Abstract: In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.
Abstract: An inversion-free iterative algorithm is presented for
solving nonlinear matrix equation with a stepsize parameter t. The
existence of the maximal solution is discussed in detail, and the
method for finding it is proposed. Finally, two numerical examples
are reported that show the efficiency of the method.
Abstract: In this paper, the nonlinear matrix equation is investigated. Based on the fixed-point theory, the boundary and the existence of the solution with the case r>-δi are discussed. An algorithm that avoids matrix inversion with the case -1
Abstract: The generalized Higham matrix is a complex symmetric matrix A = B + iC, where both B ∈ Cn×n and C ∈ Cn×n are Hermitian positive definite, and i = √−1 is the imaginary unit. The growth factor in Gaussian elimination is less than 3√2 for this kind of matrices. In this paper, we give a new brief proof on this result by different techniques, which can be understood very easily, and obtain some new findings.
Abstract: The traditional second order statistics approach of using only the hermitian covariance for non circular signals, does not take advantage of the information contained in the complementary covariance of these signals. Radar systems often use non circular signals such as Binary Phase Shift Keying (BPSK) signals. Their noncicular property can be exploited together with the dual centrosymmetry of the bistatic MIMO radar system to improve angle estimation performance. We construct an augmented matrix from the received data vectors using both the positive definite hermitian covariance matrix and the complementary covariance matrix. The Unitary ESPRIT technique is then applied to the signal subspace of the augmented covariance matrix for automatically paired Direction-of-arrival (DOA) and Direction-of-Departure (DOD) angle estimates. The number of targets that can be detected is twice that obtainable with the conventional ESPRIT approach. Simulation results show the effectiveness of this method in terms of increase in resolution and the number of targets that can be detected.
Abstract: Turbulence of the incoming wind field is of paramount
importance to the dynamic response of civil engineering structures. Hence reliable stochastic models of the turbulence should be available from which time series can be generated for dynamic response and
structural safety analysis. In the paper an empirical cross spectral
density function for the along-wind turbulence component over the wind field area is taken as the starting point. The spectrum is spatially
discretized in terms of a Hermitian cross-spectral density matrix for the turbulence state vector which turns out not to be positive
definite. Since the succeeding state space and ARMA modelling of
the turbulence rely on the positive definiteness of the cross-spectral
density matrix, the problem with the non-positive definiteness of such
matrices is at first addressed and suitable treatments regarding it are proposed. From the adjusted positive definite cross-spectral density
matrix a frequency response matrix is constructed which determines the turbulence vector as a linear filtration of Gaussian white noise.
Finally, an accurate state space modelling method is proposed which allows selection of an appropriate model order, and estimation of a state space model for the vector turbulence process incorporating its phase spectrum in one stage, and its results are compared with a conventional ARMA modelling method.
Abstract: Let F(x, y) = ax2 + bxy + cy2 be a positive definite
binary quadratic form with discriminant Δ whose base points lie on
the line x = -1/m for an integer m ≥ 2, let p be a prime number
and let Fp be a finite field. Let EF : y2 = ax3 + bx2 + cx be an
elliptic curve over Fp and let CF : ax3 + bx2 + cx ≡ 0(mod p) be
the cubic congruence corresponding to F. In this work we consider
some properties of positive definite quadratic forms, elliptic curves
and cubic congruences.
Abstract: In this paper, we present parallel alternating two-stage
methods for solving linear system Ax=b, where A is a symmetric
positive definite matrix. And we give some convergence results of
these methods for nonsingular linear system.
