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S
6,6
= - (B
mud
/C
mud
)sin Zx
m
/c
mud
(2.15q )
S
6,7
= - (B
mud
/C
mud
) cos Zx
m
/c
mud
(2.15r )
S
7,6
= 1
(2.15s )
S
7,7
= - tan Z(x
m
+x
b
)/c
mud
(2.15t )
S
7,8
= - A
a2
/A
b
(2.15u )
S
7,9
= (A
a2
/A
b
) tan Z(x
m
+x
b
)/c
mud
(2.15v )
S
8,6
= tan Z(x
m
+x
b
)/c
mud
(2.15w)
S
8,7
= 1
(2.15x )
S
8,8
= - tan Z(x
m
+x
b
)/c
mud
(2.15y )
S
8,9
= -1
(2.15z )
S
9,8
= 1
(2.15a
'
)
S
9,9
= - tan Z(x
m
+x
b
+x
a
)/c
mud
(2.15b
'
)
S
9,10
= (A
a1
/A
a2
){- 1 +
i
tan Z(x
m
+x
b
+x
a
)/c
mud
} (2.15c
'
)
S
10,8
= tan Z(x
m
+x
b
+x
a
)/c
mud
(2.15d
'
)
S
10,9
= 1
(2.15e
'
)
S
10,10
= - tan Z(x
m
+x
b
+x
a
)/c
mud
-
i
(2.15f
'
)
whereas, for the elements of the forcing function
R
, we have
R
1
= - {A
c
c
mud
'p sin Z(x
c
-x
s
)/c
mud
}/{A
p
ZB
mud
cos Zx
c
/c
mud
} (2.16a )
R
2
= - {c
mud
'p cos Z(x
c
-x
s
)/c
mud
}/{ZB
mud
cos Zx
c
/c
mud
} (2.16b )
2.4.2.6 Matrix inversion.
It is important to observe from Equation 2.14 that the coefficient matrix
S
is both
sparse
(that is, most of its terms are identically zero, and therefore need
not be stored by a custom designed algorithm) and
banded
(in other words, each
equation contains a limited, closely clustered number of unknowns, with the
overall nonzero elements of
S
located within a narrow diagonal band). To
mathematicians anyway, these properties imply significant computational
advantages, especially in view of the complex nature of the coefficients
underlying the overall system. Although the solution to the foregoing system is
trivial on workstations and mainframe computers, we emphasize that practical
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