Abstract
In today’s era of rapid development in science and technology, the development of digital technology has increasingly higher requirements for data processing functions. The matrix signal commonly used in engineering applications also puts forward higher requirements for processing speed. The eigenvalues of the matrix represent many characteristics of the matrix. Its mathematical meaning represents the expansion of the inherent vector, and its physical meaning represents the spectrum of vibration. The eigenvalue of a matrix is the focus of matrix theory. The problem of matrix eigenvalues is widely used in many research fields such as physics, chemistry, and biology. A neural network is a neuron model constructed by imitating biological neural networks. Since it was proposed, the application research of its typical models, such as recurrent neural networks and cellular neural networks, has become a new hot spot. With the emergence of deep neural network theory, scholars continue to combine deep neural networks to calculate matrix eigenvalues. This article aims to study the estimation and application of matrix eigenvalues based on deep neural networks. This article introduces the related methods of matrix eigenvalue estimation based on deep neural networks, and also designs experiments to compare the time of matrix eigenvalue estimation methods based on deep neural networks and traditional algorithms. It was found that under the serial algorithm, the algorithm based on the deep neural network reduced the calculation time by about 7% compared with the traditional algorithm, and under the parallel algorithm, the calculation time was reduced by about 17%. Experiments are also designed to calculate matrix eigenvalues with Obj and recurrent neural networks (RNNS) models, which proves that the Oja algorithm is only suitable for calculating the maximum eigenvalues of non-negative matrices, while RNNS is commonly used in general models.
1 Introduction
1.1 Background
In the information society, due to the explosive increase of data, people have higher and higher requirements for data accuracy, and the workload of data processing is also increasing. People’s lives are getting faster and faster, and the real-time requirements of data processing systems are constantly increasing. In the field of numerical calculation, the calculation of array signals is basic. Matrix eigenvalues occupy an important position in matrix analysis. The calculation of matrix eigenvalues is the basis of matrix calculations. Classical algorithms include the Jacobi and quick response methods. The relative independence of each element in the matrix calculation and the similarity of the operation process make the parallelization of matrix operations easy. The operating principle of deep neural networks is to imitate the way of thinking of the human brain. Deep neural networks have diverse applications, including computer vision, speech recognition, and other fields, and their applications in matrix theory have also been studied by more and more scholars. For example, some scholars have proposed an Ojb model based on neural networks to solve matrix eigenvalues.
1.2 Meaning
As a field of mathematics, a matrix contains rich content, including matrix eigenvalue problems, determinant problems, trace problems, and decomposition problems, and is the most practical mathematical theory. The eigenvalue problem is the main research field of matrix theory, and its research has important theoretical significance and practical value. The problem of matrix eigenvalues is an important subject of matrix theory research and has important uses in many fields. In the computational engineering of scientific research and engineering design, many complex problems can be simplified through the flexible use of matrix eigenvalues. Solving the matrix eigenvalues is relatively simple, but applying eigenvalues to other fields is not so simple. The eigenvalues of the matrix have various meanings in factor analysis, image processing, mechanics, quantum mechanics, and so on. Many problems can be solved by solving matrix eigenvalue problems, and their applications are diverse. The neural network is an engineering system that simulates the structure and intelligent action of the human brain based on our understanding of the human brain’s structure and action mechanism. It is used to solve the matrix eigenvalues to improve the calculation efficiency, so it is of great significance to study the matrix eigenvalues based on deep neural networks.
