Comparison theorems for the convergence factor of iterative methods for singular matrices

✍ Scribed by Ivo Marek; Daniel B. Szyld

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 157 KB
• Volume: 316
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00275-x

✦ Synopsis

Iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix T, i.e., by the quantity γ (T ) = max{|λ|, λ ∈ σ (T ), λ / = 1}, where σ (T ) is the spectrum of T. Theorems are presented comparing the convergence factor of two iterative methods. The comparison is based on the relationship between the matrices of the splittings. A cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison. Although this cone is based on the (unknown) projection onto the null-space of a matrix, the characterization provided in the paper allows, in specific instances, the cone to be readily computable.