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In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation AXB+CYD=E. By integrating real representation of a quaternion matrix with H-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply H-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of H-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.

In this paper, the idea of partitioning is used to solve quaternion least squares problem, we divide the quaternion Bisymmetric matrix into four blocks and study the relationship between the block matrices. Applying this relation, the real representation of quaternion, and M-P inverse, we obtain the least squares Bisymmetric solution of quaternion matrix equation $ AXB = C $ and its compatable conditions. Finally, we verify the effectiveness of the method through numerical examples.

In this paper, we investigate the best fit solution for the second order differential problem with one initial and other integral conditions. We obtain the representation of that minimizer and present an example.paration.

U okviru završnog rada potrebno je realizirati programski sustav za identifikaciju parametarskog modela vozila u kretanju. Programski sustav mora obnavljati model u stvarnom vremenu na temelju mjernih podataka o kretanju vozila te veličina s ostalih senzora i elemenata upravljačkog sustava. Model mora obuhvaćati osnovne fizikalne komponente sustava drugog reda te vanjske i unutarnje sile koje uzrokuju kretanje. Proučiti i umanjiti utjecaj šuma i mjernih nesigurnosti na identifikaciju. Fizikalni model kretanja preslikati u virtualni korisnički model, uz odabrane opcije i parametre preslikavanja. Koristiti ARM zasnovano STM32 F4 ugradbeno računalo s operacijskim sustavom za rad u stvarnom vremenu (RTOS) te komunikacijskim podsustavom. U programskom jeziku C napisati potrebnu programsku potporu. Po potrebi koristiti programsko okruženje MATLAB za provjeru i simulaciju algoritama. In the scope of BSc Thesis a software system has to be designed to identify the parametric model of a moving vehicle. Software system has to refresh the model in real time based on the data gathered from vehicle movement, sensors and other elements of the control system. The model has to encompass basic physical components of a second order system as well as external and internal forces that cause movement. Explore and decrease noise impact and measurement uncertainty on identification. Move a physical movement model into virtual user model according to a few selectable options and parameters. Use ARM based STM32F4 microcontroler with real time operating system (RTOS) and communication subsystem. Write software in programing language C. If needed use MATLAB to simulate and test algorithms.

International audience; The main result of this paper is the formulation of an explicit expression for the condition number of the truncated least squares solution of $Ax=b$. This expression is given in terms of the singular values of $A$ and the Fourier coefficients of $b$. The result is derived using the notion of the Fréchet derivative together with the product norm on the data $[A,b]$ and the 2-norm on the solution. Numerical experiments are given to confirm our results by comparing them to those obtained by means of a finite difference approach.

The stochastic gradient matching pursuit algorithm requires the sparsity of the signal as prior information. However, this prior information is unknown in practical applications, which restricts the practical applications of the algorithm to some extent. An improved method was proposed to overcome this problem. First, a pre-evaluation strategy was used to evaluate the sparsity of the signal and the estimated sparsity was used as the initial sparsity. Second, if the number of columns of the candidate atomic matrix was smaller than that of the rows, the least square solution of the signal was calculated, otherwise, the least square solution of the signal was set as zero. Finally, if the current residual was greater than the previous residual, the estimated sparsity was adjusted by the fixed step-size and stage index, otherwise we did not need to adjust the estimated sparsity. The simulation results showed that the proposed method was better than other methods in terms of the aspect of reconstruction percentage in the larger sparsity environment.

Least squares solution of linear inequalities appears in many disciplines such as linear separability problems and inconsistency correction. In this paper we consider this problem with uncertainty in its data. Then we prove that its robust counterpart is equivalent to a second order conic linear optimization problem, which can be efficiently solved using interior point methods.

In this paper, we investigate the properties of a generalized Green’s function describing the minimum norm least squares solution for a second order discrete problem with two nonlocal conditions. The properties obtained of a generalized Green’s function resemble analogous properties of an ordinary Green’s function that describes the unique exact solution if it exists. Several features are illustrated by examples.

Let X and Y be Hilbert spaces, and let T : X → Y be a bounded linear operator with closed range. In this paper, we present an optimal perturbation result on the least squares solutions to the operator equation Tx = y under the most general condition.

