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Review of matrix multiplication algorithms
Abstract
Matrix multiplication is one of the most fundamental linear algebra operations and serves as the main structural element in various algorithms, including the systems of linear equations, matrices, treatment evaluation matrix determinant, in signal processing, as well as the transitional end of the graph. In some cases, the asymptotic complexity for these algorithms depends on the complexity of the matrix multiplica ¬ tion which motivates the study of possibilities for faster matrix multiplication. In addition, the inclusion of matrix multiplication in many tests indicates its role in deter-mining factor for high-performance computing.
Algorithm of matrix multiplication is usually build as follows. The first algorithm for small a task matrix, then the tensor product construction produces a multiplication algorithm for large matrices. Several times over the years, the basic rules of building a basic algorithm were relaxed, and with more caution in construction of tensor product, it demonstrates how to use the softer core designs to give efficient algorithms for multiplication of large matrix.
Algorithms for matrix multiplication does not only depends on the algorithm, but is also dependent on the architecture machines. the way in which 2 rooms proliferate in the car, affect the algorithms. But everyone will find that adding always faster than multiplication. These algorithms use this fact to make a temporary difficulty, still most of these algorithms are not used, and a constant factor, the complexity is so great, that makes them unusable. Floating point numbers will take longer as the number of cycles is used to multiply a floating is large compared to the integers; the rest of the article is devoted to a different type of matrix multiplication algorithms.
introduction
People have talent, that it is difficult to program. Driving, recognizing the continuous speech, and play chess turned out to be much more difficult for computers than they are for the people. For this class we can probably add a optimization algorithm. How is it that a mathematician might look at a simple algorithm for complex product
U = A * C-B * D

V = A * C + B * D
And find that the number of operations can be reduced to 3? One technique that seems far from clear is
U = C * {A-B) + b * (c-D), V = D * (A + B) + b * (CD), where expression B * (C-D) is computed once and reused.
() Similarly, we have A * A-B * B, which can be computed using the 1 multiplication

* IN * B = (A + B) * (A-B)
(b) the cross product of vectors/I + A2J + ask and b/I + b2j + HK believes that:

A2 * a3 * 62-63
~ I * bi-AI * IA * 62-63 and > 2 * h
in the three target expressions. While working for the program generation symmetric scene, one would ask, there is a way to calculate the cross product in less than six multiplications, at the cost of additional adds. Yes, indeed, one finds the way to the cross product of two vectors using five multiplication.
SI = a/a, 2
■ S-2 = b2-63 "3 =" 1-3 "
S4 = 61-s2

MI = a3 * 61 = t/m3-m2
M2 = and * 63 c * s2 = M3 m4 m5 s3 * = b2 = a2 * s4
CI = m4-TI
C2 = c3 = m2 a t-Ti-M5
© harder but more interesting example is the Strassen algorithm [3], to multiply two 2 * 2 matrix, which includes 7 multiplications and 18 additions, saving one multiplication by 14 additions more ordinary matrix multiplication. This result is more than a math queer thing, because it gives the asymptotically faster algorithm for matrix multiplication. The algorithm has subsequently improved the Winograd algorithm, which reduced the number of additions from 18 to 15, thus reducing the permeability of the matrix multi ¬ plication. Although these results are easily verified, the lack of visible structure and theory behind such methods leaves doubts the optimality of such methods.
2 Background
Most of the effort invested to accelerate the practical realization of matrix multiplication has focused on well known industry standard algorithm with improvement seen when necessary domestic products are calculated in various ways, which are better suited for computer architecture. Much less effort was made to investigate alternative algorithms, asymptotic complexity less than 0 operations (N3), required the usual algo-rithm multiply N * N matrix. One of these algorithms is the Strassen algorithm introduced in 1969, which has complexity 0 (nl9 ^).