Information Technology Reference
In-Depth Information
The matrix system
A
u
D b
now has the following matrix
A
, a right-hand side
b
,
and a vector of unknowns
u
:
0
@
1
A
1 0 0 0 0
˛1
C
2˛
˛ 0 0
0
˛1
C
2˛
˛0
00
˛1
C
2˛
˛
0
A D
;
0
0
0
1
0
@
1
A
0
@
1
A
u
0
u
1
u
2
u
3
u
4
0
u
`1
1
u
`1
2
u
`1
3
0
u
D
b D
;
:
`
In the rest of this chapter, we include the
u
values at the boundaries in the linear
systems.
The more general case, with n
C
1 grid points and Dirichlet boundary conditions,
leads to a linear system
A
u
D b
with n
C
1 unknowns. The .n
C
1/
.n
C
1/ matrix
A
can be expressed as
0
@
1
A
A
0;0
00
0
:
:
:
:
A
1;1
0
A
1;0
:
:
:
:
0A
2;1
A
2;2
A
2;3
:
:
:
:
:
:
:
:
:
:
:
0
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
A D
:
(7.123)
:
:
:
:
:
0
A
i;i1
A
i;i
A
i;iC1
:
:
:
:
:
:
:
:
:
:
:
:
:
0
:
:
:
:
:
:
:
:
:
:
A
n1;n
0
0A
n;n1
A
n;n
The components A
i;j
follow from (7.119). Let us explain how we derive the A
i;j
expression in detail. We start with a general row, say, the row with index i .Thisrow
arises from the (7.119), and the matrix entries correspond to the coefficients in front
of the unknowns in this equation:
A
i;i1
D
˛;
(7.124)
A
i;iC1
D
˛;
(7.125)
A
i;i
D
1
C
2˛ :
(7.126)
