Biomedical Engineering Reference
In-Depth Information
a full matrix containing a large number of unknown variables to be solved but it is
as good as any other methods that are currently available.
The matrix obtained in the example above is in the form of a tri-diagonal matrix
which is a special case of matrices that occurs frequently. A tri-diagonal matrix has
nonzero elements only on the diagonal plus or minus one column such as:
AA
AAA
AA
0
0
0
11
12
21
22
23
A
=
0
0
32
33
000
A
nn
In this form it is advantageous to consider variants of Gaussian elimination such as
the
TriDiagonal Matrix Algorithm
(
TDMA
), also known as the Thomas algorithm.
Let us consider a general tri-diagonal form of a system of algebraic equations as:
AA
00 0
0
0
φ
φ
B
11
12
1
1
AA A
0
0
0
0
B
21
22
23
1
2
0
0
0
0
00
A AA
0
0
φ
=
B
ii
−
1
ii
ii
+
1
i
i
00 0
0
00 00
00 00 0
A
A
A
φ
φ
B
n
− −− − −
2
nn
11
nn
1
n
1
n
−
1
A
A
B
nn
−
1
nn
n
n
The TDMA like the Gaussian elimination solves the system of equations above in
two parts:
forward elimination
and
back substitution
. For the
forward elimination
process, the neighbouring entries are eliminated below the diagonal to yield zero
entries. This means replacing the elements of
A
21
,
A
32
,
A
43
,….,
A
nn-
1
with zeroes.
For the first row, the diagonal entry
A
11
is normalized to unity and the neighbouring
entry
A
12
and the matrix
B
term
B
1
are modified according to
A
B
A
'
=
12
,
B
'
=
1
(5.77)
12
1
A
A
11
11
Like the Gaussian elimination, by multiplying the first row of the matrix by
A
21
and subtracting it from the second row; all the elements in the second row are sub-
sequently modified (where
A
21
becomes zero), which also include the terms in
B
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