Right‐angled triangle property in inversion of general tridiagonal matrices

Abstract

In this study a further relationship is extended to the elements in the lower triangle of the inverse of a general tridiagonal matrix for a non‐block case. Once the upper triangle of the inverse is determined based on Huang and McColl's analytical inversion formula, the corresponding lower triangle can be calculated efficiently using two proposed theorems. Each element in the lower triangle is decomposed into two parts: one is the coefficient; the other the counterpart element in the upper triangle. The coefficient is a cross product function of the elements in the tridiagonal matrix and can be easily obtained by using the right‐angled triangle property among the coefficients. This results in a faster computation of the lower triangle of the inverse of a general tridiagonal matrix. Several examples are given to demonstrate the superiority of two theorems developed by the author to Huang and McColl's algorithm. It is shown that the algorithm based on the right‐angled triangle property outperforms Huang and McColl's with regard to the speed of computing the inverse of the tridiagonal matrix. Empirically it is shown that the improvement rates are about 20% in calculating the inverse of Hermite matrices of sizes ranging from 7 by 7 to 20 by 20 for both the clamped and natural cubic spline.