1.3 Related work
Matrix theory is the main content of linear algebra, and matrix eigenvalues are the main research points of matrix theory. Therefore, many scholars have conducted related research on matrix eigenvalues. Weiss et al. discussed the extension of eigenvalue decomposition to Hermitian matrix factorization. Through the analysis of the unit circle, Weiss et al. proved that the eigenvalue is the only convergent but possibly infinitely long Laurent series. The eigenvector can have any phase response, and it is proved to exist. The shortcoming of this research is the lack of data support [1]. Xu and Chen designed a new vibration structure mode. This model can be discretized as the inverse problem of the left and right eigenvalues of a certain structural matrix. When the structure matrix is a generalized central termination matrix, the left and right eigenvalue inverse problems with sub-matrix constraints are discussed, and the necessary and sufficient conditions for the problem to be solvable are obtained [2]. Gernandt and Trunk proved that the largest Jordan chain at each eigenvalue of sE-A may disappear, and the sum of the lengths of all broken Jordan chains is the number of eigenvalues that can be arbitrarily placed in the extended complex plane. Gernandt and Trunk proved the strict upper and lower bounds of the algebraic and geometric multiplicity of the eigenvalues under the rank one disturbance, but the research method of this experiment is more complicated [3]. Ding et al. believed that there were multiple solutions to the power flow equation in the power system, and all eigenvalues of the Jacobian matrix at the high-voltage operable solution should have negative real parts. Therefore, they re-examined the identification of Type 1 LV load flow solutions in the presence of negative reactance, and then tested the selected institute of electrical and electronics engineers standard power system model and the real-world Polish power system to validate the analysis [4]. Janssen et al. considered a multivariate heavy-tailed random volatility model and analyzed the large sample behavior of its sample covariance matrix. Janssen et al. studied its limit behavior in the case of infinite variance and obtained the result of ordered eigenvalues and corresponding eigenvectors. It is also concluded that in the case of heavy-tailed fluctuation series, the possible restriction behaviors are more diverse [5]. Mou and Anderson established a general theorem about the eigenvalue invariance of some non-homogeneous matrix products. When determining the invariance of this characteristic value, what is important is not the detailed entry but the zero–non-zero structure. Then, the theorem is applied to analyze the convergence speed of the distributed algorithm for solving linear equations on the undirected graph network. This article lacks a detailed introduction [6]. Hu et al. showed that it is possible to use the relative phase of the eigenvalues of the scattering matrix for identification. They performed multi-frequency, fully polarized abdominal SM measurements on 80 specimens of 12 insects in a microwave anechoic chamber. The relationship between the polarization direction corresponding to the maximum radar cross section and the radar frequency is analyzed, and a parallel and vertical insect identification method based on the relative phase of the SM feature value is proposed. However, the study of method proof in this article is somewhat simple [7].
1.4 Innovation
The innovation of this article is that: (1) the deep neural network theory is introduced into the solution of matrix eigenvalues, and some solution models are proposed, which is an innovation in the method. (2) Comparing the time of the traditional algorithm and the matrix eigenvalue algorithm based on a deep neural network to prove the superiority of the neural network method. Two matrix eigenvalue calculation models based on deep neural networks are used to calculate matrix values to test the feasibility of the model.
2 Matrix eigenvalue estimation and application method based on neural networks
2.1 Deep neural network
2.1.1 Neuron mathematical model
According to the analysis of human brain neuron structure, a simple mathematical model of the neuron can be constructed [8]. Although neurons have various shapes and functions, they can be roughly divided into cell bodies and protrusions in structure. The processes are divided into dendrites and axons. The basic function of neurons is to realize information exchange by receiving, integrating, conducting, and outputting information. The number of neurons in the brain is very large, and neurons can perform nonlinear processing on input signals, so it can handle very complex analysis and inference work. Figure 1 is a mathematical model of artificial neurons based on neurons of the human brain [9], where x is the input of the neuron, y is the output of the neuron, and F(x) represents the activation function.
Mathematical model of a neuron.
The activation function of the earliest proposed perceptron neuron usually takes the sign function:
In a computing network, the activation function of a node defines the output of the node under a given input or set of inputs. In neural networks, this function is also called the transfer function. There are many types of activation functions; the common ones are Sigmoid, tanh, and Relu; most of the time, the activation function will use the sigmoid function, for example:
Or
where c is a constant, and c > 0, its f(x) is usually taken as a linear function [10]:
Recurrent neural network model.
2.1.2 Neural network structure
Many neural network models have been developed in various application fields. Although these network models have different structures, operating modes, and operations, they have many common features. Specifically, the artificial neural network is composed of a very complex dynamic system, with large-scale, highly interconnected information processing units [11].
Recurrent neural network
The output of all units of the recurrent neural network will be fed back to all other units. This is a powerful learning system with a simple structure and simple programming. Figure 2 is a model of a recurrent neural network.
