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JST Vol. 21 (1) Jan. 2013 - Pertanika Journal - Universiti Putra ...

JST Vol. 21 (1) Jan. 2013 - Pertanika Journal - Universiti Putra ...

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Heng, S. C., Ibrahim, Z. B., Suleiman, M. and Ismail, F.<br />

Then, equations [8] and [9] can be rewritten in the matrix form, as follows:<br />

é f 1 2 ù<br />

n+<br />

ê1-2 h(<br />

) ú é 2ù<br />

ê ( i+ 1) ê 1 ú () i é2 0 ù () i<br />

y 1 3 úé n e ù - -<br />

1 3<br />

é<br />

n y ù é<br />

n 1 f ù éVù + ê úê + ú ê ú + ê ú n+<br />

1 1<br />

= h<br />

( i+ 1) ê úê<br />

ú+ ê () i 6 úê<br />

ú+ ê ú (12)<br />

ê () i<br />

18 6 f úê<br />

n+ 2 e ú<br />

n 2 18 êy ú<br />

n 2 0 êf ú êVú ê<br />

+ + n+<br />

2 2<br />

1 h(<br />

) úë û ê ú<br />

1 ë û<br />

ê ú<br />

11<br />

ë û ë û<br />

ê<br />

- - - ê ú<br />

11 11 y ú ê ú ë û<br />

ê ê11 ú<br />

ë n+<br />

2 úû<br />

ë û<br />

In order to solve the above matrix equation, LU decomposition, which is a matrix<br />

decomposition that rewrites the matrix in the upper triangular matrix and lower triangular<br />

matrix was used (William et al., 2007).<br />

Let<br />

é æ f ö<br />

1 2 ù<br />

ê n<br />

1 2hç<br />

+ ÷<br />

ú<br />

ê<br />

- ç ÷<br />

yn+<br />

1 3 ú<br />

ê<br />

ç è ÷ ø<br />

ú<br />

A = ê ú<br />

ê 18 6 æ f öú<br />

n+<br />

2<br />

ê - 1-<br />

h ç<br />

÷ ú<br />

ê ÷<br />

11 11 è<br />

çç<br />

y ø÷<br />

ú<br />

ë n+<br />

2 û<br />

é 2ù<br />

ê-1- ú ( i) é2 0 ù ( i)<br />

ê 3úê<br />

éy ù<br />

n+ 1 ê úé<br />

f ù é n 1 V ù<br />

+ 1<br />

B= ê úê<br />

ú+ hê<br />

( i) 6 úê<br />

ú+ ê ú<br />

18 0<br />

( i<br />

ê úê<br />

ú )<br />

2 2 2<br />

1<br />

y ê úê<br />

ú<br />

n f êVú + n+<br />

ê - úë ûú ê ê ú<br />

11úë<br />

û ë û<br />

ê<br />

ë û<br />

ë11 úû<br />

é ( i+<br />

1)<br />

e ù<br />

n+<br />

1<br />

E = ê ú<br />

ê ( i+<br />

1)<br />

e ú<br />

ë n+<br />

2 û<br />

Suppose that matrix A is a product of the two matrices,<br />

40 <strong>Pertanika</strong> J. Sci. & Technol. <strong>21</strong> (1): 283 - 298 (<strong>2013</strong>)<br />

(13)<br />

(14)<br />

(15)<br />

AE × = B<br />

(16)<br />

LU × = A<br />

(17)<br />

where L is the lower triangular and U is the upper triangular. Thus,<br />

Let<br />

Hence,<br />

( LU × ) × E= B.<br />

(18)<br />

X = U× E.<br />

(19)<br />

L× X = B.<br />

(20)

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