Biomedical Engineering Reference
In-Depth Information
method process. This leaves the option of employing iterative methods. In an it-
erative method, one guesses the solution, and uses the equation to systematically
improve the solution until it reaches a specified level of convergence. If the number
of iterations is small in achieving convergence, an iterative solver may cost less to
use than a direct method.
Of the various classes of iterative methods, the simplest is the
Jacobi
method
which solves the left hand side of the matrix expression, using previous values for
ϕ
on the right hand side. Using the system of equations,
Aϕ
=
B
, as described in
the previous section; the general form of the algebraic equation for each unknown
nodal variable of
ϕ
is written as:
i
−
1
n
∑
∑
(5.81)
AA
φ
+
φ
+
A B
φ
=
ij
j
ii
i
ij
j
i
j
=
1
ji
=+
1
In Eq. (5.81), the Jacobi method requires that the nodal variables
ϕ
j
(non-diagonal
matrix elements) are assumed at each iteration step
k
and the nodal variables
ϕ
i
are
treated as unknown at iteration step
k
+ 1. Solving for
ϕ
i
, we have
i
−
1
n
1
(
)
( )
( )
k
+
1
∑∑
k
k
φ
=
B
−
A
φ
−
A
φ
(5.82)
i
i
ij
j
ij
j
A
ii
j
=
1
ji
=+
1
The iteration process begins with an initial guess of the nodal variables
ϕ
j
(
k
= 0).
After repeated application of Eqn. (5.82) to all the
n
unknowns, the first iteration,
k
= 1, is completed. We proceed to the next iteration step,
k
= 2, by substituting the
iterated values at
k
= 1 into Eqn. (5.82) to estimate the new values at the next itera-
tion step. This process is continuously repeated for as many iterations as required to
converge to the desired solution.
A more immediate improvement to the Jacobi method is provided by the
Gauss-
Siedel
method where the updated nodal variables
φ
+
are immediately used on
the right-hand side of Eq. (5.81) as soon as they are available. In such a case, the
previous values of
(
k
1)
k
φ
that appear in the second term of the right-hand side of
Eq. (5.82) are replaced by the current values of
()
k
φ
, giving
()
i
−
1
A
n
A
B
A
ij
ij
(
k
+
1)
∑
(
k
+
1)
∑
(
k
)
i
(5.83)
φ
=− −
φ
φ
i
j
j
A
A
ii
ii
ii
j
=
1
ji
=+
1
Comparing the above two iterative procedures, the Gauss-Siedel iteration is typi-
cally twice as fast as the Jacobi iteration. After repeated applications of Eqs. (5.82)
and (5.83), convergence can be gauged in a number of ways. One method is to en-
sure that the maximum difference between each iteration,
φ
+
(
k
1)
()
k
φ
falls below
some predetermined value. If the relative change is continually increasing with each
iteration, then the solution is diverging.
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