Gram-Schmidt类型的最优算法
Optimal algorithms of Gram-Schmidt type
AI总结：
针对有限d维对称、交错或Hermite型在除环上的正交分解，提出三种Gram-Schmidt类型算法，包括实现简单的O(d^3)算法、复杂度匹配矩阵乘法的顺序最优算法以及类似复杂度的并行NC算法。
摘要：
给出了三种Gram-Schmidt类型的算法，用于在除环上产生有限d维对称、交错或Hermite型的正交分解。第一种算法使用d^3/3+O(d^2)次环运算，实现非常简单。接着，该算法被改编为两个新方向：一个是复杂度匹配矩阵乘法复杂度的最优顺序算法；另一个是具有类似复杂度的并行NC算法。
英文摘要：
Three algorithms of Gram-Schmidt type are given that produce an orthogonal decomposition of finite $d$-dimensional symmetric, alternating, or Hermitian forms over division rings. The first uses $d^3/3+O(d^2)$ ring operations with very simple implementation. Next, that algorithm is adapted in two new directions. One is an optimal sequential algorithm whose complexity matches the complexity of matrix multiplication. The other is a parallel NC algorithm with similar complexity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
