Biomedical Engineering Reference
In-Depth Information
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
B AB
(5.78)
'
ii
+
1
'
i
ii
−−
11
i
A
=
,
B
=
ii
+
1
i
A AA
'
A AA
'
ii
ii
−−
11
i
i
ii
ii
−−
11
i
i
The matrix containing the non-zero coefficients is therefore manipulated into:
B
1
A
0000 0
φ
φ
1
12
1
B
01
A
0 0 0 0
2
1
23
0
00 0
φ
=
B
00
0 1
A
0 0
i
i
ii
+
1
00
0
0
φ
φ
00
0 0 0 1
A
B
B
n
1
nn
1
n
1
00
0 0 0 0 1
n
n
The second stage involves back substitution , to evaluate ϕ nm and ϕ i as
'
'
'
ϕ
= =−
B
and
ϕ
B
ϕ +
A
(5.79)
n
n
i
i
i
1
ii
+
1
The TDMA is more economical than the Gaussian elimination because of the ab-
sence of arithmetic operations (multiplication and divisions) in obtaining ϕ i during
back substitution.
During the forward elimination stage the term A nn may consist of a zero value
and to prevent ill-conditioning of a matrix, it is necessary to ensure that
(5.80)
AA A
ii
>
+
ii
1
ii
+
1
This means that the diagonal coefficients need to be much larger than the sum of the
neighbouring coefficients. If this is not the case, matrix row swapping (otherwise
known as pivoting ) can be performed.
5.5.2
Iterative Methods
Direct methods such as Gaussian elimination can be employed to solve any system
of equations. Unfortunately, in most CHD problems that usually results in a large
system of non-linear equations. The cost of using this method is generally quite
high as it requires a lot pre-conditioning of the matrix to prepare it for the direct
 
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