since the vertices k and j are in the same irreducible diagonal block, there exists a
path from k to j. Hence, there exists a path from i to j.
Now the matrix C has irreducible diagonal blocks Cm2. The structure of C
is similar as the structure of Cm2. Similarly, we can show that the matrix C is
irreducible. n
Therefore C = A hpB is irreducibly diagonally dominant by Definition 3.2.4
and Lemma 3.2.7. Based on Theorem 3.2.5, C is nonsingular. Therefore, we have
the iteration
r+1 = C-'D ",
where C = A /B, D = AA + |(1 p)B, and C-1D is the amplification matrix.
3.3 Important Role and Approximation for Conductivity Function
The conductivity function a varies by many orders of magnitude between
the surface of the Earth and the ionosphere. Figure ?? and Figure 3-3 show
that a grows more than the factor 1010 S/m from the surface of the Earth to the
ionosphere, which leads the differential Equation (3.6) to be a very stiff equation.
For a stiff equation
dy
d -ky, (3.20)
where k is a very large positive number, its solution is y(t) = e-kty(O) and y(t)
tends to zero quickly. Let us consider the following three schemes:
1. Euler explicit scheme for Equation (3.20):
yn+1 y TL
=^ -ky".
At
Therefore, yn = (1 kAt)"y(0). When k is very large, (1 kAt) < -1 and
y" --+ +oo as n increases. Hence Euler explicit scheme does not work.
2. Euler implicit scheme for Equation (3.20)
yn+l __ n1
y+- kyn+l.
At