Feedforward neural network
The feedforward neural network is the first and simplest type of artificial neural network. In this network, information only moves forward in one direction, from the input node to the output node through the hidden node. The schematic diagram of the multilayer feedforward neural network is shown in Figure 3.
Multilayer feedforward neural network model.
In this structure, information is transmitted from the back to the front layers [12]. This algorithm is widely used in pattern recognition, signal processing, symbol theory, intelligent control, function fitting, and many other fields.
Convolutional neural network
A convolutional neural network is the first deep learning algorithm that appeared and is widely used. A convolutional neural network mainly contains three layers of structure: first, the image is input into the network through the input layer, then the convolutional layer and the pooling layer together form the second layer to process the image, and finally, the classification result is output through the fully connected layer [13]. The structure of the convolutional neural network is shown in Figure 4.
Schematic diagram of convolutional neural network structure.
2.1.3 Neural network for eigenvalue calculation
The matrix
It can be concluded that A can be diagonalized if and only if A has n linearly independent feature vectors
Under certain conditions, only the largest or smallest characteristic value needs to be found. In addition, in some application problems, the matrix A gradually changes over time [14], and it is particularly important to find the largest and smallest eigenvalues. Using the next neural network model to find the largest and smallest eigenvalues, we obtain the following equations:
where i, j=1, 2, …, n.
If the front is +, the minimum eigenvalue is obtained; and if it is −, the maximum eigenvalue is obtained. But the disadvantage of this model is that it is too complicated and will increase installation costs. It can only solve a maximum or minimum eigenvalue, and cannot solve another eigenvalue. So some scholars proposed the following neural network model [15].
2.2 Estimation and application of matrix eigenvalues
2.2.1 Matrix eigenvalues
The matrix only scales and transforms specific vectors or certain vectors and does not produce a rotation effect on these vectors. These vectors are called the eigenvectors of the matrix, and the ratio is the eigenvalue. The mathematical definition of the eigenvalues of a matrix is: for an n-th order square matrix A and an n-dimensional non-zero vector x, there is a value that makes the system of equations satisfy:
where λ is the eigenvalue of A, and the eigenvector of λ is x. Solving the characteristic equation of a matrix is a general method of solving the eigenvalue of a matrix [16], namely:
In the formula,
where
2.2.2 The basic theorem of matrix eigenvalue estimation
Here is a brief introduction to some basic theorems of matrix eigenvalue estimation.
Suppose the characteristic value of
(13)Let
(14)Let Z be a given number X, then all the eigenvalues of A are located
in the union [18].
These theorems are some of the most basic matrix eigenvalue theorems, and they have laid a good foundation for the solution of eigenvalue problems.
2.2.3 Estimation of eigenvalues of an interval matrix
If 2 ≤ j ≤ n−1, discuss the following two problems about the maximum and minimum eigenvalues of real symmetric interval matrices and find the matrix that reaches the maximum and minimum:
j = 1, n.
Let
If
2.2.4 Application of matrix eigenvalues
Linear algebra is a branch of algebra. It mainly deals with linear relationship problems, including the study of lines, surfaces, and subspaces, as well as the general properties of all vector spaces. The eigenvalue problem of a matrix is an important content of linear algebra theory and is widely used in scientific research and engineering practice. It also solves many practical problems, for example, finding the principal stress and vibration frequency and finding the principal axis of inertia. In addition, there are solutions to partial differential equations, stable distribution and convergence of Markov chains, and estimation of matrix operations errors. In addition, the Fourier transform essentially expresses the function as a set of orthogonal bases in a specific function space. These bases happen to act as eigenvectors, which can solve many related problems [19].
The most typical applications of matrix eigenvalues are to solve the general term of the Fibonacci sequence, autosomal genetic problems, and application in equations. The general term of the Fibonacci sequence is also called the golden section sequence. This sequence was introduced by the mathematician Leonardo Fibonacci using rabbit reproduction as an example, so it is also called the “rabbit sequence.” The autosomal problem refers to the problem of the probability of collection and transmission of chromosome dominance and recession. These two application losses use matrix theory to model, transform complex problems into matrix solving problems, and use matrix eigenvalues and eigenvectors to transform the matrix into a diagonal matrix and then solve it.
