Pseudo-spectral derivative of quadratic quasi-interpolant splines

354 S. Remogna
ningS1 0(∆ k), from (5) and (7) we obtain
(
(Ñ1 2 a2 − 1
)′ = 2
(
(Ñ 2 j )′ = 2
h 2
N 1 1 − a 2
h 3 + h 2
N 1 2
+ a j+1− b j
N 1 j −
h j+1 + h j
(
(Ñk+3 2 )′ = 2
)
,
c j−1
h j−1 + h j−2
N 1 j−2+ b j− c j−1
h j + h j−1
N 1 j−1
a j+1
h j+2 + h j+1
N 1 j+1
c k+2
h k+2 + h k+1
N 1 k+1 + 1−c k+2
h k+2
N 1 k+2
Evaluating (6) at the points ofT k , we have
)
, j = 2,...k+2,
)
.
Q ′ k+3
2 f(t i )= ∑ f j (Ñ 2 j) ′ (t i ), i=1,...,k+3.
j=1
Therefore the pseudo-spectral derivative at the quasi-interpolation knots can be computed
only using the values of f and(Ñ 2 j )′ atT k .
The values of (Ñ 2 j )′ atT k can be stored in a matrix D 2 ∈ R (k+3)×(k+3) : d i j =
(Ñ 2 j )′ (t i ), for i, j = 1,...,k+3, called differentiation matrix.
Setting y for the vector with components y j = f(t j ), j = 1,...,k+ 3 and y ′ for
the vector with components y ′ j = Q′ 2 f(t j), j = 1,...,k+3, we simply write:
(8) y ′ = D 2 y.
The differentiation matrix D 2 has a structure as follows:
⎛
⎞
× × ×
⎛
× × × × 0
× × × × ×
× × × × × 0
D 2 =
··· ··· ··· ··· ···
=
0 × × × × ×
× × × × ×
⎜
⎟ ⎜
⎝ 0 × × × × ⎠ ⎝
× × ×
D (1)
2
D (2)
2
D (3)
2
⎞
,
⎟
⎠
where D (1)
2 , D(3) 2
∈R 2×(k+3) , D (2)
2
∈R (k−1)×(k+3) is a banded matrix, with bandwidth
2 (i.e. 2 bands above and below the diagonal) and the elements different from zero are: