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On Fast Algorithms for Triangular and Dense Matrix Inversion

机译：关于三角和密集矩阵求逆的快速算法

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摘要

We first propose in this paper a recursive algorithm for triangular matrix inversion (TMI) based on the 'Divide and Conquer' (D&C) paradigm. Different versions of an original sequential algorithm are presented. A theoretical performance study permits to establish an accurate comparison between the designed algorithms. Our implementation is designed to be used in place of dtrtri, the level 3 BLAS TMI. Afterwards, we generalize our approach for dense matrix inversion (DMI) based on LU factorization (LUF). This latter is used in Mathematical software libraries such as LAPACK xGETRI and MATLAB inv. A = LU being the input dense matrix, xGETRI consists, once the factors L and U are known, in inverting U then solving the triangular matrix system XL = U~(-1) (i.e. L~TX~T = (U~(-1))~T, thus X = A~(-1)). Two other alternatives may be derived here (L and U being known): (ⅰ) first invert L, then solve the matrix system UX = L~(-1) for X; (ⅱ) invert both L and U, then compute the product X = U~(-1)L~(-1). Each of these three procedures involves at least one triangular matrix inversion (TMI). Our DMI implementation aims to be used in place of the level 3 BLAS TMI-DMI. Efficient results could be obtained through an experimental study achieved on a set of large sized randomly generated matrices.
机译：我们首先在本文中提出一种基于“分而治之”（D＆C）范式的三角矩阵求逆（TMI）递归算法。介绍了原始顺序算法的不同版本。理论性能研究允许在设计的算法之间建立准确的比较。我们的实现旨在代替dtrtri（3级BLAS TMI）使用。之后，我们基于LU分解（LUF）概括了用于密矩阵求逆（DMI）的方法。后者用于数学软件库，例如LAPACK xGETRI和MATLAB inv。 A = LU是输入的密集矩阵，一旦知道了因子L和U，xGETRI就会对U求逆，然后求解三角矩阵系统XL = U〜（-1）（即L〜TX〜T =（U〜（ -1））〜T，因此X = A〜（-1））。这里可以得出另外两个选择（L和U是已知的）：（ⅰ）首先求L的倒数，然后对X求解矩阵系统UX = L〜（-1）； （ⅱ）反转L和U，然后计算乘积X = U〜（-1）L〜（-1）。这三个过程中的每一个都涉及至少一个三角矩阵求逆（TMI）。我们的DMI实施旨在代替3级BLAS TMI-DMI。通过在一组大型随机生成的矩阵上进行的实验研究，可以获得有效的结果。

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