A fast cellular method of matrix multiplication

The presence of rounding errors is frequently inevitable when performing arithmetic operations in computers due to the use of floating-point number system whose elements have finite precision. Consequently, error-free transformation algorithms are proposed as solutions, and a particular instance of these is the error-free transformation algorithm for real matrix multiplication. Based on the previously mentioned algorithm, this study proposes two error-error free transformation algorithms for complex matrix multiplication. By conducting the numerical experiment, one finds that the combination of both proposed algorithms with an accurate summation algorithm generates the most accurate results, but it also suffers from a notable increase in the computing time ratio. On the other hand, the second proposed algorithm, which is simply another form of the first one with the change of order in summation, combined with the pure floating-point addition produces results that are nearly as accurate as those generated by the former combination. Since these results are achieved with significantly smaller computing time ratio, then the study concludes that the last combination is the best method for complex matrix multiplication in terms of accuracy and computing time efficiency.

We present a novel heterogeneous parallel matrix multiplication algorithm that utilizes both central processing units (CPUs) and graphics processing units (GPUs) for large-scale matrices. Based on Strassen’s method, we represent matrix multiplication work as a set of matrix addition and multiplication tasks among their sub-matrices. Then, we distribute the tasks to CPUs and GPUs while considering the characteristics of the tasks and computing resources to minimize the data communication overhead and fully utilize the available computing power. To handle a large matrix efficiently with limited GPU memory, we also propose a block-based work decomposition method. We then further improve the performance of our method by exploiting the concurrent execution abilities of a heterogeneous parallel computing system. We implemented our method on five different heterogeneous systems and applied it to matrices of various sizes. Our method generally shows higher performance than the prior GPU-based matrix multiplication methods. Moreover, compared with the state-of-the-art GPU matrix multiplication library (i.e., CUBLAS), our method achieved up to 1.97 times higher performance using the same GPUs and CPU cores. In some cases, our method using a low-performance GPU (e.g., GTX 1060, 3 GB) achieved performance comparable to that of CUBLAS using a high-performance GPU (e.g., RTX 2080, 8 GB). Also, our method continually improves performance as we use more computing resources like additional CPU cores and GPUs. We could achieve such high performance because our approach fully utilized the capacities of the given heterogeneous parallel computing systems while employing the Strassen’s method, which has a lower asymptotic complexity. These results demonstrate the efficiency and robustness of our algorithm.

Parallelizing Strassen's method for matrix multiplication on distributed-memory MIMD architectures

In the past few years, successive improvements of the asymptotic complexity of square matrix multiplication have been obtained by developing novel methods to analyze the powers of the Coppersmith-Winograd tensor, a basic construction introduced thirty years ago. In this paper we show how to generalize this approach to make progress on the complexity of rectangular matrix multiplication as well, by developing a framework to analyze powers of tensors in an asymmetric way. By applying this methodology to the fourth power of the Coppersmith-Winograd tensor, we succeed in improving the complexity of rectangular matrix multiplication. Let $\alpha$ denote the maximum value such that the product of an $n\times n^\alpha$ matrix by an $n^\alpha\times n$ matrix can be computed with $O(n^{2+\epsilon})$ arithmetic operations for any $\epsilon>0$. By analyzing the fourth power of the Coppersmith-Winograd tensor using our methods, we obtain the new lower bound $\alpha>0.31389$, which improves the previous lower bound $\alpha>0.30298$ obtained five years ago by Le Gall (FOCS'12) from the analysis of the second power of the Coppersmith-Winograd tensor. More generally, we give faster algorithms computing the product of an $n\times n^k$ matrix by an $n^k\times n$ matrix for any value $k\neq 1$. (In the case $k=1$, we recover the bounds recently obtained for square matrix multiplication). These improvements immediately lead to improvements in the complexity of a multitude of fundamental problems for which the bottleneck is rectangular matrix multiplication, such as computing the all-pair shortest paths in directed graphs with bounded weights.

This paper firstly gives a description of the traditional distribution schemes of parallel matrix multiplication and an analysis of their efficiencies. In order to reduce communication expenses between multiple processors, two solutions are offered in this paper, and the corresponding specific implementation models are also listed. Then a method of how to calculate the n-th power of massive matrix is proposed. Finally, the experimental results show that this method of calculating the n-th power of massive matrix can reduce the execution time and improve the efficiency.

