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Context. Modern radio astronomical arrays have (or will have) more than oneorder of magnitude more receivers than classical synthesis arrays, such as theVLA and the WSRT. This makes gain calibration a computationally demanding task.Several alternating direction implicit (ADI) approaches have therefore beenproposed that reduce numerical complexity for this task from $\mathcal{O}(P^3)$to $\mathcal{O}(P^2)$, where $P$ is the number of receive paths to becalibrated.  Aims. We present an ADI method, show that it converges to the optimalsolution, and assess its numerical, computational and statistical performance.We also discuss its suitability for application in self-calibration and reporton its successful application in LOFAR standard pipelines.  Methods. Convergence is proved by rigorous mathematical analysis using acontraction mapping. Its numerical, algorithmic, and statistical performance,as well as its suitability for application in self-calibration, are assessedusing simulations.  Results. Our simulations confirm the $\mathcal{O}(P^2)$ complexity andexcellent numerical and computational properties of the algorithm. They alsoconfirm that the algorithm performs at or close to the Cramer-Rao bound (CRB,lower bound on the variance of estimated parameters). We find that thealgorithm is suitable for application in self-calibration and discuss how itcan be included. We demonstrate an order-of-magnitude speed improvement incalibration over traditional methods on actual LOFAR data.  Conclusions. In this paper, we demonstrate that ADI methods are a valid andcomputationally more efficient alternative to traditional gain calibrationmethod and we report on its successful application in a number of actual datareduction pipelines.

Fast gain calibration in radio astronomy using alternating direction implicit methods: Analysis and applications