There are many applications of matrix eigenvalues in differential equations, especially for solving differential equations [20]. In solving a system of first-order constant coefficient equations, the focus is to find the basic solution group of the equation. When the characteristic roots of the coefficient matrix A of the equation group are all single roots, it can be transformed into finding the characteristic vector corresponding to the characteristic root. Regardless of whether the eigenvalues of the matrix are used in modeling or equations, the matrix is diagonalized, making the calculation easier. The eigenvalues and eigenvectors of the diagonalized matrix are known, and the power can be obtained by calculating the same power of the diagonal elements.
2.3 Matrix eigenvalues based on neural networks
The problem of matrix eigenvalue calculation is always an eye-catching research field, especially the construction of neural networks to solve the problem of maximum and minimum eigenvalues, which has attracted the attention of more scholars. Matrix eigenvalue calculation is a common concern of theoretical solutions and engineering applications.
2.3.1 Oja neural network model
Signal processing most of the time deals with signal sequences of infinite length, so if you want to get it, you have to collect all the data series before starting the calculation. But this is obviously impossible, so it is impossible to get it directly. In this case, people thought of a way to collect data to estimate the size of the calculation. After that, the eigenvalue decomposition of this estimator and the projection of the signal onto the principal component subspace with the required dimension are calculated [21]. Nevertheless, the amount of calculation is still very large, so in fact, the random approximation method is used. The random approximation is a parameter estimation method. It is a method of estimating parameters in a stepwise approach when there is random error interference. Every time a new input is made, the value is estimated, and the more you move forward, the estimated value will become more and more accurate. This method is also called an adaptive method. Most neural network algorithms belong to this type of technology. Neural networks work on infinitely expanding data and require less storage than the above steps. Each data is only used when it arrives, because it does not need to be stored for the future, so the value of the application will become higher.
Given an n-dimensional stationary random pass, set its expectation E(x) = 0, then its covariance matrix is equal to its autocorrelation matrix:
In Figure 5, the input vector X, the output layer has only one neuron, and its output is y(t)[22].
Oja neural network model.
Obviously,
2.3.2 Neural network model for calculating the maximum eigenvalue of the matrix
In order to reduce hardware costs, some experts proposed the following neural network model to calculate the maximum and minimum eigenvalues of
Similar to the Oja model, this model is only suitable for solving the largest eigenvalue of a symmetric, positive, definite matrix
where B is any positive definite matrix, and the model is globally asymptotically stable at the maximum eigenvalue. Compared with the general model, this model does not force w(t) to remain on a certain unit circle, making it more widely used [23].
2.3.3 Neural network method for finding eigenvalues of a symmetric matrix
As shown in Figure 6, the process of neuron processing information is divided into three levels: input, output, and processing. The input layer is used to input information into the system, the processing layer is to analyze the data, and the output layer is to transmit the processed information. The connection strength between different neurons can be expressed by “weight,” denoted as
Neuron model.
The basic neuron model is expressed as follows:
3 Estimation experiment of matrix eigenvalues
3.1 Matrix eigenvalue calculation time
3.1.1 Hardware and software design
This text adopts the Spartan-3E XC3S500E device to carry on the system design; its general parameters are shown in Table 1. Since Spartan-3E only provides Microblaze soft core, the system’s processors are all based on Microblaze soft core. The algorithm designed in this article is a parallel algorithm. The matrix to be calculated is input by the user, and the program first determines the symmetry of the input matrix. If the input is a symmetric matrix, select the symmetric matrix eigenvalue calculation algorithm; otherwise, select the general matrix. This article uses the ISE10.1 suite for design. ISE10.1 is installed in the default mode, and the installation directory cannot appear in unrecognizable ways such as Chinese or spaces. Taking into account the compatibility of the system, after the installation is complete, you need to download the Windows XP SP3 patch package from the Xilinx official website. In order to enhance the robustness of the program, the program needs to perform preliminary processing on the matrix input by the user. When the element is a complex number and the matrix is not a square matrix, the user is prompted to input an error, and the reason for the error is provided, and the user is required to re-enter the matrix to be calculated.
