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n−i−j−k−
1
A
1
l
⊗
A
1
n−i−j−k−l
.
1=1
(Consult Box
B.4
for the general representation of
A
mn
.)
Backward substitution:
B
=I
⇔
A
⎡
⎤
⎡
⎤
B
11
B
12
...
B
1
n−
1
B
1
n
A
11
A
12
...
A
1
n−
1
A
1
n
⎣
⎦
⎣
⎦
0B
22
...
B
2
n−
1
B
2
n
0A
22
...
A
2
n−
1
A
2
n
· ·
00
...
B
n−
1
n−
1
B
n−
1
n
00
...
·
·
...
· ·
00
...
A
n−
1
n−
1
A
n−
1
n
00
...
·
·
...
=I
.
(B.28)
0
nn
0
nn
(i) B
11
A
11
=I
2
⇒
B
11
=A
−
1
11
;
11
A
12
A
−
1[2]
B
11
A
12
A
−
1
A
−
1
(ii) B
12
A
12
+B
12
A
22
=0
⇒
B
12
=
−
22
=
−
22
;
(B
11
A
13
+B
12
A
23
)A
−
1
(iii)B
11
A
13
+B
12
A
23
+B
13
A
33
=0
⇒
B
13
=
−
33
=
(B.29)
=A
−
1
11
(A
12
A
−
1
A
13
)A
−
1
33
;
(iv)B
11
A
14
+B
12
A
24
+B
13
A
34
+B
14
A
44
=0
22
A
23
−
⇒
B
14
=
−
(B
11
A
14
+B
12
A
24
+B
13
A
34
)A
−
1
44
=
=A
−
1
11
[A
12
A
−
1
22
(A
24
−
A
23
A
−
1
33
A
34
)+A
13
A
−
1
33
A
34
−
A
14
]A
−
1
44
.
(Consult Box
B.5
for the general representation of
B
1
n
.)
Notable for the GKS algorithm is the following. In the first step or the forward substitution,
a set of equations for (
B.30
) with respect to the
Kronecker-Zehfuss product
is constructed by
substituting (
B.31
)into(
B.32
)intothepowersof(
B.33
) set up in matrix equations for the first
polynomial, the second polynomial, and finally the
n
th polynomial:
x
1
x
2
x
1
x
2
[2]
,...,
x
1
[
n−
1]
x
1
x
2
[
n
]
,
(B.30)
x
2
y
(
x
):=
y
1
=A
11
x
1
+A
12
x
1
x
1
x
2
+
⊗
···
,
(B.31)
y
2
x
2
x
2
x
(
y
):=
x
1
=B
11
y
1
+B
12
y
1
y
1
y
2
+
⊗
···
,
(B.32)
x
2
y
2
y
2
x
1
x
2
,...,
x
1
[
n
]
=
x
1
x
1
x
2
.
⊗···⊗
(B.33)
n
times
x
2
x
2
Throughout, we particularly take advantage of the fundamental Kronecker-Zehfuss product rule
(
AB
)
D
), i.e. its reduction to the Cayley product of two matri-
ces. A heavy computation of the matrices
⊗
(
BD
)=(
A
⊗
B
)(
C
⊗
{
A
22
,
A
23
, ...,
A
33
,
A
34
,...
}
is taken over by the
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