论文标题
决定因素的公式
A Formula for the Determinant
论文作者
论文摘要
我们为$ n \ times n $矩阵的决定因素提供了一个公式，并带有带有单位的通勤环的条目。该公式可以通过“直线程序”进行评估，仅执行添加，减法和环元素的乘法；特别是它不需要分区或有条件的分支（例如，如高斯消除所要求的）。执行的操作数量由$ n $，特别是$ o（n^4 \ log n）$的固定功率界定。此外，可以将操作分为“阶段”，以便在给定阶段的操作数是矩阵条目或较早阶段的操作结果，并且阶段的数量受$ n $的固定功率的限制，$ n $，尤其是$ o \ big o \ big（\ log n log n）^2 \ big big big big big big big big）
We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring elements; in particular it requires no divisions or conditional branching (as are required, for example, by Gaussian elimination). The number of operations performed is bounded by a fixed power of $n$, specifically $O(n^4\log n)$. Furthermore, the operations can be partitioned into "stages" in such a way that the operands of the operations in a given stage are either matrix entries or the results of operations in earlier stages, and the number of stages is bounded by a fixed power of the logarithm of $n$, specifically $O\big((\log n)^2\big)$.
