On recursive algorithms for inverting tridiagonal matrices

TL;DR

This paper thoroughly analyzes recursive algorithms for inverting tridiagonal matrices, identifies their limitations, and proposes new formulas to develop the fastest, most stable algorithm with minimal residual errors.

Contribution

It introduces new formulas for inverting tridiagonal matrices, enabling the construction of a highly efficient and stable recursive algorithm.

Findings

Existing recursive algorithms are often unstable or inapplicable.

New formulas significantly improve the stability and speed of inversion.

The proposed method achieves very small residual errors in practice.

Abstract

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.