Degeneration of structured integer matrices modulo an integer Original Research Article

Hensel’s lifting modulo a prime q is a customary means of the solution of an integer or rational linear system of equations. In combination with some effective numerical algorithms this technique enables solution in nearly optimal time in the case of most popular structured inputs. Practically one can further benefit from choosing q=2v for a proper positive integer v and performing binary computations within the computer precision. If the input matrix becomes singular because of the reduction modulo q, then the approach fails. For larger integers q and random integer input matrices, however, such degeneration occurs rarely according to the analysis by Brent and McKay 1987. Based on distinct techniques we show that degeneration also occurs rarely for random integer matrices with all most popular structures such as the Toeplitz, Hankel, band and rank (quasiseparable) structures. Furthermore with random small-rank modifications of an input matrix we have good chances to overcome degeneration, safely solve the new linear system, and recover the solution of the original one. The results of our extensive tests support our formal analysis.