Abstract: Support vector regression (SVR) has been regarded
as a state-of-the-art method for approximation and regression. The
importance of kernel function, which is so-called admissible support
vector kernel (SV kernel) in SVR, has motivated many studies
on its composition. The Gaussian kernel (RBF) is regarded as a
“best" choice of SV kernel used by non-expert in SVR, whereas
there is no evidence, except for its superior performance on some
practical applications, to prove the statement. Its well-known that
reproducing kernel (R.K) is also a SV kernel which possesses many
important properties, e.g. positive definiteness, reproducing property
and composing complex R.K by simpler ones. However, there are a
limited number of R.Ks with explicit forms and consequently few
quantitative comparison studies in practice. In this paper, two R.Ks,
i.e. SV kernels, composed by the sum and product of a translation
invariant kernel in a Sobolev space are proposed. An exploratory
study on the performance of SVR based general R.K is presented
through a systematic comparison to that of RBF using multiple
criteria and synthetic problems. The results show that the R.K is
an equivalent or even better SV kernel than RBF for the problems
with more input variables (more than 5, especially more than 10) and
higher nonlinearity.
Abstract: The real representation of the quaternionic matrix is
definited and studied. The relations between the positive (semi)define
quaternionic matrix and its real representation matrix are presented.
By means of the real representation, the relation between the positive
(semi)definite solutions of quaternionic matrix equations and those of
corresponding real matrix equations is established.
Abstract: Kernel function, which allows the formulation of nonlinear variants of any algorithm that can be cast in terms of dot products, makes the Support Vector Machines (SVM) have been successfully applied in many fields, e.g. classification and regression. The importance of kernel has motivated many studies on its composition. It-s well-known that reproducing kernel (R.K) is a useful kernel function which possesses many properties, e.g. positive definiteness, reproducing property and composing complex R.K by simple operation. There are two popular ways to compute the R.K with explicit form. One is to construct and solve a specific differential equation with boundary value whose handicap is incapable of obtaining a unified form of R.K. The other is using a piecewise integral of the Green function associated with a differential operator L. The latter benefits the computation of a R.K with a unified explicit form and theoretical analysis, whereas there are relatively later studies and fewer practical computations. In this paper, a new algorithm for computing a R.K is presented. It can obtain the unified explicit form of R.K in general reproducing kernel Hilbert space. It avoids constructing and solving the complex differential equations manually and benefits an automatic, flexible and rigorous computation for more general RKHS. In order to validate that the R.K computed by the algorithm can be used in SVM well, some illustrative examples and a comparison between R.K and Gaussian kernel (RBF) in support vector regression are presented. The result shows that the performance of R.K is close or slightly superior to that of RBF.
Abstract: Intelligent systems based on machine learning
techniques, such as classification, clustering, are gaining wide spread
popularity in real world applications. This paper presents work on
developing a software system for predicting crop yield, for example
oil-palm yield, from climate and plantation data. At the core of our
system is a method for unsupervised partitioning of data for finding
spatio-temporal patterns in climate data using kernel methods which
offer strength to deal with complex data. This work gets inspiration
from the notion that a non-linear data transformation into some high
dimensional feature space increases the possibility of linear
separability of the patterns in the transformed space. Therefore, it
simplifies exploration of the associated structure in the data. Kernel
methods implicitly perform a non-linear mapping of the input data
into a high dimensional feature space by replacing the inner products
with an appropriate positive definite function. In this paper we
present a robust weighted kernel k-means algorithm incorporating
spatial constraints for clustering the data. The proposed algorithm
can effectively handle noise, outliers and auto-correlation in the
spatial data, for effective and efficient data analysis by exploring
patterns and structures in the data, and thus can be used for
predicting oil-palm yield by analyzing various factors affecting the
yield.
Abstract: From a set of shifted, blurred, and decimated image , super-resolution image reconstruction can get a high-resolution image. So it has become an active research branch in the field of image restoration. In general, super-resolution image restoration is an ill-posed problem. Prior knowledge about the image can be combined to make the problem well-posed, which contributes to some regularization methods. In the regularization methods at present, however, regularization parameter was selected by experience in some cases and other techniques have too heavy computation cost for computing the parameter. In this paper, we construct a new super-resolution algorithm by transforming the solving of the System stem Є=An into the solving of the equations X+A*X-1A=I , and propose an inverse iterative method.