scholarly journals On $m$-th roots of nilpotent matrices

2021 ◽  
Vol 37 ◽  
pp. 718-733
Author(s):  
Semra Ozturk
Keyword(s):  
Nonnegative Integer ◽  
Linear Equations ◽  
Matrix Multiplication ◽  
Arbitrary Field ◽  
System Of Linear Equations ◽  
Nilpotent Matrix ◽  
Commuting Matrices ◽  
Erroneous Result ◽  
Nilpotent Matrices ◽  
Single Matrix

A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.

Fast hash functions and convolution product

2014 ◽  
Vol 8 (2) ◽  
Author(s):  
Sami Omar ◽  
Houssem Sabri
Keyword(s):  
Mathematical Problem ◽  
Linear Equations ◽  
Matrix Multiplication ◽  
Hash Functions ◽  
Software Implementation ◽  
Convolution Product ◽  
System Of Linear Equations ◽  
The One ◽  
Matrix Vector ◽  
Subsequent Result

Abstract.We propose a new simple and efficient family of hash functions based on matrix-vector multiplications with a competitive software implementation. The hash design combines a hard mathematical problem based on solving a system of linear equations with special-random requirements and the fast computation of the convolution product algorithm. Such a mixing was often unrealizable. For security, the one-way and collision resistant criteria are based on the fact that inverting the compression function for random values is infeasible in reasonable time. In a subsequent result, we conjecture a general framework for producing secure matrix multiplication hash functions.

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Khalid ◽  
M. N. Naeem ◽  
P. Agarwal ◽  
A. Ghaffar ◽  
Z. Ullah ◽  
...  
Keyword(s):  
Numerical Approximation ◽  
Analytic Solution ◽  
Linear Equations ◽  
Computational Cost ◽  
Spline Method ◽  
System Of Linear Equations ◽  
Approximation Solution ◽  
B Spline ◽  
Novel Technique ◽  
Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

On the Stresses in a Notched Strip

1952 ◽  
Vol 19 (2) ◽  
pp. 141-146
Author(s):  
Chih-Bing Ling
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Numerical Results ◽  
Theoretical Solution ◽  
Experimental Results ◽  
System Of Linear Equations ◽  
Transverse Bending ◽  
Longitudinal Tension

Abstract In a previous paper by the author (1), a theoretical solution for a notched strip under longitudinal tension is given. The result demands the solution of an infinite system of linear equations. A considerable amount of labor is involved in solving such a system. It seems, however, that the labor can be diminished by adapting to the solution a process known as the promotion of rank. In this paper such a process is described and then applied to solve the problem of a notched strip under transverse bending. The solution of this problem seems also to be new. The numerical results obtained are compared graphically with the experimental results available.

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