Implementing Hager's exchange methods for matrix profile reduction

Hager recently introduced down and up exchange methods for reducing the profile of a sparse matrix with a symmetric sparsity pattern. The methods are particularly useful for refining orderings that have been obtained using a standard profile reduction algorithm, such as the Sloan method. The running times for the exchange algorithms reported by Hager suggested their cost could be prohibitive for practical applications. We examine how to implement the exchange algorithms efficiently. For a range of real test problems, it is shown that the cost of running our new implementation does not add a prohibitive overhead to the cost of the original reordering.