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In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.

Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an $$m\times n$$ m × n globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.

Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an [Formula: see text] globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.

In this paper, we consider a new Sylvester-Kac matrix, i.e., Fibonacci-Sylvester-Kac matrix. We discuss the eigenvalues, eigenvectors and characteristic polynomial of this matrix in two categories based on whether the Fibonacci-Sylvester-Kac matrix order is odd or even. Besides, we also give the explicit formulas for its determinant and inverse.

The explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz-like matrices with perturbed rows are presented in this paper. We derive the representation of the determinants and inverses in the form of products of Mersenne numbers and some initial values from matrix transformations, which reduces the computational complexity to a certain extent. Futhermore, we make some analysis about these basic quantities to illustrate our theoretical results.

In the study of using images to display car paint defects, the current need is to use deep Convolutional Neural Networks (CNN) to identify and classify different types of car paint defects, so as to give full play to the application of image processing in the field of automatic car paint defect detection. Using the collected car paint defect images, the car paint defects dataset is established. The preprocessing process of original data and the application of three image classification models based on CNN are visually displayed. First, the dataset of 7 types of car body defects including bubble, dust, fouling, pinhole, sagging, scratch, and shrink has been established, with a total of 2468 images. The model of MobileNet-V2, Vgg16, and ResNet34 are selected for training. As a result, after 30 training iterations, the MobileNetV2 algorithm achieved 94.3% accuracy, the accuracy of the Vgg16 algorithm is as high as 99.9%, and the accuracy of the ResNet34 algorithm is maintained at 99.2%. To sum up, for car paint defect detection, deep learning has great potential and deserves further development.

In this paper, we show that there is an intimate relationship between Toeplitz matrix, tridiagonal Toeplitz matrix, the Fibonacci number, the Lucas number, and the Golden Ratio. We introduce skew Loeplitz and skew Foeplitz matrices and derive their determinants and inverses by construction. Specifically, the determiant of $ n\times n $ skew Loeplitz matrix can be expressed by the $ (n+1) $st Fibonacci number. The inverse of skew Loeplitz matrix is sparse and can be expressed by the $ n $th and $ (n+1) $st Fibonacci numbers. Similarly, the determinant of $ n\times n $ skew Foeplitz matrix also can be expressed by the $ (n+1) $st Lucas number. The inverse of skew Foeplitz matrix can be expressed by only seven elements with each element being the explicit expression of the Lucas or Fibonacci numbers. We also calculate the determinants and inverses of skew Lankel and skew Fankel matrices.

In this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

This note contributes a generalization of Tung's 1964 result on a Harnack type determinant inequality. The generalization also includes a result of Fan as a special case.

In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type $I$. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type $II$ matrix can be also obtained, which benefits from the relation between Type $I$ and $II$ matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.

Unmanned Aerial Vehicles (UAVs), in multi-UAV collaborative search systems, need to realize adaptive autonomous trajectory planning based on their own state and surrounding environmental conditions. To allow for a fast trajectory planning of multiple UAVs from the current state to the target state, this paper proposes a trajectory planning method for multi-UAV collaborative search tasks based on genetic algorithm (GA). And using genetic algorithm, the UAV's trajectory can be optimized online based on the current mission. Through the analysis of the tasks of the multi-UAV collaborative decision-making part and the autonomous trajectory planning part, the genetic algorithm is used to model and optimize the trajectory planning problem to obtain a feasible and efficient optimal search trajectory from the current state to the decision-making target state. Finally, through specific simulation experiments, the optimal trajectory from the current state to the target state of the UAV i in the multi-UAV system is obtained, which verifies the effectiveness of the algorithm.

Unmanned Aerial Vehicle (UAV) is a kind of aircraft which is pilotless, controlled by remote control or self-control. This paper proposes the design of a multi-UAV cooperative search experimental platform, which uses image detection of searching for targets in image frames to solve the problem of multi-UAV autonomous cooperation and control. This paper will introduce novel solutions to two main problems: multi-UAV autonomous cooperative platform construction and control algorithm verification. The research work is carried out in three parts: hardware and software platform design, multi-UAV autonomous cooperative search algorithm, and platform experimental verification. The experiments include image sensing for searching a target in the image frames, keyboard control, multi-UAV cooperation, and leader-follower formation.

In this paper, for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, some properties, including the determinant, the inverse matrix, the eigenvalues and the eigenvectors, are studied in detail. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the well-known Fibonacci numbers; the inverse of the PTPT matrix can also be explicitly expressed using the Lucas number and only four elements in the PTPT matrix. Eigenvalues and eigenvectors can be obtained under certain conditions. In addition, some algorithms are presented based on these theoretical results. Comparison of our new algorithms and some recent works is given. Numerical results confirm our new theoretical results and show that the new algorithms not only can obtain accurate results but also have much better computing efficiency than some existing algorithms studied recently.

In this paper, the nonsingular CUPL-Toeplitz linear system from Markov chain is solved. We introduce two fast approaches whose complexity could be considered to be O ( n log n ) based on the splitting method of the CUPL-Toeplitz matrix which equals to a Toeplitz matrix minus a rank-one matrix. Finally, we confirm the performance of the new algorithms by three numerical experiments.

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.

