Matrix Cosine Research Articles

On Explicit Formulas of Hyperbolic Matrix Functions

Hyperbolic matrix functions are essential for solving hyperbolic coupled partial differential equations. In fact the best analytic-numerical approximations for resolving these equations come from the use of hyperbolic matrix functions. The hyperbolic matrix sine and cosine sh(A), ch(A) (A∈Mr(C)) can be calculated using numerous different techniques. In this article we derive some explicit formulas of sh(tA) and ch(tA) (t∈R) using the Fibonacci-H\"{o}rner and the polynomial decomposition, these decompositions are calculated using the generalized Fibonacci sequences combinatorial properties in the algebra of square matrices. Finally we introduce a third approach based on the homogeneous linear differential equations. And we provide some examples to illustrate your methods.

Representation and finite time stability of solution and relative controllability of conformable type oscillating systems

In this paper, we firstly introduce the concept of conformable matrix sine and cosine functions, which help us to seek explicit formula of solutions to a conformable type linear oscillating system by using the variation of constants method. Secondly, we establish some sufficient conditions to guarantee the finite time stability results and give sufficient and necessary conditions to examine that a linear conformable system is relatively controllable. Further, we apply fixed point theorem to prove relatively controllable result for a semilinear conformable system. Finally, examples are presented to illustrate the validity of the main theorems.

Explicit formulas for computing matrix trigonometric functions

Various methods have been introduced for calculating the matrix sine and cosine. This paper concerns the Fibonacci-Horner decomposition of the matrix sine and cosine sin(tA), cos(tA) (A ∈ M (r; ℂ), t ∈ ℝ), which is derived from the combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. Finally we derive some formulas for the trigonometric functions of f(A) where f is an homographic function.

Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic for optimal efficiency so they do not conveniently extend to an arbitrary precision environment. We develop an algorithm for computing the matrix cosine that takes the unit roundoff of the working precision as input, and so works in an arbitrary precision. The algorithm employs a Taylor approximation with scaling and recovering and it can be used with a Schur decomposition or in a decomposition-free manner. We also derive a framework for computing the Fréchet derivative, construct an efficient evaluation scheme for computing the cosine and its Fréchet derivative simultaneously in arbitrary precision, and show how this scheme can be extended to compute the matrix sine, cosine, and their Fréchet derivatives all together. Numerical experiments show that the new algorithms behave in a forward stable way over a wide range of precisions. The transformation-free version of the algorithm for computing the cosine is competitive in accuracy with the state-of-the-art algorithms in double precision and surpasses existing alternatives in both speed and accuracy in working precisions higher than double.

New Hermite series expansion for computing the matrix hyperbolic cosine

There are, currently, very few implementations to compute the hyperbolic cosine of a matrix. This work tries to fill this gap. To this end, we first introduce both a new rational-polynomial Hermite matrix expansion and a formula for the forward relative error of Hermite approximation in exact arithmetic with a sharp bound for the forward error. This matrix expansion allows obtaining a new accurate and efficient method for computing the hyperbolic matrix cosine. We present a MATLAB implementation, based on this method, which shows a superior efficiency and a better accuracy than other state-of-the-art methods. The algorithm developed on the basis of this method is also able to run on an NVIDIA GPU thanks to a MEX file that connects the MATLAB implementation to the CUDA code.

Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.

Computing the matrix sine and cosine simultaneously with a reduced number of products

A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be used in single and double precision arithmetic. The resulting algorithms are more efficient than schemes based on Padé approximations for a wide range of norm matrices.

Explicit solution and dynamical properties of atmospheric Ekman flows with boundary conditions

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with the constant eddy viscosity. Different from the previous work, we modify the boundary conditions and derive the explicit solution by using the notation of matrix cosine and matrix sine. For the arbitrary height-dependent eddy viscosity, we get the solution of the classical problem with zero velocity and acceleration at the bottom of the layer. In addition, uniqueness is shown and dynamical properties of solution are characterized.

A Note on Relative Controllability of Higher Order Linear Delayed Discrete Systems

This article is concerned with relative controllability of higher order linear discrete delayed systems with a single delay. By a formula that uses special matrix functions (discrete delayed matrix sine and cosine) to compute the solution of the initial problem, a criterion is found for relative controllability. In terms of special matrix functions, a control function is constructed to solve the problem of relative controllability. It is shown that the transformation of a given system into a higher dimensional system without delays leads to a nonequivalent problem of relative controllability. The results are illustrated by an example.

On Bernoulli series approximation for the matrix cosine

This paper presents a new series expansion based on Bernoulli matrix polynomials to approximate the matrix cosine function. An approximation based on this series is not a straightforward exercise since there exist different options to implement such a solution. We dive into these options and include a thorough comparative of performance and accuracy in the experimental results section that shows benefits and downsides of each one. Also, a comparison with the Padé approximation is included. The algorithms have been implemented in MATLAB and in CUDA for NVIDIA GPUs.