Spartan-3E XC3S500E basic parameters
| Type | Parameter | Numerical value |
|---|---|---|
| Technical parameter | Operating temperature | −40 to 100℃ |
| Supply voltage | 1.14–1.26 V | |
| Appearance | Length | 17 mm |
| Width | 17 mm | |
| Height | 1 mm | |
| Installation parameters | Installation mode | Surface Mount |
3.1.2 Calculation time under serial algorithm
Serial algorithm refers to work on a computer, the tasks in the computer are executed one by one.
3.1.2.1 Serial algorithm design
The design of the algorithm needs to be based on a single-core system. This article uses the Xilinx Platform Studio software under the ISE10.1 suite to build a single-core system based on the Microblaze soft core, and then develops software on the basis of the single-core system.
The creation of a single-core system requires three steps: new projects, selection of necessary peripherals and configuration parameters, and configuration of startup memory. First, create a new project. In this step, open the Xilinx Platform Studio software; next, enter the file storage path; then select the option to create a new project, and select the appropriate layout in the next window; finally, select the relevant system resources, such as the processor type and the size of the local memory. The second step is to select the necessary peripherals and configure the parameters. Spartan-3E provides three I/O interfaces: RS232_DTE, RS232_DCE, and LEDs_8Bit. This article uses the RS232_DCE serial port to connect to the computer, so you only need to select the RS232_DCE interface, configure the serial port bit rate to be 9,600, data bits to 8, and no parity. Then, select the necessary external equipment according to the needs in order to achieve the needs of subsequent program design. Finally, configure the startup memory, select the test sample, follow the default memory test configuration, and finally generate the project. The memory address allocation is shown in Table 2.
Memory allocation address
| Example | Name | Base address | Size (kb) |
|---|---|---|---|
| Dlmb_cntlr | C_BASEADDR | 0 × 00000000 | 8 |
| ilmb_cntlr | C_BASEADDR | 0 × 00000000 | 8 |
| Debug_module | C_BASEADDR | 0 × 84400000 | 64 |
| Rs232_DCE | C_BASEADDR | 0 × 84400000 | 64 |
The software development based on Xilinx FPGA embedded system is also realized by Xilinx Platform Studio. Software development is mainly divided into four steps: adding applications, writing programs, writing user constraint files, and modifying maximum segment size (MSS) driver files. To add an application project is to select “Add Software Application Project” to select a processor for the application. To edit a source file is to open the source file that needs to be edited and edit the source code; to write a user constraint file is to edit to complete pin constraints, timing constraints, area constraints, etc.; modifying the MSS driver file refers to modifying part of the MSS information to enable the functions in the application program to have the authority to call system files.
3.1.2.2 Experimental results and analysis
In this experiment, we perform the calculation time statistics of the matrix eigenvalue algorithm based on the deep neural network and the traditional matrix eigenvalue algorithm and compare the calculation time of these two algorithms to compare their calculation speed, and the result is shown in Figure 7.
The calculation time of the two algorithms under the serial algorithm.
It can be seen from earlier that under the serial algorithm, the calculation time of matrix eigenvalues increases with the increase of the matrix, and the increase is larger. Algorithms based on deep neural networks always take less time than traditional algorithms, which are faster. After calculation, the calculation time of the algorithm based on a deep neural network is reduced by about 7% compared with the traditional algorithm when the matrix size is between 4 × 4 and 20 × 20.
3.1.3 Computation time under parallel algorithm
Parallel calculations are calculations performed by parallel computers. This is the same as the state of nature in which there are multiple simultaneous and complex related events. In a parallel system, a large task is broken down into multiple small subtasks, and these subtasks are assigned to different processors. Due to the synergy between the processors, these subtasks are executed at the same time to increase the speed of problem solving, or to expand the scale of problem solving at the same time.
3.1.3.1 Parallel algorithm design
The design of parallel algorithms needs to be based on a dual-core system, and a dual-core system is a dual-core processor, which refers to the addition of two independent physical central processing units to a single computing component, and the dual-core system is realized by adding an internet protocol (IP) core to the single-core system. These two cores can run program instructions independently, use parallel computing capabilities to speed up the program’s running speed, and provide multitasking capabilities. There are dual-core systems with different systems based on the difference in communication mechanisms. This experiment is mainly based on the Mailbox communication mechanism to design a dual-core system. The steps of dual-core system design based on the Mailbox system are as follows: creating a single-core system; configuring the connection of each IP core; and making the necessary memory address allocation. After completing the above steps, the construction of a dual-core system based on the Mailbox system is basically realized. The memory address allocation is shown in Table 3.
