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

ITR/ACS：协作研究 LinBox：半数值黑盒线性代数通用库

• 批准号: 0113121
• 负责人: Erich Kaltofen
• 金额: $ 17.02万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2004-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0113121&HistoricalAwards=false
• 关键词: ITR; ACS; Collaborative; Research; LinBox

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.

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，他们提出研究线性代数的高效算法设计，在软件库中实现这些算法，以及如何将库与广泛使用的科学计算软件相结合。算法将被实现，并设计新的算法，用于矩阵的黑盒表示-因此命名为LinBox-在入口域上，这些入口域要么是符号的，即精确的，要么是浮点的，即不精确的。该库一般被编程为具有抽象底层算法的C++模板类;它们可以与各种快速库一起编译，用于基本字段，浮点和多项式运算。一个服务器/客户端接口无缝地将库连接到通用符号系统Maple和Mathematica以及数字系统MatLab。所实现的算法的并行执行是便利的。 黑盒矩阵被存储为函数（实际上是线性运算符）：矩阵是一个过程，它以任意向量作为输入，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、最小多项式和特征多项式的快速方法。最后，LinBox将包含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积分解和计算整数矩阵的Smith标准形。