Biomedical Engineering Reference
In-Depth Information
In the worked example in Sect. 7.2.3 the matrix obtained 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:
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:
⎡
⎤
⎡
⎤
⎡
⎤
A
11
A
12
0
0
0
0
0
φ
1
φ
1
···
φ
i
···
φ
n
−
1
φ
n
B
1
B
2
···
B
i
···
B
n
−
1
B
n
⎣
⎦
⎣
⎦
⎣
⎦
A
21
A
22
A
23
0
0
0
0
0
···
···
···
0
0
0
00
A
ii
−
1
A
ii
A
ii
+
1
0
0
=
00 0
···
···
···
0
00 0 0
A
nn
−
2
A
n
−
1
n
−
1
A
n
−
1
n
00 0 0 0
A
nn
−
1
A
nn
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 zeros.
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
12
A
11
,
B
1
A
11
A
12
=
B
1
=
(7.50)
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 subsequently
modified (where
A
21
becomes zero), which also include the terms in
B
on the right
hand side of the equations. Applying the same procedure to the rest of the rows of
the matrix, the neighbouring element entries and the matrix
B
terms in general form
are:
A
ii
−
1
B
i
−
1
A
ii
+
1
A
ii
−
A
ii
−
1
A
i
−
1
i
B
i
−
A
ii
+
1
=
B
i
=
,
(7.51)
A
ii
−
1
A
i
−
1
i
A
ii
−
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