矩阵伴随计算的Kaltofen无除法行列式微分算法
Kaltofen's division-free determinant algorithm differentiated for matrix adjoint computation
摘要：
Kaltofen在1992年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。对于抽象环上的矩阵，根据Baur和Strassen的结果，行列式算法，实际上是一个直线程序，导致计算矩阵伴随的算法具有相同的复杂性。而后者的伴随算法是通过自动微分的逆模式得到的，因此不“显式”。我们提出了一种替代（仍然密切相关）的算法的伴随，可以直接实现，我们的意思是不诉诸自动转换。该算法是将程序微分技术“手工”应用于Kaltofen方法而推导出来的，并作了完整的描述。作为子问题，我们研究的分化的程序，计算最小多项式的线性生成的序列，我们使用一个懒惰的多项式评估机制，以减少斯特拉森的避免司在我们的情况下的成本。
英文摘要：
Kaltofen has proposed a new approach in 1992 for computing matrix determinants without divisions. 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 ring, by the results of Baur and Strassen, the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix. However, the latter adjoint algorithm is obtained by the reverse mode of automatic differentiation, hence somehow is not "explicit". We present an alternative (still closely related) algorithm for the adjoint thatcan be implemented directly, we mean without resorting to an automatic transformation. The algorithm is deduced by applying program differentiation techniques "by hand" to Kaltofen's method, and is completely decribed. As subproblem, we study the differentiation of programs that compute minimum polynomials of lineraly generated sequences, and we use a lazy polynomial evaluation mechanism for reducing the cost of Strassen's avoidance of divisions in our case.
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