同时计算行秩和列秩轮廓
Simultaneous computation of the row and column rank profiles
AI总结：
提出一种新的主元选择策略，通过PLUQ三角分解同时恢复输入矩阵及其所有前导子矩阵的行秩和列秩轮廓，并给出一个秩敏感的四次递归算法，其时间复杂度为O(mnr^{ω-2})。
摘要：
全主元高斯消去生成一个PLUQ矩阵分解。根据搜索主元时使用的策略，置换矩阵可以揭示矩阵的行或列秩轮廓的一些信息。我们提出了一种新的主元选择策略，使得能够同时恢复输入矩阵及其任何前导子矩阵的行和列秩轮廓。我们提出了一种秩敏感的四次递归算法，该算法计算一个m×n秩为r的矩阵的PLUQ三角分解，需要O(mnr^{ω-2})次域运算，其中ω是矩阵乘法的指数。与Malashonock的LEU分解相比，该算法具有相似的递归结构，但其时间复杂度是秩敏感的，并且具有更低的领先常数。在字长有限域上，该算法还提高了先前已知实现的实用效率。
英文摘要：
Gaussian elimination with full pivoting generates a PLUQ matrix decomposition. Depending on the strategy used in the search for pivots, the permutation matrices can reveal some information about the row or the column rank profiles of the matrix. We propose a new pivoting strategy that makes it possible to recover at the same time both row and column rank profiles of the input matrix and of any of its leading sub-matrices. We propose a rank-sensitive and quad-recursive algorithm that computes the latter PLUQ triangular decomposition of an m \times n matrix of rank r in O(mnr^{ω-2}) field operations, with ωthe exponent of matrix multiplication. Compared to the LEU decomposition by Malashonock, sharing a similar recursive structure, its time complexity is rank sensitive and has a lower leading constant. Over a word size finite field, this algorithm also improveLs the practical efficiency of previously known implementations.
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