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284 Polynomial Approximation of Differential Equations
Then, we can write
(13.2.3) −D ⋆ n ¯pn = ¯ fn,
where, according to the notations of section 7.2, the matrix D ⋆ n for n = 4 turns out
to be
⎡
⎢
⎣
2 ˜ d (2)
11
˜d (2)
21
˜d (2)
31
˜d (2)
12
˜d (2)
11 + ˜ d (2)
22
˜d (2)
32
˜d (2)
13
˜d (2)
12 0 0 ˜ d (2)
13 0 0
˜d (2)
23 0 d ˜(2) 12 0 0 d ˜(2) 13
˜d (2)
11 + ˜ d (2)
33 0 0 d ˜(2) 12 0 0 d ˜(2) 13
˜d (2)
21 0 0 d ˜(2) 11 + ˜ d (2)
22
0
d ˜(2) 21 0 d ˜(2) 21
0 0 ˜ d (2)
21
˜d (2)
31
˜d (2)
12
2 ˜ d (2)
22
˜d (2)
32
˜d (2)
13
0
˜d (2)
23 0 0
˜d (2)
23 0 d ˜(2) 23
˜d (2)
22 + ˜ d (2)
33 0 0 ˜ d (2)
23
˜d (2)
31 0 0 d ˜(2) 32 0 0 d ˜(2) 11 + ˜ d (2)
33
0
d ˜(2) 31 0 0 d ˜(2) 32 0 d ˜(2) 21
0 0 d ˜(2) 31 0 0 d ˜(2) 32
˜d (2)
31
˜d (2)
12
˜d (2)
22 + ˜ d (2)
33
˜d (2)
32
0
˜d (2)
13
˜d (2)
23
2 ˜ d (2)
33
Such a matrix can be decomposed as D ⋆ n = An + KnAnKn, where An (n = 4) is the
block diagonal matrix
(13.2.4) An :=
⎡
Ân 0 0
⎢ 0
⎣
Ân 0
0 0 Ân
⎤
⎥
⎦ , with Ân :=
and Kn (n = 4) is the permutation matrix (note that Kn = K −1
n ):
⎡
⎢
⎣
˜d (2)
11
˜d (2)
21
˜d (2)
31
˜d (2)
12
˜d (2)
22
˜d (2)
32
˜d (2)
13
˜d (2)
23
˜d (2)
33
⎤
⎥
⎦ ,
⎤
⎥
⎦