Spatial addresses of dual-core systems based on Mailbox
| Example | Name | Base address | Size |
|---|---|---|---|
| dlmb_cntlr0 | C_BASEADDR | 0 × 00000000 | 8 kb |
| ilmb_cntlr0 | C_BASEADDR | 0 × 00000000 | 8 kb |
| plb_plb_bridge | C_BRIDGE_BASEADDR | 0 × 85000000 | 4 kb |
| Xps_mailbox_0 | C_SPLBO_BASEADDR | 0 × 81e00000 | 256 kb |
3.1.3.2 Experimental results and analysis
Mailbox is used to complete the parallel calculation of matrix eigenvalues of the dual-core system. In this system, we conducted a comparative experiment between the neural network-based matrix eigenvalue algorithm and the traditional matrix eigenvalue algorithm and explored which algorithm has a faster calculation time. The experimental results obtained are shown in Figure 8.
The calculation time of the two algorithms under the parallel algorithm.
It can be seen from the figure that in the parallel algorithm, no matter what matrix eigenvalue algorithm is used, the calculation time of the matrix eigenvalue increases with the increase of the matrix. However, in the case of the same matrix size, the calculation time of the matrix eigenvalue algorithm based on the neural network is always shorter than the traditional matrix eigenvalue algorithm, that is, the calculation is faster. After calculation, the neural network-based matrix eigenvalue algorithm is compared with the traditional matrix eigenvalue algorithm, and the calculation time is reduced by 17% within the matrix size of 4 × 4 to 20 × 20.
Through the above two experiments, it can be seen that dual-core parallel computing can reduce the time complexity of the algorithm and reduce the time spent on calculations. With the expansion of the matrix scale, the superiority of parallel computing becomes more and more obvious.
3.2 Numerical experiment of matrix eigenvalue based on deep neural network
3.2.1 Numerical design
This experiment uses the MATLAB platform to calculate the eigenvalue matrix design of the matrix as follows:
5 × 5 symmetric positive definite matrix:
Any symmetric positive definite matrix:
This experiment uses a 5 × 5 matrix, which is smaller and easier to calculate, which is convenient for the experiment.
3.2.2 Matrix eigenvalue calculation based on Oja
We use the Oja model introduced earlier to calculate matrix eigenvalues. The matrix eigenvalues in the above numerical design are, respectively, {0.8392, 0.7315, 0.4793, 0.4266, 0.0218}; the initial value w(0) is randomly selected, and the Oja algorithm calculates the maximum eigenvalue to obtain a value of 0.8372. The convergence curve is shown in the left figure of Figure 9. Taking
Oja algorithm eigenvalue calculation numerical curve.
3.2.3 Matrix eigenvalue calculation based on RNNS
This experiment uses the RNNS model introduced in the previous article to calculate the matrix eigenvalues. The result of the eigenvalue calculation is shown in the left figure of Figure 10, and the result of the eigenvalue vector w(t) is shown in the right figure of Figure 10.
RNNS algorithm matrix eigenvalues and eigenvalue vector calculation results.
It can be seen from the figure that under the RNNS algorithm, the minimum eigenvalue calculation result obtained is 0.0134. For general matrices, the RNNS model is effective, and the eigenvalues of the matrix can be calculated the fastest. The performance effect is better than that of the Ojb model.
It can be seen from the calculation result curves of the above two models that the Oja model is only suitable for the calculation of the maximum eigenvalue of a non-negative matrix and cannot calculate negative matrices and other eigenvalues. The RNNS model is suitable for the calculation of general matrices and is more practical.