The intersection of climate change and human activities exacerbates the prominent contradiction between environmental resources and economic development. The synergistic development of water resources, economy and society, and ecological environment is particularly urgent. Therefore, accurately identifying its influencing factors and related pathways holds significant practical importance in resolving the contradiction between resources and economic development. This study constructs a system of influencing factors for the synergistic development of “water resources, economic society and ecological environment” from three dimensions, reveals the influencing relationship between indicators through the Decision Making Trial and Evaluation Laboratory (DEMATEL), determines the hierarchical structure of the influencing factors and the related paths through the Interpretive Structural Model (ISM), and classifies the influencing factors through the Matrix Multiplication Method of Cross Influence Matrix Multiplication (MICMAC). The results show that water resource factors are important cause factors and economic, social and ecological factors are essential effect factors. The factors influencing the synergistic development of “water resources, economic society and ecological environment” can be divided into four levels, of which the total water resources, precipitation and NDVI are at the bottom and are the most fundamental factors. The percentage of groundwater supply, total water supply, water resource utilization rate and energy consumption of 10,000 yuan of GDP are the deep core elements. Total grain output, water resources per capita, and water use per capita are the direct influencing factors. Finally, taking the Central Plains Urban Agglomeration as an example, selecting the identified important influencing factors and analyzing the level of synergistic development of its coupled system found that the degree of coupling coordination increased from 0.324 in 2011 to 0.978 in 2020. The water resources system showed a fluctuating change, and it is still the key to synergistic development. Analyzing the relationship between subsystems and clarifying the key factors for synergistic development of coupled systems are important for policy formulation, improving environmental governance and promoting high-quality economic development.

In this paper, an easy method is proposed to reduce the computational complexity of matrix-vector multiplication that is required in solving integral equations using method of moments. This method approximates the Green’s function using a Fourier series. It has the advantage that its formulation separates the observation point from the source point effortlessly and regardless of the Green’s function. Since the fast multipole method uses this idea, the two methods are discussed and compared theoretically and numerically by solving the electromagnetic wave scattering from perfect conductor of three-dimensional basic canonical shape (sphere). The results showed that the proposed method is accurate, and for large number of unknowns in the problem, it has a computational complexity less than the method of moments and over the conventional fast multipole method. Hence, the generality and simplicity of this method compare with the conventional fast multipole method and the efficiency compare with the method of moments, which will be achieved.

For many scientific applications, dense matrix multiplication is one of the most important and computation intensive linear algebra operations. An efficient matrix multiplication on high performance and parallel computers requires optimizations on how matrices are decomposed and exchanged between computational nodes to reduce communication and synchronization overhead, as well as to efficiently exploit the memory hierarchy within a node to improve both spatial and temporal data locality. In this paper, we presented our studies of performance, cache behavior, and energy efficiency of multiple parallel matrix multiplication algorithms on a multicore desktop computer and a medium-size shared memory machine, both being considered as referenced sizes of nodes to create a medium- and largescale computational clusters for high performance computing used in industry and national laboratories. Our results highlight both the performance and energy efficiencies, and also provide implications on the memory and resources pressures of those algorithms. We hope this could help users choose the appropriate implementations according to their specific data sets when composing larger-scale scientific applications that use parallel matrix multiplication kernels on a node.

We consider the techniques behind the current best algorithms for matrix multiplication. Our results are threefold. (1) We provide a unifying framework, showing that all known matrix multiplication running times since 1986 can be achieved from a single very natural tensor - the structural tensor T_q of addition modulo an integer q. (2) We show that if one applies a generalization of the known techniques (arbitrary zeroing out of tensor powers to obtain independent matrix products in order to use the asymptotic sum inequality of Schonhage) to an arbitrary monomial degeneration of T_q, then there is an explicit lower bound, depending on q, on the bound on the matrix multiplication exponent omega that one can achieve. We also show upper bounds on the value alpha that one can achieve, where alpha is such that n * n^alpha * n matrix multiplication can be computed in n^{2+o(1)} time. (3) We show that our lower bound on omega approaches 2 as q goes to infinity. This suggests a promising approach to improving the bound on omega: for variable q, find a monomial degeneration of T_q which, using the known techniques, produces an upper bound on omega as a function of q. Then, take q to infinity. It is not ruled out, and hence possible, that one can obtain omega=2 in this way.

This paper considers the coherent integration problem for maneuvering target detection and motion parameters estimation, involving range migration (RM) and Doppler frequency migration (DFM) within the coherent pulse interval. A fast non-searching method based on adjacent cross correlation function (ACCF) and Lv's distribution (LVD) is proposed, where the adjacent correlation operation is first employed to remove the RM and reduce the order of DFM. After that, LVD is applied to realize the coherent integration, target detection and parameters estimation. In addition, at the cost of some performance loss, another fast method via ACCF iteratively is also introduced to further reduce the computational complexity and obtain the motion parameters estimation. The proposed two methods are fast in that they can be easily implemented by using complex multiplications, the fast Fourier transform (FFT) and inverse FFT (IFFT). Compared with the existing methods, the presented algorithms can obtain the motion parameters estimation without any searching procedure and can achieve a good balance between the computational cost and the detection ability as well as parameters estimation performance. Finally, several simulation experiments are provided to demonstrate the effectiveness.

