8.18 SIMULTANEOUS LINEAR EQUATIONS
the nine elements of (8.126) to those of the 3×3 matrixA. This is done in the
particular order illustrated in the example below.
Once the matricesLandUhave been determined, one can use the decompositionto solve the set of equationsAx=bin the following way. From (8.125), we have
LUx=b,but this can be written astwotriangular sets of equations
Ly=b and Ux=y,whereyis another column matrix to be determined. One may easily solve the first
triangular set of equations fory, which is then substituted into the second set.
The required solutionxis then obtained readily from the second triangular set
of equations. We note that, as with direct inversion, once theLUdecomposition
has been determined, one can solve for various RHS column matricesb 1 ,b 2 ,...,
with little extra work.
UseLUdecomposition to solve the set of simultaneous equations (8.123).We begin the determination of the matricesLandUby equating the elements of the
matrix in (8.126) with those of the matrix
A=
243
1 − 2 − 2
−33 2
.
This is performed in the following order:
1st row: U 11 =2, U 12 =4, U 13 =3
1st column: L 21 U 11 =1, L 31 U 11 =− 3 ⇒L 21 =^12 ,L 31 =−^32
2nd row: L 21 U 12 +U 22 =− 2 L 21 U 13 +U 23 =− 2 ⇒U 22 =−4,U 23 =−^72
2nd column: L 31 U 12 +L 32 U 22 =3 ⇒L 32 =−^94
3rd row: L 31 U 13 +L 32 U 23 +U 33 =2 ⇒U 33 =−^118Thus we may write the matrixAas
A=LU=
100
1
2 10
−^32 −^941
24 3
0 − 4 −^72
00 −^118
.
We must now solve the set of equationsLy=b,whichread
100
1
2 10
−^32 −^941
y 1
y 2
y 3
=
4
0
− 7
.
Since this set of equations is triangular, we quickly find
y 1 =4,y 2 =0−( 21 )(4) =− 2 ,y 3 =− 7 −(−^32 )(4)−(−^94 )(−2) =−^112.These values must then be substituted into the equationsUx=y,whichread
24 3
0 − 4 −^72
00 −^118
x 1
x 2
x 3