the coefficient matrix A is of order 3x3,

find only (3–1)2 = 4 number of unknown

and U matrices, which turn out as, l32 = 0.26,

12, u33 = 8.84. Thus the coefficient matrix

as A = LU,

=

Having carried out the above decomposition,

=

After having carried out the above

decomposition, we solve Ly = b for y, where,

y5]T, and b = [– 4 –2 4 6 3]

T. Thus we obtain,

2.91 – 1.44]T. Thereafter, solving Ux = y

solution to the given system of equations as,

3. Improved Crout’s Method

The standard Crout’s method was formulated

American mathematician Prescott Durand

1984)[6]–[12]

. As in the case of improved Doolittle’s

here also, after having carried out pivoting,

equation is divided by the coefficient of

Here all aij are known while lij and uij are

are to be evaluated. For this purpose we

hand side matrices L and U, and equate

elements on both sides, i.e., set aij

Applied and Computational Mathematics 2015; 4(3): 207-213

Equations in Three

equations,

and then divide the

as required by the

decompose the coefficient

scheme described above.

3x3, so we need to

elements of the L

0.26, u22 = –19, u23 =

matrix gets decomposed

decomposition, we solve Ly

= b and obtain y = [–2 29 35.37]

get the solution to the given system

x = [2

2.2. Solution of a System of Five

Unknowns

We consider the following system

5x1 + 10x2 + 15x3 – 20x

2x1 + x2 + 5x3 + 3x4 +

3x1 + 4x2 + x3 + 2x4 +

4x1 + x2 + 3x3 + 5x4 +

2x1 + 4x2 – x3 + 2x4 +

We divide the pivot equation

and then decompose the coefficient

above and obtain (5–1)2 = 16 unknown

U matrices as, l32 = 0.67, l42 =

0.95, l54 = – 0.34, and, u22 =–3,

= – 7.33, u34 = 6.67, u35 = 1, u44

0.72. Accordingly, we obtain,

x

above mentioned

where, y = [y1 y2 y3 y4

obtain, y = [–4 6 12 –

y for x, we get the

as, x = [1 2 –1 1 2]T.

formulated by the

Durand Crout (1907 –

Doolittle’s method,

pivoting, the pivot

x1. Thereafter, the

main idea is to write out the system

the pivot equation has its pivot

the main considerations for

coefficient matrix A, as A = LU,

(i) As in the case of improved

li1 = ai1, for all i.

(ii) Same as in the case of improved

retain u1j = a1j, for all j.

(iii) The diagonal elements

unity, i.e., uii = 1 for all i.

Accordingly, we decompose

as,

=

are unknown which

multiply the right

the corresponding

= (lu)ij . This

correspondence enables us to

elements lij and uij.

We see that for an nxn order

required to evaluate only (n–1)

209

35.37]T. Next solving Ux = y we

system of equations as,

1 4]T

Five Equations in Five

system of linear equations,

20x4 + 5x5 = – 20

+ 2x5 = –2

+ 4x5 = 4

+ x5 = 6

+ 5x5 = 3

equation by 5 in order to make a11 = 1,

coefficient matrix as described

unknown elements of the L and

= 2.33, l32 = 0, l43 = 0.91, l53 =

u23 = –1, u24 = 11, u25 = 0, u33

44 = – 10.73, u45 = – 3.91, u55 =

system Ax = b in such a way that

pivot element a11 = 1. Furthermore,

the decomposition of the

LU, are as under:

improved Doolittle’s method, we keep

improved Doolittle’s method we

of the matrix U are kept as

the coefficient matrix A = LU

to evaluate all the unknown

order coefficient matrix A, it is

1)2 unknown elements lij, and uij


of the L and U matrices respectively. It may

that by following the above mentioned

decomposition, the number of operations

reduced. As described above, after having

above mentioned decomposition, we solve

followed by Ux = y for x, to obtain the required

the given system of linear equations. Here

examples to illustrate the above mentioned

3.1. Solution of a System of Four Equations

Unknowns


5x1 + 10x2 + 15x3 – 20x4 = –10

=

Next, solving Ly = b, where b is the column

12 8]T we obtain, y = [–2 –2 –1.91 1]

T, and

y, we get the solution to above system as x

3.2. Solution of a System of Five Equations

Unknowns


16x1 – 8x2 + 8x3 + 16x4 + 24x5

3x1 + 2x2 + 2x3 + 2x4 + 4x5 = 9

3x1 + 3x2 – 3x3 + 2x4 – 4x5 = 7

Having carried out the above mentioned

we first solve Ly = b, where b = [2.5 9 7

[2.5 0.43 0.47 5.84 1]T. Next, we solve Ux

solution to the given system of equations

1]T.