Robust visual tracking via part-based model

In this paper, we propose a novel visual object tracking method using a part based appearance model. First, a local kernel feature is developed to encode edge information of patches. Next, bounding box of the target is divided into multiple parts. Then, each part uses correlation filter based tracking to predict position in the next frame. The matrix cosine similarity is utilized to measure reliabilities of the patches. Finally, optimal target location is predicted via maximizing the likelihood, which is obtained by adaptively fusing the reliable patches locations. Experimental results illustrate that our algorithm outperforms state-of-the-art tracking methods significantly in terms of accuracy and robustness.

Simulation of harmonic oscillators on the lattice

This work deals with the simulation of a two‐dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second‐order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state‐of‐the‐art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two‐dimensional lattice and check its energy conservation.

Fabric defect detection using saliency of multi‐scale local steering kernel

Fabric defect detection (FDD) plays an important role in the quality control in textile industry. In this study, the authors propose an efficient FDD method by using the saliency analysis of multi-scale local steering kernel (LSK). In the proposed method, a given RGB fabric image is first converted into the Commission International Eclairage (CIE) L*a*b colour space and then the LSK in each colour channel is computed by the singular value decomposition and the centre surrounding definition. Next, the matrix cosine similarity is employed to measure the similarity between different LSK features for generating the desired defective maps. Finally, a multi-scale averaging fusion scheme is applied to integrate the obtained defective maps at different scales for the final defective map. The experimental results indicate that the proposed method achieves the state-of-the-art performance on FDD compared to the other competitors.

A novel fractional delayed matrix cosine and sine

We introduce the fractional delayed matrix cosine and sine functions and use the variation of constants method to obtain an explicit formula for a solution to the initial value problem for a Riemann–Liouville sequential fractional linear time-delay system of order 1<2α≤2. Our results extend those for fractional linear time-delay systems and novel for the classical case α=1.

Fast Taylor polynomial evaluation for the computation of the matrix cosine

In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and a mixed forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the matrix cosine function more efficiently than using Paterson–Stockmeyer method. A sequential Matlab implementation of the new algorithm is provided, giving better efficiency and accuracy than state-of-the-art algorithms. Moreover, we provide an implementation in Matlab that can use NVIDIA GPUs easily and efficiently.

An efficient and accurate algorithm for computing the matrix cosine based on new Hermite approximations

In this work we introduce new rational-polynomial Hermite matrix expansions which allow us to obtain a new accurate and efficient method for computing the matrix cosine. This method is compared with other state-of-the-art methods for computing the matrix cosine, including a method based on Padé approximants, showing a far superior efficiency, and higher accuracy. The algorithm implemented on the basis of this method can also be executed either in one or two NVIDIA GPUs, which demonstrates its great computational capacity.

Efficient evaluation of matrix polynomials

This paper presents a new family of methods for evaluating matrix polynomials more efficiently than the state-of-the-art Paterson–Stockmeyer method. Examples of the application of the methods to the Taylor polynomial approximation of matrix functions like the matrix exponential and matrix cosine are given. Their efficiency is compared with that of the best existing evaluation schemes for general polynomial and rational approximations, and also with a recent method based on mixed rational and polynomial approximants. For many years, the Paterson–Stockmeyer method has been considered the most efficient general method for the evaluation of matrix polynomials. In this paper we show that this statement is no longer true. Moreover, for many years rational approximations have been considered more efficient than polynomial approximations, although recently it has been shown that often this is not the case in the computation of the matrix exponential and matrix cosine. In this paper we show that in fact polynomial approximations provide a higher order of approximation than the state-of-the-art computational methods for rational approximations for the same cost in terms of matrix products.

Two algorithms for computing the matrix cosine function

The computation of matrix trigonometric functions has received remarkable attention in the last decades due to its usefulness in the solution of systems of second order linear differential equations. Several state-of-the-art algorithms have been provided recently for computing these matrix functions. In this work, we present two efficient algorithms based on Taylor series with forward and backward error analysis for computing the matrix cosine. A MATLAB implementation of the algorithms is compared to state-of-the-art algorithms, with excellent performance in both accuracy and cost.

Stability of delay differential equations via delayed matrix sine and cosine of polynomial degrees

In this paper, we study the finite time stability of delay differential equations via a delayed matrix cosine and sine of polynomial degrees. Firstly, we give two alternative formulas of the solutions for a delay linear differential equation. Secondly, we obtain a norm estimation of the delayed matrix sine and cosine of polynomial degrees, which are used to establish sufficient conditions to guarantee our finite time stability results. Meanwhile, a numerical example is presented demonstrating the validity of our theoretical results. Finally, we extend our study to the same issue of a delay differential equation with nonlinearity by virtue of the Gronwall inequality approach.

Representation of Solutions of Linear Differential Systems of the Second Order with Constant Delays

We deduce representations for the solutions of initial-value problems for n-dimensional differential equations of the second order with delays: $$ x^{{\prime\prime} }(t)=2 Ax^{\prime}\left( t-\tau \right)-\left({A}^2+{B}^2\right) x\left( t-2\tau \right) $$ and $$ x^{{\prime\prime} }(t)=\left( A+ B\right) x^{\prime}\left( t-\tau \right)- A B x\left( t-2\tau \right) $$ by using special delay matrix functions. Here, A and B are commuting (n × n)-matrices and τ > 0. Moreover, a formula connecting the delay matrix exponential function with delayed matrix sine and delayed matrix cosine is obtained. We also discuss common features of the considered equations.