Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices

Matrices represented as a sum of diagonal and semiseparable ones are considered here. These matrices belong to the class of structured matrices which arises in numerous applications. Fast O(N) algorithms for their inversion were developed before under additional restrictions which were a source of instability. Our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms. In this paper, the case of semiseparable matrices of order one is considered.