A stochastic conjugate gradient method for approximation of a function is proposed. The proposed method avoids computing and storing the covariance matrix in the normal equations for the least squares solution. In addition, the method performs the conjugate gradient steps by using an inner product that is based stochastic sampling. Theoretical analysis shows that the method is convergent in probability. The method has applications in such fields as predistortion for the linearization of power amplifiers., 21 pages, 5 figures

A large amount of today’s telecommunication consists of mobile and short distance wireless applications, where the effect of the channel is unknown and changing over time, and thus needs to be described statistically. Therefore the received signal can not be accurately predicted and has to be estimated. Since telecom systems are implemented in real-time, the hardware in the receiver for estimating the sent signal can for example be based on a DSP where the statistic calculations are performed. A fixed-point DSP with a limited number of bits and a fixed binary point causes larger quantization errors compared to floating point operations with higher accuracy. The focus on this thesis has been to build a library of functions for handling fixed-point data. A class that can handle the most common arithmetic operations and a least squares solver for fixed-point have been implemented in MATLAB code. The MATLAB Fixed-Point Toolbox could have been used to solve this task, but in order to have full control of the algorithms and the fixed-point handling an independent library was created. The conclusion of the simulation made in this thesis is that the least squares result are depending more on the number of integer bits then the number of fractional bits. En stor del av dagens telekommunikation består av mobila trådlösa kortdistanstillämpningar där kanalens påverkan är okänd och förändras över tid. Signalen måste därför beskrivas statistiskt, vilket gör att den inte kan bestämmas exakt, utan måste estimeras. Eftersom telekomsystem arbetar i realtid består hårdvaran i mottagaren av t.ex. en DSP där de statistiska beräkningarna görs. En fixtals DSP har ett bestämt antal bitar och fast binärpunkt, vilket introducerar ett större kvantiseringsbrus jämfört med flyttalsoperationer som har en större noggrannhet. Tyngdpunkten på det här arbetet har varit att skapa ett bibliotek av funktioner för att hantera fixtal. En klass har skapats i MATLAB-kod som kan hantera de vanligaste aritmetiska operationerna och lösa minsta-kvadrat-problem. MATLAB:s Fixed-Point Toolbox skulle kunna användas för att lösa den här uppgiften men för att ha full kontroll över algoritmerna och fixtalshanteringen behövs ett eget bibliotek av funktioner som är oberoende av MATLAB:s Fixed-Point Toolbox. Slutsatsen av simuleringen gjord i detta examensarbete är att resultatet av minsta-kvadrat-metoden är mer beroende av antalet heltalsbitar än antalet binaler. fixtal, telekommunikation, DSP, MATLAB, Fixed-Point Toolbox, minsta-kvadrat-lösning, flyttal, Householder QR faktorisering, saturering, kvantiseringsbrus

We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using Henon's map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor as well as for an almost periodic example. For the boundary value and the least squares approach, we analyze in detail errors that occur, when restricting the infinite dimensional problem to a finite interval.

There was proposed a new method of estimation of multiprobe microwave multimeter algorithms precision by mean of accumulation of partial error. The least square solution is used for variance and covariance matrix definition. The weighed coefficient is obtained from algorithms derivatives with respect to intermediate variable. Substituting the expression for standard deviation (variance) end weighed coefficient in formula for partial error accumulation we can compare different algorithms, study frequency properties of algorithms and make conclusion about its applicability in multiprobe microwave multimeter for passing power and reflection coefficient definition. It was proved that solutions (i.e. algorithms) are the same for analytic and numerical (least square) method. So it is possible apply least square solution for three equation with three unknown variable.

A perturbation result concerning the upper and lower bounds on relative errors of solutions to linear algebraic equations is given under any norm for consistent systems and under the Euclidean norm for general systems.

This paper gives some closed-form formulas for computing the maximal and minimal ranks and inertias of P−X with respect to X, where P∈CHn is given, and X is a Hermitian least squares solution to the matrix equation AXB=C. We derive, as applications, necessary and sufficient conditions for X⩾(⩽,>

This paper investigates the plane problem of a frictional receding contact formed between an elastic functionally graded layer and a homogeneous half space, when they are pressed against each other. The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson's ratio. A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction. The entire contact zone thus formed between the layer and the homogeneous medium can transmit both normal and tangential traction. It is assumed that the contact region is under sliding contact conditions with the Coulomb's law used to relate the tangential traction to the normal component. Employing Fourier integral transforms and applying the necessary boundary conditions, the plane elasticity equations are reduced to a singular integral equation in which the unknowns are the contact pressure and the receding contact lengths. Ensuring mechanical equilibrium is an indispensable requirement warranted by the physics of the problem and therefore the global force and moment equilibrium conditions for the layer are supplemented to solve the problem. The Gauss-Chebyshev quadrature-collocation method is adopted to convert the singular integral equation to a set of overdetermined algebraic equations. This system is solved using a least squares method coupled with a novel iterative procedure to ensure that the force and moment equilibrium conditions are satisfied simultaneously. The main objective of this paper is to study the effect of friction coefficient and nonhomogeneity factor on the contact pressure distribution and the size of the contact region. (C) 2014 Elsevier Ltd. All rights reserved., The main author is grateful for the support of Texas A&M University at Qatar and also for the funding provided to their laboratory by the Tunisian Ministry of Higher Education and Scientific Research. The third author is supported by a Fulbright Visiting Scholar Program Award to the University of Arizona.