fastest matrix multiplication algorithm

presented that the number of operations required to perform a matrix multiplication could be reduced by consideringIn 1981, Schonhage developed a sophisticated theory involving the bilinear complexity of rectangular matrix multiplication that showed that approximate bilinear algorithms are even more powerful. He wrote down \(C_{ij} \)s in terms of block matrices \(P_{k} \)s. Each \(P_{k} \) could be calculated simply from products and sums of sub-blocks of A and B.Oh, God! This algorithm looks so natural and trivial that it is very hard to imagine that there is a better way to multiply matrices. D’Alberto and Nicolau have written a very efficient adaptive matrix multiplication code, which switches from Strassen’s to the cubic algorithm at the optimal point.Thus, the naive algorithm will likely be your best bet unless your matrices are very large.Will we ever achieve the hypothesized \( O(n^{2}) \)?If you like my post, please share it with your friends, and also read In the base case, when the number of rows is equal to 1, the algorithm performs just one scalar multiplication. The Coppersmith Winograd algorithm had been the world champion for 20 years until finally being improved by Stothers in 2010.

Kak, S. (2014) Efficiency of matrix multiplication on the cross-wired mesh array.
Using just a bit of algebra, he was able to reduce the number of sub-calls to matrix-multiplies to 7. However, the order can have a considerable impact on practical performance due to the The optimal variant of the iterative algorithm for In the idealized cache model, this algorithm incurs only which works for all square matrices whose dimensions are powers of two, i.e., the shapes are which consists of eight multiplications of pairs of submatrices, followed by an addition step. He used the following factoring scheme. The divide and conquer algorithm computes the smaller multiplications accounting for the eight recursive calls on matrices of size A variant of this algorithm that works for matrices of arbitrary shapes and is faster in practiceThe cache miss rate of recursive matrix multiplication is the same as that of a The number of cache misses incurred by this algorithm, on a machine with Algorithms exist that provide better running times than the straightforward ones. It is obvious.In the recursive case, total time is computed as the sum of the partitioning time ( dividing matrices into 4 parts), the time for all the recursive call, and the time to add the matrices resulting from the recursive calls:\begin{equation*}T(n)=\Theta(1)+8T(\frac{n}{2})+\Theta(n^{2}) = 8T(\frac{n}{2})+\Theta(n^{2})\end{equation*}Surprise! The elementary algorithm for matrix multiplication can be implemented as a tight product of three nested loops:By analyzing the time complexity of this algorithm, we get the number of multiplications M(n,n,n) given by the following summation:\begin{equation*}M(n,n,n) = \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}1\end{equation*}Sums get evaluated from t… Where is the improvement? This is not the exact “crossover point” since its value is highly system dependent. Armed with these equations, now we devise a simple, recursive algorithm:Let’s investigate this recursive version of the matrix multiplication algorithm.

Exploiting the full parallelism of the problem, one obtains an algorithm that can be expressed in On modern architectures with hierarchical memory, the cost of loading and storing input matrix elements tends to dominate the cost of arithmetic. For almost a decade after his discovery, no further progress was made.In 1978, Pan found explicit algorithms to further reduce ω with a technique called as In 1979, Bini et al. To see what this means, carve A into four n/2 × n/2 blocks, and also square matrices B and C :Then their product can be expressed in terms of these blocks and is exactly as if the blocks were single elementsFrom this, we have obtained 8 smaller matrix multiplications and 4 additions. The question typically translates to determining the exponent of matrix multiplication: the smallest real number \( \omega \) such that the product of two matrices of size n x n over a field F can be determined using \( n^{\omega} \) operations over F. Many works has been invested in making matrix multiplication algorithms efficient over the years, but the bound is still between  \(2 \leq \omega \leq 3 \).The elementary algorithm for matrix multiplication can be implemented as a tight product of three nested loops:By analyzing the time complexity of this algorithm, we get the number of multiplications M(n,n,n) given by the following summation:\begin{equation*}M(n,n,n) = \sum_{i=1}^{n}\sum_{j=1}^{n}n\end{equation*}The sum of n ns is just as simple, \(n \times n = n^{2} \), so\begin{equation*}M(n,n,n) = \sum_{i=1}^{n}n^{2}\end{equation*}Finally, the sum of \(n^{2} \)s is \(n \times n^{2}= n^{3} \).Thus the running time of this square matrix multiplication algorithm is \(O(n^{3}) \). On a single machine this is the amount of data transferred between RAM and cache, while on a distributed memory multi-node machine it is the amount transferred between nodes; in either case it is called the There are a variety of algorithms for multiplication on The result is even faster on a two-layered cross-wired mesh, where only 2What is the fastest algorithm for matrix multiplication?Henry Cohn, Chris Umans.

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