Biomedical Engineering Reference
In-Depth Information
includes the terms in
B
on the right hand side of the equations. The other elements
A
31
,
A
41
,
...
,
A
n
1
in the first column of matrix
A
are treated similarly by repeating
this process down the first column (e.g. Row3
×
A
31
/
A
11
), all the elements
below
A
11
are reduced to zero. The same procedure is then applied for the second
column, (for all elements below
A
22
) and so forth until the process reaches the
n
−
Row1
1
column. Note that at each stage we need to divide by
A
nn
and therefore it is imperative
that the value is non-zero. If it is, then row exchange with another row below that
has a non-zero needs to be performed.
After this process is complete, the original matrix
A
becomes an
upper triangular
matrix
that is given by:
−
All the elements in the matrix
U
except the first row differ from those in the original
matrix
A
and our systems of equations can be rewritten in the form:
Uφ
=
B
The upper triangular system of equations can now be solved by the
Back Substitution
process. It is observed that the last row of the matrix
U
contains only one non-zero
coefficient,
A
nn
, and its corresponding variable
φ
n
is solved by
B
n
U
nn
The second last row in matrix
U
contains only the coefficients
A
n-
1,
n
and
A
nn
and,
once
φ
n
is known, the variable
φ
n-
1
can be solved. By proceeding up the rows of the
matrix we continue substituting the known variables and
φ
i
is solved in turn. The
general form of equation for
φ
i
can be expressed as:
φ
n
=
n
B
i
−
A
ij
φ
j
j
=
i
+
1
φ
i
=
(7.49)
A
ii
It is not difficult to see that the bulk of the computational effort is in the
forward
elimination
process; the back substitution process requires less arithmetic operations
and is thus much less costly. Gaussian elimination can be expensive especially for 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.
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