The Hessian ∇2A is a block diagonal matrix of block size 3 × 3. For a given sequence, the entries of the 3 × 3 block will be the same if that nucleotide belongs to the consensus pattern (Ck). The gradient system is mainly obtained for enabling us to identify the stability boundaries and stability regions on the likelihood surface. The theoretical details of these concepts are published in [20]. The stability region of each local maximum is an approximate convergence zone of the EM algorithm. If we can identify all the saddle points on the stability boundary of a given local maximum, then we will be able to find all the corresponding Tier-1 local maxima. Tier-1 local maximum is defined as the new local maximum that is connected to the original local maximum through one decomposition point. Similarly, we can define Tier-2 and Tier-k local maxima that will take 2 and k decomposition points respectively. However, finding every saddle point is computationally intractable and hence we have adopted a heuristic by generating the eigenvector directions of the PSSM at the local maximum. Also, for such a complicated likelihood function, it is not efficient to compute all saddle points on the stability boundary. Hence, one can obtain new local maxima by obtaining the exit points instead of the saddle points. The point along a particular direction where the function has the lowest value starting from the given local maximum is called the exit point. The next section details our approach and explains the different phases of our algorithm.