Krylov subspace methods are methods to find solutions to high-dimensional linear systems efficiently. One of those methods is the Induced Dimension Reduction method, a method that has been implemented in a parallel Fortran package. To ensure the efficiency of this package, it is important that low-level computations go fast, like creating the LU decomposition.

In this paper, a parallel algorithm for the LU decomposition is developed and improved. Later, the algorithm is extended to work efficiently for matrices with a special structure, band matrices. From this, it follows that the algorithms created do show an increase in efficiency and a decrease in computational time. Furthermore, initial testing after integration in the IDR package also shows an improvement in computational time.