scholarly journals Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Wenai Xu ◽  
Zhaolin Jiang
Keyword(s):  
Kronecker Product ◽  
Hadamard Product ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Frobenius Norm

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

scholarly journals Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Zhaolin Jiang ◽  
Yunlan Wei
Keyword(s):  
Differential Equations ◽  
Research Area ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Frobenius Norm ◽  
Lucas Numbers ◽  
Skew Circulant ◽  
Fibonacci And Lucas Numbers

Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

scholarly journals Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers

2021 ◽  
pp. 131-151
Author(s):  
Yüksel Soykan
Keyword(s):  
Circulant Matrix ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Hadamard Products ◽  
Tribonacci Numbers

In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.

scholarly journals VanderLaan Circulant Type Matrices

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Hongyan Pan ◽  
Zhaolin Jiang
Keyword(s):  
Lower Bound ◽  
Complex Systems ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Control Methods ◽  
Tridiagonal Matrices ◽  
Transformation Matrices

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, andg-circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaang-circulant matrix.

scholarly journals On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix

2018 ◽  
Vol 6 (1) ◽  
pp. 23-36
Author(s):  
Jorma K. Merikoski ◽  
Pentti Haukkanen ◽  
Mika Mattila ◽  
Timo Tossavainen
Keyword(s):  
Lower Bounds ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Frobenius Norm

Abstract Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

scholarly journals On k-circulant matrices with the generalized third-order Pell numbers

2021 ◽  
Vol 27 (4) ◽  
pp. 187-206
Author(s):  
Yüksel Soykan ◽  
Keyword(s):  
Circulant Matrix ◽  
Numerical Results ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Third Order ◽  
Pell Numbers

In this paper, we obtain explicit forms of the sum of entries, the maximum column sum matrix norm, the maximum row sum matrix norm, Euclidean norm, eigenvalues and determinant of k-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this k-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

scholarly journals On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zhaolin Jiang ◽  
Jinjiang Yao ◽  
Fuliang Lu
Keyword(s):  
Fibonacci Numbers ◽  
Spectral Norm ◽  
Matrix Norm ◽  
Circulant Matrices ◽  
Frobenius Norm ◽  
Transformation Matrices ◽  
Skew Circulant

Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.

scholarly journals On k-circulant matrices with the Lucas numbers

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 4037-4046
Author(s):  
Biljana Radicic
Keyword(s):  
Complex Number ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Nonzero Complex Number

Let k be a nonzero complex number. In this paper, we determine the eigenvalues of a k-circulant matrix whose first row is (L1,L2,..., Ln), where Ln is the nth Lucas number, and improve the result which can be obtained from the result of Theorem 7. [28]. The Euclidean norm of such matrix is obtained. Bounds for the spectral norm of a k-circulant matrix whose first row is (L-11, L-12,..., L-1n ) are also investigated. The obtained results are illustrated by examples.

scholarly journals DIVERSITY AES IN MIXCOLUMNS STEP WITH 8X8 CIRCULANT MATRIX

2021 ◽  
Vol 8 (9) ◽  
pp. 19-35
Author(s):  
Yan-Wen Chen ◽  
Jeng-Jung Wang ◽  
Yan-Haw Chen ◽  
Chong-Dao Lee
Keyword(s):  
Embedded Systems ◽  
Efficient Method ◽  
Matrix Multiplication ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Matrix Operation ◽  
Branch Number ◽  
Involutory Matrix ◽  
Software Implementations ◽  
Encryption Decryption

In AES MixColumns operation, the branch number of circulant matrix is raised from 5 to 9 with 8´8 circulant matrices that can be enhancing the diffusion power. An efficient method to compute the circulant matrices in AES MixColumns transformation for speeding encryption is presented. Utilizing 8´8 involutory matrix multiplication is required 64 multiplications and 56 additions in in AES Mix-Columns transformation. We proposed the method with diversity 8´8 circulant matrices is only needed 19 multiplications and 57 additions. It is not only to encryption operations but also to decryption operations. Therefore, 8´8 circlant matrix operation with AES key sizes of 128bits, 192bits, and 256 bits are above 29.1%, 29.3%, and 29.8% faster than using 4´4 involutory matrix operation (16 multiplications, 12 additions), respectively. 8´8 circulant matrix encryption/decryption speed is above 78% faster than 8´8 involutory matrix operation. Ultimately, the proposed method for evaluating matrix multiplication can be made regular, simple and suitable for software implementations on embedded systems.

scholarly journals Accelerating Deep Neural Networks by Combining Block-Circulant Matrices and Low-Precision Weights

Electronics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 78 ◽  
Author(s):  
Zidi Qin ◽  
Di Zhu ◽  
Xingwei Zhu ◽  
Xuan Chen ◽  
Yinghuan Shi ◽  
...  
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Matrix Vector Multiplication ◽  
Processing Power ◽  
Measurement Results ◽  
Block Circulant Matrix ◽  
Storage Complexity ◽  
Fully Connected

As a key ingredient of deep neural networks (DNNs), fully-connected (FC) layers are widely used in various artificial intelligence applications. However, there are many parameters in FC layers, so the efficient process of FC layers is restricted by memory bandwidth. In this paper, we propose a compression approach combining block-circulant matrix-based weight representation and power-of-two quantization. Applying block-circulant matrices in FC layers can reduce the storage complexity from O ( k 2 ) to O ( k ) . By quantizing the weights into integer powers of two, the multiplications in the reference can be replaced by shift and add operations. The memory usages of models for MNIST, CIFAR-10 and ImageNet can be compressed by 171 × , 2731 × and 128 × with minimal accuracy loss, respectively. A configurable parallel hardware architecture is then proposed for processing the compressed FC layers efficiently. Without multipliers, a block matrix-vector multiplication module (B-MV) is used as the computing kernel. The architecture is flexible to support FC layers of various compression ratios with small footprint. Simultaneously, the memory access can be significantly reduced by using the configurable architecture. Measurement results show that the accelerator has a processing power of 409.6 GOPS, and achieves 5.3 TOPS/W energy efficiency at 800 MHz.

Sign in / Sign up

Export Citation Format

Share Document