really wanted. Practical implementations of Strassen's algorithm switch to standard methods of matrix multiplication for small enough submatrices, for which those algorithms are more efficient. The particular crossover point for which Strassen's algorithm is more efficient depends on the specific implementation and hardware. Earlier authors had estimated that Strassen's algorithm is faster for matrices with widths from 32 to 128 for optimized implementations. However, it has been observed that this crossover point has been increasing in recent years, and a 2010 study found that even a single step of Strassen's algorithm is often not beneficial on current architectures, compared to a highly optimized traditional multiplication, until matrix sizes exceed 1000 or more, and even for matrix sizes of several thousand the benefit is typically marginal at best (around 10% or less). A more recent study (2016) observed benefits for matrices as small as 512 and a benefit around 20%. Winograd form It is possible to reduce the number of matrix additions by instead using the following form discovered by Winograd: where u = (c - a)(C - D), v = (c + d)(C - A), w = aA + (c + d - a)(A + D - C). This reduces the number of matrix additions and subtractions from 18 to 15. The number of matrix multiplications is still 7, and the asymptotic complexity is the same. Asymptotic complexity The outline of the algorithm above showed that one can get away with just 7, instead of the traditional 8, matrix-matrix multiplications for the sub-blocks of the matrix. On the other hand, one has to do additions and subtractions of blocks, though this is of no concern for the overall complexity: Adding matrices of size requires only operations whereas multiplication is substantially more expensive (traditionally addition or multiplication operations). The question then is how many operations exactly one needs for Strassen's algorithms, and how this compares with the standard matrix multiplication that takes approximately (where ) arithmetic operations, i.e. an asymptotic complexity . The number of additions and multiplications required in the Strassen algorithm can be calculated as follows: let be the number of operations for a matrix. Then by recursive application of the Strassen algorithm, we see that , for some constant that depends on the number of additions performed at each application of the algorithm. Hence , i.e., the asymptotic complexity for multiplying matrices of size using the Strassen algorithm