4 Discussion
Linear algebra is a relatively young subject that began in the nineteenth century, and its original role was to solve linear equations. When there are many unknowns, the equation assembly is particularly large. However, with this matrix calculation tool, the calculation of the equation becomes very simple. With the development of this field, for example, when the number of unknowns is more than the number of equations or there is no solution, etc., its applications become more extensive. The eigenvectors and eigenvalues of matrices are important tools for studying linear algebra, and they play an important role in many fields such as image processing and data processing. With the development of artificial intelligence, deep neural networks have entered the field of people’s vision. Combining the theory of deep neural networks to solve matrix eigenvalues shas become an important research direction. This article designs a hardware and software system, implements the designed algorithm on the system, completes the calculation of matrix eigenvalues, and uses a variety of matrix verification algorithms of different orders. In this paper, the traditional matrix eigenvalue algorithm and the matrix eigenvalue algorithm based on deep neural network are compared to understand the difference in processing time between them, and an experiment is designed to study the characteristics of the matrix eigenvalue algorithm based on deep neural network.
5 Conclusion
This article introduces and explains the related theories of deep neural networks, explains the estimation and application-related methods of matrix eigenvalues, and introduces the method of solving matrix eigenvalues based on deep neural networks. In addition, this article did two experiments: the first experiment is to test the calculation time of the matrix eigenvalue algorithm based on the neural network and the traditional matrix eigenvalue algorithm under the parallel algorithm and the serial algorithm. Compare the calculation times of different matrix eigenvalue algorithms and draw a conclusion: (1) when the matrix size is between 4 × 4 and 20 × 20, under the serial algorithm, the algorithm based on a deep neural network reduces the calculation time by about 7% compared with the traditional algorithm. (2) Under the parallel algorithm, the calculation time is reduced by 17%. It is proved that the matrix eigenvalue algorithm based on a deep neural network takes less time and is faster. (3) The calculation time of the serial algorithm is always lower than that of the parallel algorithm, which proves that the running time of the parallel algorithm is shorter and the speed is faster. The second experiment is a numerical experiment on matrix eigenvalues based on deep neural networks, which calculate matrix eigenvalues based on the Obj model and RNNS model, respectively. It is found that the Oja algorithm is only suitable for the calculation of the maximum eigenvalue of a non-negative definite matrix, while RNNS is effective for general matrices and can calculate the eigenvalue of the matrix the fastest.
Acknowledgements
This work was supported by the 2021 university-level regular scientific research project of Xi ‘an Fanyi University: Estimation of eigenvalues of matrices with unknown elements and its application (21A04).
-
Funding information: This work was supported by the 2021 university level regular scientific research project of Xi ‘an Fanyi University: Estimation of eigenvalues of matrices with unknown elements and its application (21A04).
-
Conflict of interest: Author states no conflict of interest.
References
[1] Weiss S, Pestana J, Proudler IK. On the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix. IEEE Trans Signal Process. 2018;66(10):2659–72.10.1109/TSP.2018.2812747Search in Google Scholar
[2] Xu WR, Chen GL. Submatrix constrained inverse eigenvalue problem involving generalised centrohermitian matrices in vibrating structural model correction. East Asian J Appl Mathematics. 2016;6(1):42–59.10.4208/eajam.200715.181115aSearch in Google Scholar
[3] Gernandt H, Trunk C. Eigenvalue placement for regular matrix pencils with rank one perturbations. SIAM J Matrix Anal Appl. 2017;38(1):134–54.10.1137/16M1066877Search in Google Scholar
