Abstract
Several new fast direct methods for solving constant coefficient elliptic boundary value problems are presented. The methods make extensive use of the algebraic properties of the modified Chebyshev polynomials $S_n (x)$ and $C_n (x)$, which allow us to obtain operation counts of $O(n^2 )$ or $O(n^2 \log _2 ({n / k}))$ for solving problems on an $n \times n$ grid. The algorithms are shown to be numerically stable by giving a Wilkinson-style error analysis. Previously studied fast direct methods, and shooting and multiple shooting techniques are related to the algorithms.
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