Untitled - Cdm.unimo.it

72 Polynomial Approximation of Differential Equations
vector multiplications with (4.3.1). As we mentioned before, it is convenient to operate
with complex numbers. Thus, let i denote the complex unity, i.e., i 2 = −1. Then, we
consider the transform
(4.3.3) γm =
n−1
j=0
δj e 2imjπ/n , 0 ≤ m ≤ n − 1.
Here γm, 0 ≤ m ≤ n − 1, and δj, 0 ≤ j ≤ n − 1, are complex variables, related by
the standard complex FFT. We study later how to efficiently compute (4.3.3). We now
establish a relation between (4.3.3) and (4.1.4) in the Chebyshev case. To this end,
the papers of cooley, lewis and welsh(1970), and brachet et al.(1983) give useful
indications. Assume that n is even. We first define the input data as follows:
(4.3.4) δj
⎧
⎨p(ξ
:=
⎩
(n)
2j+1 ) n 0 ≤ j ≤ 2 − 1,
≤ j ≤ n − 1.
p(ξ (n)
2n−2j )
With this choice, we compute γm, 0 ≤ m ≤ n − 1. Finally, one verifies that
(4.3.5) cm = (−1)m
n
·
γ0
n
2
if m = 0,
γm e imπ/2n + γn−m e −imπ/2n if 1 ≤ m ≤ n − 1.
Of course, this final computation has a cost only proportional to n. The procedure can
be further improved by virtue of the analysis presented in swarztrauber(1986). Since
the δj’s are real, one can split the data in a suitable way, to allow the computation of
two complex FFTs of length n
2
instead of n. We skip the details for simplicity.
The inverse transform (see (4.1.6)) is clearly treated in a very similar way.
The next problem is related to matrix-vector multiplications involving (4.3.2). We
now define
(4.3.6) δj :=
⎧
⎪⎨
⎪⎩
p(η (n)
0 ) j=0,
p(η (n)
2j ) + i
p(η (n)
2j+1 ) − p(η(n) 2j−1 )
1 ≤ j ≤ n
2
p(η (n)
n ) j = n
2 ,
p(η (n)
2n−2j ) + i
p(η (n)
2n−2j−1 ) − p(η(n) 2n−2j+1 )
n
2
− 1,
+ 1 ≤ j ≤ n − 1.