. Author manuscript; available in PMC: 2014 Nov 20.

Published in final edited form as: SIAM J Sci Comput. 2014;36(2):C95–C118. doi: 10.1137/120866580

Algorithm 3.

Chebyshev semi-iterative (CS) method (computes x ≈ A^†b).

Given A ∈ ℝ^{m × n}, b ∈ ℝ^m, and a tolerance ε > 0, choose 0 < σ_L ≤ σ_U such that all nonzero singular values of A are in [σ_L, σ_U] and let $d = (σ_{U}^{2} + σ_{L}^{2}) / 2$ and $c = (σ_{U}^{2} - σ_{L}^{2}) / 2$ .
Let x = 0, υ = 0, and r = b.
for
$k = 0, 1, \dots, ⌈ (log ε - log 2) / log \frac{σ_{U} - σ_{L}}{σ_{U} + σ_{L}} ⌉$
do
$β \leftarrow {\begin{matrix} 0 & if k = 0, \\ \frac{1}{2} {(c / d)}^{2} & if k = 1, \\ {(α c / 2)}^{2} & otherwise, \end{matrix} α \leftarrow {\begin{matrix} 1 / d & if k = 0, \\ d - c^{2} / (2 d) & if k = 1, \\ 1 / (d - α c^{2} / 4) & otherwise . \end{matrix}$
υ ← βυ + A^Tr.
x ← x + αυ.
r ← r − αAυ.
end for