The paper presents exact analytical solutions of one-dimensional periodic heat conduction through an inhomogeneous slab for a certain class of thermal conductivity profiles (including linear and exponential). The exact analytical solutions for some of these profiles have been compared with those obtained by considering the slab to be made up of a number of homogeneous layers with different thermal conductivities varying from layer to layer and using the layered structure (or matrix multiplication) method. The numerical results arrived at by the layered-structure method converge rapidly (with increasing number of layers considered) to the values obtained from the exact analytical solutions. This gives confidence in the application of the layered-structure method to periodic heat conduction through inhomogeneous slabs. The numerical results have been presented in the form of elements of a 2 × 2 matrix, relating the sinusoidal steady-state temperature and heat flux on the two sides of the slab.

On fast matrix–vector multiplication in wavelet Galerkin BEM

This work presents an algorithm to reduce the multiplicative computational complexity in the creation of digital holograms, where an object is considered as a set of point sources using mathematical symmetry properties of both the core in the Fresnel integral and the image. The image is modeled using group theory. This algorithm has multiplicative complexity equal to zero and an additive complexity (k-1)N2 for the case of sparse matrices or binary images, where k is the number of pixels other than zero and N2 is the total of points in the image.

As feature sizes and pitches continue to decrease, more complex correction algorithms are needed to solve increasingly difficult geometric configurations. Usage of these more complex algorithms results in unacceptably long time-to-mask when applied to an entire design. In many cases, the more complex algorithms are only required in a small percentage of areas of the entire design, and these areas are not always known prior to tapeout. Hotspot fixing (HSF) flows are increasingly used to fix these hotspot areas to minimize errors and decrease time-to-mask. These flows involve “recorrecting” a design, using the previous correction output as the input to the HSF flow. This input file contains a hierarchy that was optimized for the original correction. Hotspot areas are frequently smaller than the original correction areas and frequently repeat in unique cell outputs of the original correction, so the optimal hierarchy for a HSF fix flow may be very different from the original correction. A new hierarchy, optimized for HSF, is difficult to form from the corrected output. This paper describes the usage of pattern-matching to regain hierarchical compression for identical hotspot areas that are not repeating cells in the original correction. Using this pattern-matching HSF flow, turnaround time for the hotspot fixing can be more than 50X faster than re-using the original correction’s hierarchy for complex HSF methods. These significant gains can be achieved in spite of the additional complexity it can add to the flow. In the case where simpler/faster HSF correction methods are used, significant turnaround time gains can still be made by using this pattern matching technique. Hotspot fixing methodologies are increasingly deployed during tapeouts as a means to optimize the tradeoffs between complex, highly accurate correction methods and faster methods that are sufficient for most pattern areas.1 However, pattern database hierarchies may not be optimum for these hotspot fixing flows, as they are optimized for the initial correction run or method. This paper examines the usage of pattern matching to regain hierarchy and significantly reduce turn-around-time for complex hotspot fixing methods. Gains in turn-around-time can be well over 50 times faster than reusing the original correction’s hierarchy.

The usual solution methods of discretised partial differential equations are based exclusively on matrix-vector multiplications as basis operation. On the one hand, this is supported by the use of sparse matrices (cf. §1.3.2.5); on the other hand, one tries to apply fast iterative methods (e.g., the multigrid method [124], [119, §12]) whose basic steps are matrix-vector multiplications. Krylov methods are based on the same concept. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation1 and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. However, there are interesting problems which require the solution of a linear or nonlinear matrix equation and cannot be solved via the matrix-vector multiplication. Examples are the linear Lyapunov and Sylvester equations as well as the quadratic Riccati equation, which arise, e.g., in optimal control problems for partial differential equations and in model reduction methods. The $$\mathcal{H}$$ -matrix arithmetic allows the solution of these matrix equations efficiently. Here, the use of hierarchical matrix operations and matrix-valued functions is only one part of the solution method. Another essential fact is that the solution $$ X \in \mathbb{R}^{I \times I}(n==\!\!\!\!\!/ \!\!\!\!/I)$$ can be replaced by an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ . If one considers the equation $$f(X)\;=\;0$$ as a system of n 2 equation for the n 2 components of X, even an optimal solution method has complexity $$\mathcal{O}(n^2)$$ , since this is linear complexity in a number of unknowns (cf. Remark 1.1). Using traditional techniques, the solution of large-scale matrix equations is not feasible. Only an $$\mathcal{H}$$ -matrix $$X_{\mathcal{H}}$$ with $$\mathcal{O}(n\; \mathrm{log}^*n)$$ data instead of n 2 admits a solution with a cost almost linear with respect to n. Section 15.1 introduces Lyapunov and Sylvester equations and discusses their solution. In Section 15.2 we consider quadratic Riccati equation. An interesting approach uses the matrix version of the sign function from §14.1.1. General nonlinear matrix equations may be solved iteratively by Newton’s method or related methods (cf. Section 15.3). As an example, computing the square root of a positive definite matrix is described in §15.3.1. The influence of the truncation error introduced by H-matrix arithmetic is investigated in Section 15.3.