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In this paper, we present a novel numerical algorithm to handle system of second-order, three-point boundary value problems in a favorable reproducing kernel Hilbert space. The exact and approximate solutions of the system are given in a series form in the space W23 [0,1] with easily computable components using symbolic computation software. Numerical simulations are performed to guarantee the procedure, and to verify the theoretical statements. The numerical results demonstrate that the algorithm is higher accurate and efficient in obtaining series solutions for system of second-order, three-point boundary value problems.In this paper, we present a novel numerical algorithm to handle system of second-order, three-point boundary value problems in a favorable reproducing kernel Hilbert space. The exact and approximate solutions of the system are given in a series form in the space W23 [0,1] with easily computable components using symbolic computation software. Numerical simulations are performed to guarantee the procedure, and to verify the theoretical statements. The numerical results demonstrate that the algorithm is higher accurate and efficient in obtaining series solutions for system of second-order, three-point boundary value problems.

This paper is devoted to the so-called semianalytical structural analysis based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). Boundary problems of three-dimensional theory of elasticity are under consideration. In accordance with the method of extended domain, the given domain is embordered by extended one. The field of application of DCFEM comprises structures with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM implies finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. Corresponding discrete and discrete-continual approximation models for subdomains and coupled multilevel approximation model for extended domain are under consideration. The paper presents brief review of software and verification samples.

Curve reconstructions are widely been used in reverse engineering problems. The reverse engineering process is the creation process of computer model from its original real-life model or its prototypes. The purpose of the reverse engineering is to improve the visualization of two-dimensional data from a series of data point. This paper presents a curve fitting of cubic Bezier curve with parameter optimization by using Differential Evolution. In this research, differential evolution algorithm is used to optimize the parametric value t associated with each point so that the distance between the original images and the parametric curve is minimize. In addition, Sum Square Error (SSE) has been used to calculate the distance between the original images and the parametric curve. The proposed technique that we used in this paper provides the comparison of numerical result to solve curve fitting problem.

The aim of this paper is to study the fuzzy logistic equation. In this paper, the problem of best approximation is considered for fuzzy function, by interpolation to obtain a polynomials. This problem is solved numerically using the extended Runge-Kutta-like fourth order method which use less function evaluations. The extended Runge-kutta-like fourth order method will be briefly described. Polynomial operation with degenerate interpolation will eventually involve Vandermonde matrices. Finally, this paper shows that fuzzy logistic equation gives acceptable result.

In this paper, we propose an algorithm of two-point block method to solve the nonlinear system of third-order boundary value problems directly. The proposed method is presented in a simple form of Adams type and two approximate solutions will be obtained simultaneously with the block method using variable step size strategy. The method will be implemented with the multiple shooting technique via the three-step iterative method to generate the missing initial value. Most of the existence method will reduce the third-order boundary value problems to a system of first order equations where the systems of six equations need to be solved. The method we proposed in this paper will solve the third-order boundary value problems directly. Two numerical examples are given to illustrate the efficiency of the proposed method.

This paper is devoted to a general approach to the parameters estimation of dynamical systems in the case of incomplete experimental data. We analyze a particular case of linear dependence of reduced equation on a group of unknown coefficients which is especially important for different applications. Considered are two typical examples of chemical kinetics equations: the Lotka-Volterra system and canonical reaction of the enzyme synthesis. The methodology of parameters estimation discussed in this paper is based on the solution of overdetermined system using one of minimization method (for example the least squares method). It should be noted that for determination of the unknown parameters we use the reduced equations but not their solutions which in many cases do not exist in the closed form. All the discussed theoretical methods are illustrated with numerical examples which facilitate their practical application.

Resequencing is a common issue in information systems where the order of jobs upon arrival has to be preserved upon departure. In this paper M/M/n/∞ queueing system with reordering buffer of infinite capacity is being considered. Focus is given to the study of joint stationary distribution of the total number of customers in queue and total number of customers in reordering buffer. Using developed analytical method there was obtained system of equilibrium equations which is presented in the paper allows recursive computation of joint stationary probabilities. Numerical examples illustrating obtained results are given.

This work’s content finds its root in the material given in the first three companion papers on the very newly proposed method called “Separate Node Ascending Derivatives Expansion (SNADE) for Univariate Functions”. Those three and the present companion papers are to appear in this proceedings. This work focuses on the determination of the nodal values which make the Euclidean distance between the target function and the SNADE truncation polynomial under consideration. The minimization procedure uses certain elements of the mathematical fluctuation theory. We obtain nonlinear equations after the minimization of the Euclidean distance mentioned above. The solutions of these equations can be numerically obtained unless the target function has a very specific structure. This is a so-called “baby age” theory and needs very specific care for robustness and sophistication. Here, the purpose is just formalism and conceptuality. Practicality has been left to future works.

