14.2 Fundamental Equations and Computation
169
of the eigenvalues for all four cases on symmetric tridiagonal matrices. For
ce n (z, q):
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
√
2q 0 0 · · ·
√
2q 4 q 0 · · ·
0
q 16 q · · ·
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
q (2r) 2 q
q
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
˜
A
(2n)
0
A
(2n)
2
A
(2n)
4
. . .
. . .
A
(2n)
2r
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
= a 2k
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
˜
A
(2n)
0
A
(2n)
2
A
(2n)
4
. . .
. . .
A
(2n)
2r
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(14.16a)
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 + q q 0 0 · · ·
q 9 q 0 · · ·
0 q 25 q · · ·
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
q (2r − 1) 2 ) q
q
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
A
(2n+1)
1
A
(2n+1)
3
A
(2n+1)
5
. . .
. . .
A
(2n+1)
2r+1
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
= a 2k+1
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
A
(2n+1)
1
A
(2n+1)
3
A
(2n+1)
5
. . .
. . .
A
(2n+1)
2r+1
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(14.16b)
and for se n (z, q):
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
4 q 0 0 · · ·
q 16 q 0 · · ·
0 q 36 q · · ·
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
q (2r) 2 q
q
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
B
(2n+2)
2
B
(2n+2)
4
B
(2n+2)
6
. . .
. . .
B
(2n+2)
2r
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
= b 2k
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
B
(2n+2)
2
B
(2n+2)
4
B
(2n+2)
6
. . .
. . .
B
(2n+2)
2r
. . .
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
(14.17a)