4. Improved Cholesky’s Method

The usual Cholesky’s method for symmetric


may be appreciated

mentioned scheme of LU-

operations are substantially

having carried out the

solve Ly = b for y,

required solution to

Here we discuss few

method.

Equations in Four

equations,

10

2x1 + x2 + 5x3 + 3x

3x1 + 4x2 + x3 + 2x

4x1 + x2 + 3x3 + 5x


then decompose the coefficient

Thus we obtain a total of (n

elements of the L and U matrices

l33 = –7.33, l43 = –0.33, l44 = –10.73,

u33 = 9, u34 = –0.91. Thus the

obtained as,

x

column vector, [–2 2

then solving Ux =

= [1 2 –1 1]T.

Equations in Five

equations,

5 = 40

9

7

5x1 – x2 – 1x3 + 3x

7x1 + x2 + 3x3 + 5x


= 1, and then decompose the coefficient

above. Thus we obtain (n – 1)

terms of the L and U matrices as:

l52 = 4.5, l33 = – 5.14, l43 = 3.71,

0.78 l55 = – 9.34, u23 = 0.14, u24

0.14, u35 = 1.53, and, u45 = 4.84.

desired LU decomposition of the

x

mentioned decomposition,

1 5]T. It gives y =

= y, and obtain the

as, x = [1 1 – 1 1

Method

symmetric and positive

definite coefficient matrix was

officer, geodesist and mathematician

(1876–1918)[1], [12]

. As discussed

algorithms pivoting is necessary

diagonally dominant, or symmetric

Here, in our improved method

in the case of improved Doolittle’s

decompose the coefficient matrix

=

��

1 0 0 … 0�� 1 0 … 0�� c 1 … 0… … … … … … … … … …… … … … … … … … … … .… … … … … … … … … … . .�� c� c� … 1

��

��1 a� a�0 c c0 0 c… … … … … …… … … … … …… … … … … … …0 0


3x4 = 2

2x4 = 12

5x4 = 8

equation by 5 to make a11 = 1, and

nt matrix, as illustrated above.

(n–1)2 =(4–1)

2 = 9 unknown

matrices as, l22 = –3, l32 = –2, l42 = –7,

10.73, u23 = 0.33, u24 = – 3.67,

the decomposition A = LU, is

3x4 – 7x5 = 1

5x4 – 5x5 = 5

equation throughout by 16, to make a11

coefficient matrix as enunciated

1)2

= (5 – 1)2 = 16 unknown

as: l22 = 3.5, l32 = 4.5, l42 = 1.5,

3.71, l53 = 1.14, l44 = –1.78, l54 = –

24 = – 0.29, u34 = 0.056, u25 = –

4.84. Accordingly, we get the

the coefficient matrix as,

developed by French military

mathematician Andre-Louis Cholesky

discussed above, for all practical

necessary unless the matrix is

symmetric and positive definite.

we follow the same scheme as

Doolittle’s method. Accordingly, we

matrix A = LU as,

� … a��c … c�

… c�… … … … …

… … … … .… … … … . . 0 … c��

��

Applied and Computational Mathematics 2015;

Here all aij are known while cij are unknown

be evaluated. For this purpose we multiply

side matrices L and U, and equate the

elements on both sides. This correspondence

evaluate all the unknown elements cij.

We see that for an nxn order coefficient

to evaluate only (n–1)2 unknown elements

matrices respectively. It may be appreciated

the above mentioned scheme of LU-decomposition,

number of operations are substantially reduced.

above, after having carried out the

decomposition, we solve Ly = b for y, followed

x, to obtain the required solution to the given

equations. Here we discuss few example

above mentioned method.

=

Next, solving Ly = b, where b is the column

35 94 1]T we obtain, y = [15 –20 39 –16]

T

Ux = y, we get the solution to the given system

as x = [2 –3 4 –1]T.

4.2. Solution of a System of Five Equations

Unknowns


12x1 + 18x2 + 24x3 + 30x4 + 12x

3x1 – x2 + x3 – 2x4 + x5 = 9

4x1 + x2 + 2x3 + x4 – x5 = 5

�� 1 1.5 2 2.5 13 � 1 1 � 2 14 1 2 1 � 15 � 2 1 2 12 1 � 1 1 1

�� =

��

1 3 4 5 2

After carrying out the above mentioned

first we solve Ly = b, where b = [2 9 5 8 –1]

–5.73 –5.75 6.68]T. Thereafter, we solve Ux

the solution to the given system of equations

x = [1 –1 2 –1 1]T.

5. An Innovative LU-Decomposition

Method

We consider a linear system of n equations

Ax = b, where A is the coefficient matrix of

x and b are column vectors of order nx1, each.

carry out pivoting unless the coefficient matrix

Applied and Computational Mathematics 2015; 4(3): 207-213

unknown, which are to

multiply the right hand

the corresponding

correspondence enables us to

matrix A, we have

cij of the L and U

appreciated that by following

decomposition, the

reduced. As described

the above quoted

followed by Ux = y for

given system of linear

examples to illustrate the

4.1. Solution of a System of Four

Unknowns

We consider the following system

Decomposing the coefficient

1)2 = 9 unknown elements of the

0.5, l42 = 0, l43 = – 0.33, u22 =

u34 = – 3, u44 = 16. Thus the decomposition

as,

x

column vector, [15 –T, and then solving

system of equations

Equations in Five

equations,

12x5 = 24

5x1 – 2x2 + x3 + 2x

2x1 + x2 – x3 + x4 +


required by the method discussed

we decompose the coefficient

described above. In this case we

unknown terms of the L and U

1.73, l52 = 0.36, l43 = 0.25, l53 =

5.5, u23 = –5, u24 = – 9.5, u25 =

u35 = – 3.18, u44 = 6, u45

decomposition is given as,

� 0 0 0 1 0 0 0.91 1 0 1.73 0.25 1 0.36 2.19 0.042

�� x

�� 1 1.5 2

0 � 5.5 � 0 0 � 1.450 0 00 0 0

mentioned decomposition,

1]T, to get y = [2 3

Ux = y, and obtain

equations as,

Decomposition

equations in n unknowns,

of order nxn, while

each. First of all we

matrix is diagonally

dominant or symmetric and positive

necessitates to have the coefficients

as unity, i.e., ai1 = 1 for all i. Any

condition may be divided throughout

for the needful. Thereafter, we

matrix as, A = LU, with the following

(i) The first column of the

[lij] is kept the same as

1, for all i.

(ii) The first row of the upper

is also kept the same as

for j ≥ 2. The elements

matrix L are transformed

(iii) for i ≥ 2.

(iv) The diagonal elements

211

Four Equations in Four

system of equations,

coefficient matrix we obtain (n–1)2 =(4–

the L and U matrices as, l32 = –

4, u23 = – 2, u24 = 0, , u33 = 9,

decomposition A = LU, is given

2x4 + x5 = 8

+ x5 = – 1

equation by 12 to set a11 = 1, as

discussed in this paper. Thereafter,

coefficient matrix as, A = LU, as

we need to evaluate (5–1)2 = 16

U matrices as, l32 = 0.91, l42 =

= 2.19, l54 = 0.042, and u22 = –

= –2, u33 = – 1.45, u34 = –0.36,

45 = 0.25, u55 = 6.68. This

2 2.5 1� 5 � 9.5 � 2

45 � 0.36 � 3.180 6 0.250 0 6.68

��

positive definite. This method

coefficients of x1 in all the equations

Any equation not fulfilling this

throughout by the coefficient of x1

can decompose the coefficient

following provisions,

the lower triangular matrix L =

that of matrix A, i.e., li1= ai1 =

upper triangular matrix U = [uij]

as that of matrix A, i.e., u1j = a1j,

elements of the second column of

transformed as, li2 = ai2 – a12,

elements of the matrix U are kept as


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


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


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


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.

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[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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[12] Petre Teodorescu, Nicolae-Doru Stanescu, Nicolae Pandrea: Numerical Analysis with Applications in Mechanics and Engineering, John Wiley (2013).

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