The paper presents one important and interesting property of strictly convex functions. On the base of previously proved properties of these functions, we can generalize some global-geometric conclusions, without the even assumption of their differentiability, and also point out some geometric characteristics of the sets of all solutions of parametric optimization problems. Two specific nonlinear parametric optimizations of nonlinear problems – the minimization and the maximization problems – exist for them. They express a pair of dual optimization problems. The set of all optimal points has an important geometric interpretation. This paper connects the proved properties of a specific case and a geometric sense.

In this paper, we focus on the inverse problem associated with IFSP: Given a target measure μ, find an IFSP (w,p), such that its invariantmeasure μ is sufficiently close to μ in Monge-Kantorovich distance, i.e., dMK(μ,μ) < e. We extend themoment-basedmethod developed by Forte and Vrscay (1995) along two different directions. First, we search for a set of probabilities pi that not only minimizes the collage error but also maximizes the entropy of the iterated function system. Second, we include an extra term in the minimization process which takes into account the sparsity of the set of probabilities. In our new formulations, the minimization of collage error (for moments) can be understood as amulti-criteria problem: At least three different and conflicting criteriamust be considered, i.e., collage error, entropy and sparsity. In this paper, we employ scalarization to reduce the multi-criteria program to a single-criteria program by combining all objective functions with different trade-off weights. The...

This paper is concerned with the uniqueness of the Maximum a Posteriori estimate for restoration problems of data corrupted by Poisson noise, when we have to minimize a combination of the generalized Kullback‐Leibler divergence and a regularization penalty function. The aim of this paper is to prove the uniqueness result for 2D and 3D problems for several penalty functions, such as an edge preserving functional, a simple case of the class of Markov Random Field (MRF) regularization functionals and the classical Tikhonov regularization.

In this paper, we present a new technique for non linear reduced order system known as PWP (Piecewise Polynomial) which is based on the combination between polynomial representation and Trajectory Piecewise linear approach TPWL. The non linear system is initially approximated in a series of polynomial representation at several points xi. Every representation is then reduced with the TPWL approach. This technique offers a good approximation for both weak and strong non linear systems. Ultimately, we end up the current paper with the study of stability and introduce a stability condition for the reduced system.

In this paper it is investigated if a Jacobian‐free Newton‐Krylov (JFNK) solution method can speed up unsteady flow computations at high Reynolds numbers. Preconditioning of the linear systems that arise after Newton linearization is commonly performed with matrix‐free preconditioners or approximate factorizations based on crude approximations of the Jacobian. Approximate factorizations based on a Jacobian that matches the target residual operator are unpopular because these preconditioners consume a large amount of memory and can suffer from robustness issues. However, these preconditioners remain appealing because they closely resemble A−1.In this paper it is shown that a JFNK solution method with an approximate factorization preconditioner based on a Jacobian that approximately matches the target residual operator enables a speed up of a factor 2.5 to 12 over nonlinear multigrid for two dimensional high Reynolds number flows. The solution method performs equally well as nonlinear multigrid for three di...

GMBACK is a Krylov solver for large linear systems, which is based on backward error minimization properties. The minimum backward error is guaranteed (in exact arithmetic) to decrease when the subspace dimension is increased. In this paper we consider two test problems which lead to nonlinear systems which we solve by the Newton‐GMBACK. We notice that in floating point arithmetic the mentioned property does not longer hold; this leads to nonmonotone behavior of the errors, as reported in a previous paper. We also propose a remedy, which solves this drawback.

A long standing mystery in using Maximum Entropy (MaxEnt) is how to deal with constraints whose values are uncertain. This situation arises when constraint values are estimated from data, because of finite sample sizes. One approach to this problem, advocated by E.T. Jaynes [1], is to ignore this uncertainty, and treat the empirically observed values as exact. We refer to this as the classic MaxEnt approach. Classic MaxEnt gives point probabilities (subject to the given constraints), rather than probability densities. We develop an alternative approach that assumes that the uncertain constraint values are represented by a probability density {e.g: a Gaussian), and this uncertainty yields a MaxEnt posterior probability density. That is, the classic MaxEnt point probabilities are regarded as a multidimensional function of the given constraint values, and uncertainty on these values is transmitted through the MaxEnt function to give uncertainty over the MaXEnt probabilities. We illustrate this approach by explicitly calculating the generalized MaxEnt density for a simple but common case, then show how this can be extended numerically to the general case. This paper expands the generalized MaxEnt concept introduced in a previous paper [3].

In this paper a third eddy current benchmark problem is considered. The objective of the eddy current benchmark problem is to determine optimal operating frequency and size of the pancake coil designated for testing tubes made of Inconel. The problem considered in this paper is a true three‐dimensional one and should be modeled using numerical methods, e.g. finite element method, however, because the solution times involved are very large, we present a simplified analytical approach. