  For large-scale data fitting, the least-squares progressive iterative
approximation is a widely used method in many applied domains because of its
intuitive geometric meaning and efficiency. In this work, we present a
randomized progressive iterative approximation (RPIA) for the B-spline curve
and surface fittings. In each iteration, RPIA locally adjusts the control
points according to a random criterion of index selections. The difference for
each control point is computed concerning the randomized block coordinate
descent method. From geometric and algebraic aspects, the illustrations of RPIA
are provided. We prove that RPIA constructs a series of fitting curves (resp.,
surfaces), whose limit curve (resp., surface) can converge in expectation to
the least-squares fitting result of the given data points. Numerical
experiments are given to confirm our results and show the benefits of RPIA.
