Kaltofen无除法行列式算法的微分
Differentiation of Kaltofen's division-free determinant algorithm
摘要：
Kaltofen在[Kaltofen 1992]中提出了一种计算矩阵行列式的新方法。该算法基于Krylov子空间的婴儿步骤/巨步骤构造，并计算行列式作为特征多项式的常数项。对于抽象域上的矩阵，根据Baur和Strassen 1983的结果，行列式算法实际上是一个直线程序，导致了一个计算矩阵伴随的算法具有相同的复杂性[Kaltofen 1992]。然而，后者是通过自动微分的反向模式获得的，并且在某种程度上不是“显式的”。我们研究这种伴随算法，展示它是如何实现的（而不诉诸自动变换），并演示其使用多项式矩阵。
英文摘要：
Kaltofen has proposed a new approach in [Kaltofen 1992] for computing matrix determinants. The algorithm is based on a baby steps/giant steps construction of Krylov subspaces, and computes the determinant as the constant term of a characteristic polynomial. For matrices over an abstract field and by the results of Baur and Strassen 1983, the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix [Kaltofen 1992]. However, the latter is obtained by the reverse mode of automatic differentiation and somehow is not ``explicit''. We study this adjoint algorithm, show how it can be implemented (without resorting to an automatic transformation), and demonstrate its use on polynomial matrices.
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