Matrix multiplication algorithm

The definition of matrix multiplication is that if for an matrix and an matrix , then is an matrix with entries :c_{ij} = \sum_{k=1}^m a_{ik} b_{kj}. From this, a simple algorithm can be constructed which loops over the indices from 1 through and from 1 through , computing the above using a nested loop: • Input: matrices and • Let be a new matrix of the appropriate size • For from 1 to : • For from 1 to : • Let • For from 1 to : • Set • Set • Return This algorithm takes time (in asymptotic notation). Cache behavior The three loops in iterative matrix multiplication can be arbitrarily swapped with each other without an effect on correctness or asymptotic running time. However, the order can have a considerable impact on practical performance due to the memory access patterns and cache use of the algorithm; which order is best also depends on whether the matrices are stored in row-major order, column-major order, or a mix of both. In particular, in the idealized case of a fully associative cache consisting of bytes and bytes per cache line (i.e. cache lines), the above algorithm is sub-optimal for and stored in row-major order. When , every iteration of the inner loop (a simultaneous sweep through a row of and a column of ) incurs a cache miss when accessing an element of . This means that the algorithm incurs cache misses in the worst case. , the speed of memories compared to that of processors is such that the cache misses, rather than the actual calculations, dominate the running time for sizable matrices. The optimal variant of the iterative algorithm for and in row-major layout is a tiled version, where the matrix is implicitly divided into square tiles of size by : • Input: matrices and • Let be a new matrix of the appropriate size • Pick a tile size • For from 1 to in steps of : • For from 1 to in steps of : • For from 1 to in steps of : • Multiply and into , that is: • For from to : • For from to : • Let • For from to : • Set • Set • Return In the idealized cache model, this algorithm incurs only cache misses; the divisor amounts to several orders of magnitude on modern machines, so that the actual calculations dominate the running time, rather than the cache misses. ==Divide-and-conquer algorithm==

An alternative to the iterative algorithm is the divide-and-conquer algorithm for matrix multiplication. This relies on the block partitioning :C = \begin{pmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \\ \end{pmatrix},\, A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{pmatrix},\, B = \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \\ \end{pmatrix}, which works for all square matrices whose dimensions are powers of two, i.e., the shapes are for some . The matrix product is now :\begin{pmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \\ \end{pmatrix} = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{pmatrix} \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \\ \end{pmatrix} = \begin{pmatrix} A_{11} B_{11} + A_{12} B_{21} & A_{11} B_{12} + A_{12} B_{22}\\ A_{21} B_{11} + A_{22} B_{21} & A_{21} B_{12} + A_{22} B_{22}\\ \end{pmatrix} 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 as its base case. The complexity of this algorithm as a function of is given by the recurrence Splitting a matrix now means dividing it into two parts of equal size, or as close to equal sizes as possible in the case of odd dimensions. • Inputs: matrices of size , of size . • Base case: if is below some threshold, use an unrolled version of the iterative algorithm. • Recursive cases: :* If , split horizontally: ::C = \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} {B} = \begin{pmatrix} A_1 B \\ A_2 B \end{pmatrix} :* Else, if , split vertically: ::C = A \begin{pmatrix} B_1 & B_2 \end{pmatrix} = \begin{pmatrix} A B_1 & A B_2 \end{pmatrix} :* Otherwise, . Split vertically and horizontally: ::C = \begin{pmatrix} A_1 & A_2 \end{pmatrix} \begin{pmatrix} B_1 \\ B_2 \end{pmatrix} = A_1 B_1 + A_2 B_2 Cache behavior The cache miss rate of recursive matrix multiplication is the same as that of a tiled iterative version, but unlike that algorithm, the recursive algorithm is cache-oblivious: there is no tuning parameter required to get optimal cache performance, and it behaves well in a multiprogramming environment where cache sizes are effectively dynamic due to other processes taking up cache space. (The simple iterative algorithm is cache-oblivious as well, but much slower in practice if the matrix layout is not adapted to the algorithm.) The number of cache misses incurred by this algorithm, on a machine with lines of ideal cache, each of size bytes, is bounded by :\Theta \left(m + n + p + \frac{mn + np + mp}{b} + \frac{mnp}{b\sqrt{M}} \right) ==Sub-cubic algorithms==

