Search

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.

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.

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.