This paper describes some algorithmic enhancements to a frequency domain linearized Euler solver for prediction of turbomacinery noise radiating through turbofan engine exhausts. The solver is based on a direct solution approach for preventing temporal growth of jet shear layer instabilities commonly seen in radiation problems through exhaust jets. Direct solutions, however, require excessively large amount of computer memory for realistic problems which in turn limits the feasibility of the direct methods. One way of dealing with this is to increase the discretization accuracy without much penalty on the structure of the coefficient matrix of the resultant linear system of equations. This way the requirement on the number of grid points per wavelength can be relaxed. Spectral like resolutions are provided by the B-spline Galerkin discretization approach. Linear and quadratic variants are implemented. Another effort is spent on developing a hybrid approach. With this approach the regions in which temporal growth of instabilities is expected, a direct solution is sought while in the other regions an iterative solution is sufficient. This way the size of the linear equation system to be inverted directly is reduced, lowering the memory requirement but at an expense of increased solution time. Computations are presented using the B-spline Galerkin schemes and compared with those obtained using the standard 4th-order scheme. The computational resource requirements are discussed. The hybrid approach is still under implementation and only limited results from its iterative part are presented. © 2010 by Yusuf Ozyoruk.