On the stability of the cyclic reduction without back substitution for tridiagonal systems

Componentwise error analysis for a modification of the cyclic reduction without back substitution for a tridiagonal system is presented. We consider relative roundoff errors and equivalent perturbations, so the main supposition is that all the data is nonzero. First, backward analysis for the computation of each component of the solution in separate is presented. Bounds on the relative equivalent perturbations are obtained depending on two constants. From these bounds it is easy to obtain a componentwise forward error analysis. Then the two constants are defined for some special classes of matrices, i.e. diagonally dominant (row or column), symmetric positive definite, totally nonnegative andM-matrices, and it is shown that the bounds for these classes of matrices are small.The author was supported by Grants MM-211/92 and MM-434/94 from the National Scientific Research Fund of the Bulgarian Ministry of Education and Science.