精确反演稀疏带状矩阵的线性缩放直接方法
Linearly scaling direct method for accurately inverting sparse banded matrices
AI总结：
提出一种新的O(n)算法，通过解析递归表达式直接求解带状矩阵系统，并扩展至带外少量非零元及全逆矩阵计算，在精度和效率上优于高斯消元法。
摘要：
在计算物理和化学的许多问题中，存在一种特殊的稀疏矩阵，称为“带状矩阵”。这些矩阵定义为仅在主对角线给定距离内具有非零元素，通常需要求逆以求解相关的线性方程组。本文提出一种新的O(n)算法来求解此类系统，其中n×n为矩阵大小。我们给出了直接获得解的解析递归表达式，以及计算机实现的伪代码。此外，我们回顾了可能并行化该方法的不同选项，描述了处理带外少量非零元素的带状矩阵的扩展，并利用相同思想提出了一种获得完整逆矩阵的方法。最后，我们通过使用大型随机带状矩阵以及通过有限差分求解一维泊松方程时出现的矩阵，证明了新算法在精度和数值效率上与基于高斯消元的标准方法相比具有竞争力。
英文摘要：
In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main diagonal, need often to be inverted in order to solve the associated linear system of equations. In this work, we introduce a new O(n) algorithm for solving such a system, being n X n the size of the matrix. We produce the analytical recursive expressions that allow to directly obtain the solution, as well as the pseudocode for its computer implementation. Moreover, we review the different options for possibly parallelizing the method, we describe the extension to deal with matrices that are banded plus a small number of non-zero entries outside the band, and we use the same ideas to produce a method for obtaining the full inverse matrix. Finally, we show that the New Algorithm is competitive, both in accuracy and in numerical efficiency, when compared to a standard method based in Gaussian elimination. We do this using sets of large random banded matrices, as well as the ones that appear when one tries to solve the 1D Poisson equation by finite differences.
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