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On the Inversion of Adjacent Tridiagonal and Pentadiagonal Matrices

机译：关于邻对角线和对角线矩阵的求逆

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Pentadiagonal as well as tridiagonal matrices have a wide number of applications in various fields of science, like mechanics, image processing, mathematical chemistry, etc.. For example, in fluid mechanics which is a commonly used subject, the number of meshes necessary to obtain reasonably good results is at times expressible in millions. Powerful techniques were developed to solve such systems. In the most common of these methods, inverses of tridiagonal and pentadiagonal matrices are encountered. Numerical inverses of band matrices as well as full matrices are amongst the topics for which serious difficulties in computations arise, perhaps not from a theoretical point of view but from the point of view of the computational time required. A rather inexpensive method was suggested by Huang and McColl initially for the inversion of symmetric tridiagonal and later for the general tridiagonal matrices. It turned out to be employable for the case of strictly diagonally dominant matrices which are quite widely encountered within many applications in literature. Needless to say it is very well known that such matrices can be shown to be non-singular and hence invertable. A similar but of course slightly more complicated method was developed for the inversion of adjacent pentadiagonal matrices by Kanal and Baykara. Kanal has also developed a parallel algorithm for the suggested method. Other methods to the same end were developed by Zhao and Huang and also by Hadj and Elouafi. The method of Zhao and Huang seems to somewhat suffer from computational complexity since it is of 0(N3). Recently, the mathematical structure of the method suggested by Kanal and Baykara was investigated in detail and it was also shown that it is faster than the method of Hadj and Elouafi.
机译：五角形和三角形矩阵在各种科学领域中都有广泛的应用，例如力学，图像处理，数学化学等。例如，在流体力学（这是一个常用的主题）中，获得所需的网格数相当好的结果有时可以表示成百万。开发了强大的技术来解决此类系统。在这些方法中，最常见的是遇到三对角和五对角矩阵的逆。带矩阵和全矩阵的数值逆是其中出现严重计算困难的主题，也许不是从理论的角度而是从所需的计算时间的角度来看。 Huang和McColl提出了一种相当便宜的方法，最初用于对称三对角矩阵的反演，后来又用于一般三对角矩阵。事实证明，它可用于严格对角占优势的矩阵，这在文献中的许多应用中都非常普遍。不用说，众所周知，这样的矩阵可以显示为非奇异的，因此是可逆的。 Kanal和Baykara开发了一种类似的方法，但是当然稍微复杂一些，用于反转相邻的五角形矩阵。 Kanal还为建议的方法开发了并行算法。为此，Zhao和Huang以及Hadj和Elouafi也开发了其他方法。 Zhao和Huang的方法似乎具有一定的计算复杂性，因为它的值为0（N3）。最近，详细研究了Kanal和Baykara提出的方法的数学结构，并且还表明它比Hadj和Elouafi的方法更快。

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