Fast algorithms for discrete polynomial transforms

Abstract

Consider the Vandermonde-like matrix P := ( P k ( cos ⁡ j π N ) ) j , k = 0 N {\mathbf {P}}:=(P_k(\cos \frac {j\pi }{N}))_{j,k=0}^N , where the polynomials P k P_k satisfy a three-term recurrence relation. If P k P_k are the Chebyshev polynomials T k T_k , then P {\mathbf {P}} coincides with C N + 1 := ( cos ⁡ j k π N ) j , k = 0 N {\mathbf {C}}_{N+1}:= (\cos \frac {jk\pi }{N})_{j,k=0}^N . This paper presents a new fast algorithm for the computation of the matrix-vector product P a {\mathbf {Pa}} in O ( N log 2 ⁡ N ) O(N \log ^2N) arithmetical operations. The algorithm divides into a fast transform which replaces P a {\mathbf {Pa}} with C N + 1 a ~ {\mathbf {C}}_{N+1} {\mathbf {\tilde a}} and a subsequent fast cosine transform. The first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications. Numerical tests demonstrate that our fast polynomial transform realizes P a {\mathbf {Pa}} with almost the same precision as the Clenshaw algorithm, but is much faster for N ≥ 128 N\ge 128 .