7 research outputs found
A deterministic Kaczmarz algorithm for solving linear systems
We propose a deterministic Kaczmarz algorithm for solving linear systems
A\x=\b. Different from previous Kaczmarz algorithms, we use reflections in
each step of the iteration. This generates a series of points distributed with
patterns on a sphere centered at a solution. Firstly, we prove that taking the
average of points leads to an effective approximation of the
solution up to relative error , where is a parameter depending
on and can be bounded above by the square of the condition number. We also
show how to select these points efficiently. From the numerical tests, our
Kaczmarz algorithm usually converges more quickly than the (block) randomized
Kaczmarz algorithms. Secondly, when the linear system is consistent, the
Kaczmarz algorithm returns the solution that has the minimal distance to the
initial vector. This gives a method to solve the least-norm problem. Finally,
we prove that our Kaczmarz algorithm indeed solves the linear system
A^TW^{-1}A \x = A^TW^{-1} \b, where is the low-triangular matrix such
that . The relationship between this linear system and the
original one is studied.Comment: 31 pages, 30 figures (some typoes are fixed in the previous version