2015/09/30 by Paweł Keller, Keller, Paweł, Iwona Wróbel +1
Computer Science · Engineering · Mathematics · #15A09 #65F05 #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1509.09264
openalex publication_date 2015/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.