Tridiagonal matrix algorithm

Abstract: We propose an exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations. Each equation of the system contains five unknown values of the exact solution and its first derivative at three grid points on the interval. In constructing the scheme, the principle of superposition of solutions was used. Partial sums of the functional series representing independent solutions provide schemes of arbitrary order of accuracy for the boundary-value poblem and for the spectral one. To solve systems of linear equations, the modified tridiagonal matrix algorithm is proposed.

Abstract: A well-known property of an $M$-matrix is that its inverse is elementwise nonnegative, which we write as $M^{-1} \geq 0$. In a previous paper [Linear Algebra Appl., 434 (2011), pp. 131--143], we gave sufficient bounds on single element perturbations so that monotonicity persists for a perturbed tridiagonal $M$-matrix. Here we extend these results, presenting the actual maximum upper bounds on single element perturbations, as well as sufficient and necessary conditions for the maximum allowable higher rank perturbations. Perturbed Toeplitz tridiagonal $M$-matrices are considered as a special case. We compare our results to existing normwise bounds due to Bouchon and an iterative algorithm provided by Buffoni. We demonstrate the utility of these results by considering an application: ensuring a nonnegative solution of a discrete analogue of an integro-differential population model.