scholarly journals Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Zhaolin Jiang ◽  
Nuo Shen ◽  
Juan Li
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Lucas Sequences ◽  
Special Pattern ◽  
Generalized Lucas Numbers

The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

scholarly journals Determinants, Norms, and the Spread of Circulant Matrices with Tribonacci and Generalized Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Juan Li ◽  
Zhaolin Jiang ◽  
Fuliang Lu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Partial Differential ◽  
Tribonacci Numbers ◽  
Generalized Lucas Numbers

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.

scholarly journals The Determinants, Inverses, Norm, and Spread of Skew Circulant Type Matrices Involving Any Continuous Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Jin-jiang Yao ◽  
Zhao-lin Jiang
Keyword(s):  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Skew Circulant ◽  
The Relationship

We consider the skew circulant and skew left circulant matrices with any continuous Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrices and present the determinant and the inverse matrices by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the skew left circulant matrices by utilizing the relationship between skew left circulant matrices and skew circulant matrix, respectively. Finally, the four kinds of norms and bounds for the spread of these matrices are given, respectively.

scholarly journals A New Method of Matrix Decomposition to Get the Determinants and Inverses of r -Circulant Matrices with Fibonacci and Lucas Numbers

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jiangming Ma ◽  
Tao Qiu ◽  
Chengyuan He
Keyword(s):  
Fibonacci Numbers ◽  
Matrix Decomposition ◽  
Circulant Matrix ◽  
New Method ◽  
Circulant Matrices ◽  
Computational Time ◽  
Lucas Numbers ◽  
Fibonacci And Lucas Numbers

We use a new method of matrix decomposition for r -circulant matrix to get the determinants of A n = Circ r F 1 , F 2 , … , F n and B n = Circ r L 1 , L 2 , … , L n , where F n is the Fibonacci numbers and L n is the Lucas numbers. Based on these determinants and the nonsingular conditions, inverse matrices are derived. The expressions of the determinants and inverse matrices are represented by Fibonacci and Lucas Numbers. In this study, the formulas of determinants and inverse matrices are much simpler and concise for programming and reduce the computational time.

scholarly journals On Jacobsthal and Jacobsthal-Lucas Circulant Type Matrices

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Yanpeng Gong ◽  
Zhaolin Jiang ◽  
Yun Gao
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Circulant Matrix ◽  
Inverse Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Fractional Order Differential Equations

Circulant type matrices have become an important tool in solving fractional order differential equations. In this paper, we consider the circulant and left circulant andg-circulant matrices with the Jacobsthal and Jacobsthal-Lucas numbers. First, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix. Furthermore, the invertibility of the left circulant andg-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant andg-circulant matrices by utilizing the relation between left circulant,g-circulant matrices, and circulant matrix, respectively.

scholarly journals Circulant Type Matrices with the Sum and Product of Fibonacci and Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Zhaolin Jiang ◽  
Yanpeng Gong ◽  
Yun Gao
Keyword(s):  
Differential Equations ◽  
Circulant Matrix ◽  
Inverse Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Fibonacci And Lucas Numbers

Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant andg-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant andg-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant andg-circulant matrices by utilizing the relation between left circulant, andg-circulant matrices and circulant matrix, 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 The Invertibility, Explicit Determinants, and Inverses of Circulant and Left Circulant andg-Circulant Matrices Involving Any Continuous Fibonacci and Lucas Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Zhaolin Jiang ◽  
Dan Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Circulant Matrix ◽  
Circulant Matrices ◽  
Lucas Numbers ◽  
Transformation Matrices ◽  
Fibonacci And Lucas Numbers ◽  
Delay Differential ◽  
The Relationship

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.

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.

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