ITR/ASC: Collaborative Research - Linbox: A Generic Library for Seminumeric Black Box Linear Algebra

LinBox小组由来自三个国家（美国、法国、加拿大）的12名研究人员组成，他们提出了线性代数高效算法的设计、在软件库中的实现以及如何将该库与广泛使用的科学计算软件接口的研究。算法将被实现，新的算法将被设计，用于矩阵的黑盒子表示——因此命名为LinBox——在符号域（即精确域）或浮点域（即不精确域）上。该库通常被编程为具有抽象底层算法的c++模板类；它们可以用各种用于基本字段、浮点数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Mathematica以及数字系统MatLab。促进了所实现算法的并行执行。黑箱矩阵被存储为函数（实际上是线性运算符）：矩阵是一个过程，它将任意向量作为输入，并有效地计算矩阵乘以向量的乘积。黑箱线性代数推广了稀疏性。LinBox库将包含求解线性方程的奇异和非奇异系统的算法，其系数矩阵以黑箱表示。在此基础上，提出了快速求解黑箱矩阵秩、最小和特征多项式的方法。最后，LinBox将包含黑盒矩阵线性丢芬图问题的方法，例如计算具有整数条目的线性系统的积分解和计算整数矩阵的史密斯范式。

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.