AI 中文总结
本文提出实Cayley-Dickson代数乘法的拟线性算法，将乘法化为虚子空间交错积并用两次递归求值，实现$O(N\log N)$复杂度，在$N\ge32$时优于现有方法，并已实现为fastCD库。
AI 中文摘要
在实Cayley-Dickson代数$A_n$（维数$N=2^n$）中直接求值定义乘积的算术复杂度为二次。本文给出了一种统一的乘法算法，使用$O(N\log N)$次实算术运算和$O(N)$辅助存储。该算法将乘法归结为虚子空间上的交错积，然后通过在一个固定二次系数扩张上的两次递归调用求值该积。对于$n\ge1$，所得双线性算法至多使用$(9n-15)2^{n-1}+10$次依赖输入的实乘法；对于$n\ge3$，指定的算术调度总共使用$(34n-83)2^{n-1}+50$次实运算。在此计数约定下，对于$N\ge16$，拟线性调度比直接乘法使用更少的运算；对于$N\ge32$，比统一Cariow-Cariowa方法使用更少的运算。该算法已在MIT许可的C11库fastCD中实现，并带有NumPy支持的Python接口，其结果已与定义递归的独立实现进行核对。在与直接乘法和统一Cariow-Cariowa方法的单核基准测试中，对于每个测试的维数$N\ge32$，无论是单次还是批量乘积，拟线性实现的平均时间均为三者中最低，并且在$N=1024$时比直接乘法快约$16$倍。
英文摘要
Direct evaluation of the defining product in the real Cayley--Dickson algebra $A_n$, of dimension $N=2^n$, has quadratic arithmetic complexity. This paper gives a uniform algorithm for multiplication using $O(N\log N)$ real arithmetic operations and $O(N)$ auxiliary storage. The algorithm reduces multiplication to the alternating product on the imaginary subspace, then evaluates that product by a two-call recursion over one fixed quadratic coefficient extension. For $n\ge1$, the resulting bilinear algorithm uses at most $(9n-15)2^{n-1}+10$ input-dependent real multiplications, and for $n\ge3$, the specified arithmetic schedule uses $(34n-83)2^{n-1}+50$ real operations in total. Under this counting convention, the quasilinear schedule uses fewer operations than direct multiplication for $N\ge16$ and than the uniform Cariow--Cariowa method for $N\ge32$. The algorithm is implemented in the MIT-licensed C11 library fastCD, with a NumPy-backed Python interface, and its results are checked against an independent implementation of the defining recursion. In single-core benchmarks against direct multiplication and the uniform Cariow--Cariowa method, the quasilinear implementation had the lowest mean time of the three at every tested dimension $N\ge32$, for both single and batched products, and was roughly $16$ times faster than direct multiplication at $N=1024$.
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