关于递归算法求解三对角矩阵逆的问题
On recursive algorithms for inverting tridiagonal matrices
AI总结：
本文探讨了递归算法在求解三对角矩阵逆时的失效原因，提出新的公式以构建高效算法，确保残差误差极小。
摘要：
如果A是三对角矩阵，那么定义A的逆X的方程AX=I和XA=I实际上是X中每个行和列元素的二阶递推关系。因此，递归算法应是求解三对角矩阵逆的自然且常用方法——但事实并非如此。尽管已提出多种此类算法，但 none 可用于数值求解任意三对角矩阵的逆。此外，一些方法存在严重的不稳定性问题。本文深入研究这些问题。我们定位并解释了递归算法求解此类矩阵逆失效的不同原因，然后提出新的X=A⁻¹元素公式，以构建渐进最快的算法，确保残差误差||AX-I||和||XA-I||始终非常小。
英文摘要：
If $A$ is a tridiagonal matrix, then the equations $AX=I$ and $XA=I$ defining the inverse $X$ of $A$ are in fact the second order recurrence relations for the elements in each row and column of $X$. Thus, the recursive algorithms should be a natural and commonly used way for inverting tridiagonal matrices -- but they are not. Even though a variety of such algorithms were proposed so far, none of them can be applied to numerically invert an arbitrary tridiagonal matrix. Moreover, some of the methods suffer a huge instability problem. In this paper, we investigate these problems very thoroughly. We locate and explain the different reasons the recursive algorithms for inverting such matrices fail to deliver satisfactory (or any) result, and then propose new formulae for the elements of $X=A^{-1}$ that allow to construct the asymptotically fastest possible algorithm for computing the inverse of an arbitrary tridiagonal matrix $A$, for which both residual errors, $\|AX-I\|$ and $\|XA-I\|$, are always very small.
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