Akra–Bazzi method
In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi. Formulation. The Akra–Bazzi method applies to recurrence formulas of the form The conditions for usage are: The asymptotic behavior of formula_13 is found by  determining the value of formula_14 for which formula_15 and plugging that value into the equation (see Θ). Intuitively, formula_17 represents a small perturbation in the index of formula_18. By noting that formula_19 and that the absolute value of formula_20 is always between 0 and 1, formula_17 can be used to ignore the floor function in the index. Similarly, one can also ignore the ceiling function. For example, formula_22 and formula_23 will, as per the Akra–Bazzi theorem, have the same asymptotic behavior. Example. Suppose formula_24 is defined as 1 for integers formula_25 and formula_26 for integers formula_27. In applying the  Akra–Bazzi method, the first step is to find the value of formula_14 for which formula_29. In this example, formula_30. Then, using the formula, the asymptotic behavior can be determined as follows: Significance. The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort, the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as formula_32  and for integers formula_34, and can thus be computed using the Akra–Bazzi method to be formula_35.