Rounding error analysis in solving \(M\)-matrix linear systems of block Hessenberg form
The authors analyze the backward error of an algorithm, proposed by them in [SIAM J. Matrix Anal. 24, 852--876 (2003; Zbl 1038.65022)], to solve block lower Hessenberg systems \(Ax=b\). The analysis uses the relation with block LU factorization proving that the proposed algorithm is backward stable for row diagonal dominant matrices and hence for \(M\)-matrices. The obtained solution is the exact solution of a perturbed system where the perturbation in the vector \(b\) is bounded by the product of the infinite norm of the matrix \(A\), the computed solution, the machine precision and a constant depending on the block size and the number of blocks of the Hessenberg matrix \(A\).
scientific article; zbMATH DE number 2093350