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For this recurrence relation, the master theorem for divide-and-conquer recurrences gives the asymptotic bound () = (). It follows that, for sufficiently large n , Karatsuba's algorithm will perform fewer shifts and single-digit additions than longhand multiplication, even though its basic step uses more additions and shifts than the ...
which consists of eight multiplications of pairs of submatrices, followed by an addition step. The divide-and-conquer algorithm computes the smaller multiplications recursively, using the scalar multiplication c 11 = a 11 b 11 as its base case. The complexity of this algorithm as a function of n is given by the recurrence [6]
Karatsuba multiplication is an O(n log 2 3) ≈ O(n 1.585) divide and conquer algorithm, that uses recursion to merge together sub calculations. By rewriting the formula, one makes it possible to do sub calculations / recursion.
A similar complex multiplication algorithm multiplies two complex numbers using 3 real multiplications instead of 4; Toom-Cook algorithm, a faster generalization of the Karatsuba algorithm that permits recursive divide-and-conquer decomposition into more than 2 blocks at a time
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
Strassen's algorithm improves on naive matrix multiplication through a divide-and-conquer approach. The key observation is that multiplying two 2 × 2 matrices can be done with only 7 multiplications, instead of the usual 8 (at the expense of 11 additional addition and subtraction operations).