  For large-scale data fitting, the least-squares progressive-iterative
approximation (LSPIA) methods were proposed by Lin et al. (SIAM Journal on
Scientific Computing, 2013, 35(6):A3052-A3068) and Deng et al. (Computer-Aided
Design, 2014, 47:32-44), where the constant step sizes were used. In this work,
we further accelerate the LSPIA method in the sense of a Chebyshev
semi-iterative scheme and present an asynchronous LSPIA (ALSPIA) method to fit
data points. The control points in ALSPIA are updated by utilizing an
extrapolated variant and an adaptive step size is chosen according to the roots
of Chebyshev polynomials. Our convergence analysis reveals that ALSPIA is
faster than the original LSPIA method in both cases of singular and nonsingular
least-squares fittings. Numerical examples show that the proposed algorithm is
feasible and effective.