Algorithms exist that provide better running times than the straightforward ones. The first to be discovered was Strassen's algorithm, devised by Volker Strassen in 1969 and often referred to as "fast matrix multiplication". It is based on a way of multiplying two 2×2 matrices which requires only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations. Applying this recursively gives an algorithm with a multiplicative cost of O( n^{\log_{2}7}) \approx O(n^{2.807}). Strassen's algorithm is more complex, and the numerical stability is reduced compared to the naïve algorithm, but it is faster in cases where or so It is very useful for large matrices over exact domains such as finite fields, where numerical stability is not an issue. Since Strassen's algorithm is actually used in practical numerical software and computer algebra systems, improving on the constants hidden in the big-O notation has its merits. A table that compares key aspects of the improved version based on recursive multiplication of 2×2-block matrices via 7 block matrix multiplications follows. As usual, n gives the dimensions of the matrix and M designates the memory size. It is known that a Strassen-like algorithm with a 2×2-block matrix step requires at least 7 block matrix multiplications. In 1976 Probert showed that such an algorithm requires at least 15 additions (including subtractions); however, a hidden assumption was that the blocks and the 2×2-block matrix are represented in the same basis. Karstadt and Schwartz computed in different bases and traded 3 additions for less expensive basis transformations. They also proved that one cannot go below 12 additions per step using different bases. In subsequent work Beniamini et el. applied this base-change trick to more general decompositions than 2×2-block matrices and improved the leading constant for their run times. It is an open question in theoretical computer science how well Strassen's algorithm can be improved in terms of asymptotic complexity. The matrix multiplication exponent, usually denoted \omega, is the smallest real number for which any n\times n matrix over a field can be multiplied together using n^{\omega + o(1)} field operations. The current best bound on \omega is \omega , by Alman, Duan, Williams, Xu, Xu, and Zhou. The conceptual idea of these algorithms is similar to Strassen's algorithm: a way is devised for multiplying two -matrices with fewer than multiplications, and this technique is applied recursively. However, the constant coefficient hidden by the big-O notation is so large that these algorithms are only worthwhile for matrices that are too large to handle on present-day computers. Victor Pan proposed so-called feasible sub-cubic matrix multiplication algorithms with an exponent slightly above 2.77, but in return with a much smaller hidden constant coefficient. Freivalds' algorithm is a simple Monte Carlo algorithm that, given matrices , and , verifies in time if . AlphaTensor In 2022, DeepMind introduced AlphaTensor, a neural network that used a single-player game analogy to invent thousands of matrix multiplication algorithms, including some previously discovered by humans and some that were not. Operations were restricted to the (normal arithmetic) and finite field \mathbb Z/2\mathbb Z (mod 2 arithmetic). The best "practical" (explicit low-rank decomposition of a matrix multiplication tensor) algorithm found ran in O(n2.778). Finding low-rank decompositions of such tensors (and beyond) is NP-hard; optimal multiplication even for 3×3 matrices remains unknown, even in commutative field. in normal arithmetic. Some algorithms were completely new: for example, (4, 5, 5) was improved to 76 steps from a baseline of 80 in both normal and mod 2 arithmetic. ==Parallel and distributed algorithms==

Shared-memory parallelism The divide-and-conquer algorithm sketched earlier can be parallelized in two ways for shared-memory multiprocessors. These are based on the fact that the eight recursive matrix multiplications in :\begin{pmatrix} A_{11} B_{11} + A_{12} B_{21} & A_{11} B_{12} + A_{12} B_{22}\\ A_{21} B_{11} + A_{22} B_{21} & A_{21} B_{12} + A_{22} B_{22}\\ \end{pmatrix} can be performed independently of each other, as can the four summations (although the algorithm needs to "join" the multiplications before doing the summations). Exploiting the full parallelism of the problem, one obtains an algorithm that can be expressed in fork–join style pseudocode: Communication-avoiding and distributed algorithms On modern architectures with hierarchical memory, the cost of loading and storing input matrix elements tends to dominate the cost of arithmetic. 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 communication bandwidth. The naïve algorithm using three nested loops uses communication bandwidth. Cannon's algorithm, also known as the 2D algorithm, is a communication-avoiding algorithm that partitions each input matrix into a block matrix whose elements are submatrices of size by , where is the size of fast memory. The naïve algorithm is then used over the block matrices, computing products of submatrices entirely in fast memory. This reduces communication bandwidth to , which is asymptotically optimal (for algorithms performing computation). In a distributed setting with processors arranged in a by 2D mesh, one submatrix of the result can be assigned to each processor, and the product can be computed with each processor transmitting words, which is asymptotically optimal assuming that each node stores the minimum elements. This algorithm transmits words per processor, which is asymptotically optimal. On modern distributed computing environments such as MapReduce, specialized multiplication algorithms have been developed. Algorithms for meshes There are a variety of algorithms for multiplication on meshes. For multiplication of two n×n on a standard two-dimensional mesh using the 2D Cannon's algorithm, one can complete the multiplication in 3n-2 steps although this is reduced to half this number for repeated computations. The standard array is inefficient because the data from the two matrices does not arrive simultaneously and it must be padded with zeroes. The result is even faster on a two-layered cross-wired mesh, where only 2n-1 steps are needed. The performance improves further for repeated computations leading to 100% efficiency. The cross-wired mesh array may be seen as a special case of a non-planar (i.e. multilayered) processing structure. In a 3D mesh with n3 processing elements, two matrices can be multiplied in \mathcal{O}(\log n) using the DNS algorithm. ==See also==