In this paper, we present an innovative patterned matrix, RFPL-Toeplitz matrix, is neither the extension of Toeplitz matrix nor its special case. We show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular Toeplitz matrices. First, the explicit expression of the group inverse of an RFPL-Toeplitz matrix is obtained. Second, the decomposition of the group inverse is given. Finally, an example demonstrates availability of the two methods for the group inverse.

In this paper, we consider the conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices. It is proved that the inverse of these special matrices can be expressed as the sum of products of lower and upper triangular matrices. Firstly, we get access to the explicit inverse of conjugate-Toeplitz matrix. Secondly, the decomposition of the inverse is obtained. Similarly, the formulae and the decomposition on inverse of conjugate-Hankel are provided. Thirdly, the stability of the inverse formulae of CT and CH matrices are discussed. Finally, examples are provided to verify the feasibility of the algorithms provided in this paper.

Circulant and block circulant type matrices are important tools in solving networked systems. In this paper, based on the style spectral decomposition of the basic circulant matrix and the basic skew circulant matrix, the block style spectral decomposition of the BCSCB matrix is obtained. And then, the structure perturbation is analysed, which includes the condition number and relative error of the BCSCB linear system. Then the optimal backward perturbation bound of the BCSCB linear system is discussed. Simultaneously, the algorithm for the optimal backward perturbation bound is given. Finally, a numerical example is provided to verify the effectiveness of the algorithm.

It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant,g-circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonaccig-circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.

In this paper, we preset some improved techniques for proofs within Zhou and Jiang (2014). Meanwhile, focusing on n and r are co-prime, we generalize those results to investigate formulae of even-order r -circulant matrices with alternately positive and negative entrywise. And we deduce explicit formulae for their spectral norms with the entries in the first row.

In this study, a new fast algorithm for optimal design of block digital filters is proposed based on the skew circulant matrix, the Toeplitz and the skew shift cyclic matrices. The developed skew-cyclic filter, compared with the cyclic filter, not only achieves the same computational complexity of the design process and the memory requirements, but is more efficient with only approximately half of the computational cost for the real signals.

In this paper, we investigate explicit formulae of spectral norms for g -circulant matrices and prove them in details. The entries in the first row of the g -circulant matrices are a i = F i s and a i = L i s ( s = 1 , 2 ) . F i and L i denotes the Fibonacci number and Lucas number, respectively. Employing these approaches, some numerical tests are presented to verify the results.

The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, andg-circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results.

Circulant matrices have become important tools in solving integrable system, Hamiltonian structure, and integral equations. In this paper, we prove that Gaussian Fibonacci circulant type matrices are invertible matrices forn>2and give the explicit determinants and the inverse matrices. Furthermore, the upper bounds for the spread on Gaussian Fibonacci circulant and left circulant matrices are presented, respectively.

Circulant matrices play an important role in solving delay differential equations. In this paper, circulant type matrices including the circulant and left circulant andg-circulant matrices with any continuous Fibonacci and Lucas numbers are considered. Firstly, the invertibility of the circulant matrix is discussed and the explicit determinant and the inverse matrices by constructing the transformation matrices are presented. Furthermore, the invertibility of the left circulant andg-circulant matrices is also studied. We obtain the explicit determinants and the inverse matrices of the left circulant andg-circulant matrices by utilizing the relationship between left circulant,g-circulant matrices and circulant matrix, respectively.

Circulant matrices may play a crucial role in solving various differential equations. In this paper, the techniques used herein are based on the inverse factorization of polynomial. We give the explicit determinants of the RFPrLrR circulant matrices and RLPrFrL circulant matrices involving Fibonacci, Lucas, Pell, and Pell-Lucas number, respectively.

Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degreen, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

Circulant matrices have important applications in solving ordinary differential equations. In this paper, we consider circulant-type matrices with thek-Fibonacci andk-Lucas numbers. We discuss the invertibility of these circulant matrices and present the explicit determinant and inverse matrix by constructing the transformation matrices, which generalizes the results in Shen et al. (2011).

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLSR-factor block circulant (retrocirculant) matrix over fieldF. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a fieldFis nonsingular. Fast algorithms for solving the unique solution of the inverse problem ofAX=bin the class of the level-2 FLS(R,r)-circulant(retrocirculant) matrix of type(m,n)over fieldFare given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.

Comprehensive two-dimensional liquid chromatographic separation by means of a combination of ion-exchange chromatography (IEC) and reversed-phase liquid chromatography (RPLC) has been realized and applied to the analysis of components of a traditional Chinese medicine Flos Lonicera. In the first dimension the components of a Flos Lonicera extract is separated on strong anion-exchange (SAX) column according to the charge-exchange properties of the components. In the second dimension the components are further fractionated by reversed-phase liquid chromatography (on ODS) on the basis of their hydrophobicity. The RPLC effluent is desalted and split and the components are identified on-line by use of both diode-array detection and mass spectrometry with atmospheric pressure chemical ionization (APCI–MS). In this way, the extract of Flos Lonicera can be rapidly separated, desalted, and analyzed, with determination of molecular weight. More than 58 components of an extract of Flos Lonicera were well resolved and six were tentatively identified from their UV and mass spectra.

In this paper, a new kind of matrices, i.e., II-circulant matrices is considered. Algorithms for computing minimal polynomial of this kind of matrices are presented by means of the algorithm for the Grobner basis of the ideal in the polynomial ring. Two algorithms for finding the inverses of such matrices are also presented based on the Buchberger's algorithm.