  This paper is concerned with algorithms for calibration of direction
dependent effects (DDE) in aperture synthesis radio telescopes (ASRT). After
correction of Direction Independent Effects (DIE) using self-calibration,
imaging performance can be limited by the imprecise knowledge of the forward
gain of the elements in the array. In general, the forward gain pattern is
directionally dependent and varies with time due to a number of reasons. Some
factors, such as rotation of the primary beam with Parallactic Angle for
Azimuth-Elevation mount antennas are known a priori. Some, such as antenna
pointing errors and structural deformation/projection effects for
aperture-array elements cannot be measured {\em a priori}. Thus, in addition to
algorithms to correct for DD effects known a priori, algorithms to solve for DD
gains are required for high dynamic range imaging. Here, we discuss a
mathematical framework for antenna-based DDE calibration algorithms and show
that this framework leads to computationally efficient optimal algorithms which
scale well in a parallel computing environment. As an example of an
antenna-based DD calibration algorithm, we demonstrate the Pointing SelfCal
algorithm to solve for the antenna pointing errors. Our analysis show that the
sensitivity of modern ASRT is sufficient to solve for antenna pointing errors
and other DD effects. We also discuss the use of the Pointing SelfCal algorithm
in real-time calibration systems and extensions for antenna Shape SelfCal
algorithm for real-time tracking and corrections for pointing offsets and
changes in antenna shape.

Instrumentation and Methods for Astrophysics