4 Implementations

The details of the maximum entropy solver that we have used is derived from the algorithm of Cornwell & Evans (1985) with modifications by Sault (1990). It is based on a modified Newton-Raphson iterative scheme: the algorithm proceeds by evaluating first- and second-order derivatives of the objective function, and, from these, deducing changes to make to the images and the Lagrange multipliers. For reference, we give the derivatives of the entropy measure for the joint approach in an appendix (this corrects some apparent errors in Holdaway & Wardle 1990). In computing these changes, the matrix of second derivatives (the Hessian matrix) needs to be inverted. The size of this matrix is $(N_{\rm pix}\cdot N_{\rm pol}+N_{\rm con})\times (N_{\rm pix}\cdot N_{\rm pol}+N_{\rm con})$,where $N_{\rm pix}$ is the number of pixels per image, $N_{\rm pol}$ is the number of polarizations (i.e. 1 for separate, and up to 4 for joint deconvolution) and $N_{\rm con}$ is the number of Lagrange multipliers (i.e. 1, 2 or 3). To make this inversion feasible, we follow Cornwell & Evans and approximate the point-spread function in the Hessian as a delta function. Additionally, in the joint deconvolution, we ignore the mixed second-order derivatives of H (i.e. $\partial^2H/\partial I\,\partial Q$, etc.). As with Sault (1990), this makes the Hessian a sufficiently simple form that inversion is straightforward and the changes can be determined.