There are many ways of numerically treating the modal equation. Generally, core storage is not a problem so that the algorithms can be impartially compared by setting an accuracy threshold and seeing which method requires the least execution time to meet that criterion on a typical set of test problems. The algorithm used in KRAKEN was chosen by just such a fly-off of numerous different algorithms. This included 3- and 5-point difference schemes, Numerov's method, iterated defect corrections and layer methods. Richardson extrapolation was also studied using polynomial and rational (Padé) approximations with the various algorithms. Several root-finders were also compared including newton, secant, and bisection schemes. The algorithm described below was the most rapid across the suite of test problems; however, it should be noted that the optimal algorithm is partly a function of the types of problems forming the test ensemble.
In the following sections we shall describe the algorithm which emerged as the fastest technique. Sections 3.1--3.3 describe the finite-difference discretization which leads to an algebraic eigenvalue problem (EVP). The EVP is solved using what is sometimes called the determinant search method as described in Sect. 3.4. Section 3.5 describes the treatment of elastic media and Sect. 3.6 discusses the use of Richardson extrapolation which provides an adaptive technique for controlling the error as well as an efficient means of obtaining high-accuracy.