[4] Ding T, Li C, Yang Y, Bo R, Blaabjerg F. Negative reactance impacts on the eigenvalues of the Jacobian matrix in power flow and type-1 low-voltage power-flow solutions. IEEE Trans Power Syst. 2017;32(5):3471–81.10.1109/TPWRS.2016.2645608Search in Google Scholar
[5] Janssen A, Mikosch TM, Rezapour Xie X. The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model. Bernoulli. 2018;24(2):1351–93.10.3150/16-BEJ901Search in Google Scholar
[6] Mou S, Anderson B. Eigenvalue invariance of inhomogeneous matrix products in distributed algorithms. IEEE Control Syst Lett. 2017;1(1):8–13.10.1109/LCSYS.2017.2698179Search in Google Scholar
[7] Hu C, Li W, Wang R, Long T, Drake VA. Discrimination of parallel and perpendicular insects based on relative phase of scattering matrix eigenvalues. IEEE Trans Geosci Remote Sens. 2020;58(6):3927–40.10.1109/TGRS.2019.2959622Search in Google Scholar
[8] Hanbay K, Talu MF. A novel active contour model for medical images via the Hessian matrix and eigenvalues. Comput Math Appl. 2018;75(9):3081–104.10.1016/j.camwa.2018.01.033Search in Google Scholar
[9] Shi Z, Yang G, Y, Xiao. A limited memory BFGS algorithm for non-convex minimization with applications in matrix largest eigenvalue problem. Math Methods Oper Res. 2016;83(2):243–64.10.1007/s00186-015-0527-8Search in Google Scholar
[10] Weder R. The number of eigenvalues of the matrix Schrödinger operator on the half line with general boundary conditions. J Math Phys. 2017;58(10):1–11.10.1063/1.5008655Search in Google Scholar
[11] Zhang W, Sun J, Xiong H, Chen D. A new joint eigenvalue distribution of finite random matrix for cognitive radio networks. IET Commun. 2016;10(13):1584–9.10.1049/iet-com.2015.0869Search in Google Scholar
[12] Kryzhanovsky BV, Litinskii LB. Connection-matrix eigenvalues in the Ising model: taking into account interaction with next-nearest neighbors. Doklady Phys. 2019;64(11):414–7.10.1134/S1028335819110065Search in Google Scholar
[13] Yang XP, Hao Z. Supereigenvalue problem to addition-min fuzzy matrix with application in P2P file sharing system. IEEE Trans Fuzzy Syst. 2020;28(8):1640–51.10.1109/TFUZZ.2019.2920806Search in Google Scholar
[14] Fukada Y. Support-free robust topology optimization based on pseudo-inverse stiffness matrix and eigenvalue analysis. Struct Multidiscip Optim. 2020;61(1):59–76.10.1007/s00158-019-02345-0Search in Google Scholar
[15] Yang P, Shang P. Recurrence quantity analysis based on matrix eigenvalues. Commun Nonlinear Sci Numer Simul. 2018;59(Jun):15–29.10.1016/j.cnsns.2017.11.001Search in Google Scholar
[16] Li J, Li W, Duan X, Xiao M. Newton’s method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils. Adv Comput Math. 2021;47(2):1–50.10.1007/s10444-021-09855-wSearch in Google Scholar
[17] Tian B, Zhang Y, Zhou W. Tracy–Widom law for the largest eigenvalue of sample covariance matrix generated by VARMA. Random Matrices Theory Appl. 2021;10(02):333–40.10.1142/S2010326321500222Search in Google Scholar
[18] Gil M. Conservation of the number of the eigenvalues of two-parameter matrix problems in bounded domains under perturbations. Annali dell’Università di Ferrara Sez 7: Scienze matematiche. 2019;66(3–4):1–9.10.1007/s11565-019-00332-3Search in Google Scholar
[19] Tseng CC, Lee SL. Design of orthogonal graph filter bank with known eigenvalues of Laplacian matrix. IET Signal Process. 2019;13(5):551–61.10.1049/iet-spr.2018.5291Search in Google Scholar
[20] Feng J, Yan S, Qin S, Han W. A neurodynamic approach to compute the generalized eigenvalues of symmetric positive matrix pair. Neurocomputing. 2019;359(SEP.24):420–6.10.1016/j.neucom.2019.06.016Search in Google Scholar
[21] Liu J, Zhang J, Huang H. The eigenvalue product bounds of the Lyapunov matrix differential equation and the stability of a class of time-varying nonlinear system. J Inequal Appl. 2019;2019(1):1–18.10.1186/s13660-019-2119-2Search in Google Scholar
[22] Wang X, Wang L, Hao Y. A construction of multi-sender authentication codes from eigenvalues and eigenvectors of the matrix over finite fields. J Harbin Inst Technol (N Ser). 2019;26(01):51–60.Search in Google Scholar
[23] Melman A. Eigenvalue bounds for matrix polynomials in generalized bases. Math Comput. 2018;87(312):1935–48.10.1090/mcom/3252Search in Google Scholar
[24] Kalinina EA. On multiple eigenvalues of a matrix dependent on a parameter. KI-Künstliche Intell. 2016;18(3):1–7.10.1007/978-3-319-45641-6_20Search in Google Scholar
© 2022 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.