ITR/ACS: Collaborative Research LinBox: A Generic Library for Seminumeric Black Box Linear Algebra

由来自三个国家(美国、法国、加拿大)的12名研究人员组成的LinBox小组建议研究线性代数的有效算法的设计，在软件库中的实现，以及如何将库与广泛使用的科学计算软件接口。将实现算法，并设计新的算法，用于矩阵的黑盒表示-因此被命名为LinBox-关于符号(即，精确)或浮点(即，不精确)的条目域。该库一般被编程为具有抽象底层算术的C++模板类；它们可以使用各种用于基本域、浮点和多项式运算的快速库进行编译。服务器/客户端接口将库无缝地连接到通用符号系统Maple和MATHEMICA以及数字系统MATLAB。促进了所实现算法的并行执行。黑盒矩阵存储为函数(实际上是线性运算符)：矩阵是将任意向量作为输入并高效地计算矩阵乘以向量乘积的过程。黑盒线性代数是稀疏性的推广。LinBox库将包含求解奇异和非奇异线性方程组的算法，其系数矩阵以黑盒表示形式给出。此外，还提出了求黑盒矩阵的秩及最小特征多项式的快速方法。最后，LinBox将包含具有黑盒矩阵的线性丢番图问题的方法，例如计算具有整数项的线性系统的整数解和计算整数矩阵的Smith范式。

The LinBox group of twelve researchers in three countries (USA, France, Canada) proposes research in the design of efficient algorithms for linear algebra, in their implementation in a software library, and in how to interface the library to widely-used scientific computing software. Algorithms will be implemented, and new algorithms designed, for the black box representation of matrices---hence the name LinBox---over entry domains that are either symbolic, that is, exact, or floating point, that is, inexact. The library is generically programmed as C++ template classes with abstract underlying arithmetics; they can be compiled with a variety of fast libraries for the basic field, floating point, and polynomial operations. A server/client interface seamlessly attaches the library to the common general purpose symbolic systems Maple and Mathematica and to the numeric system MatLab. Parallel execution of the implemented algorithms is facilitated. Black box matrices are stored as functions (as linear operators in effect): the matrix is a procedure that takes an arbitrary vector as input and efficiently computes the matrix-times-vector product. Black box linear algebra generalizes sparsity. The LinBox library will contain algorithms for solving singular and non-singular systems of linear equations whose coefficient matrix is given in black box representation. Furthermore, it is proposed to develop fast methods for the rank and the minimal and characteristic polynomial of a black box matrix. Finally, LinBox will contain methods for linear Diophantine problems with black box matrices, such as computing an integral solution to a linear system with integer entries and computing the Smith normal form of an integer matrix.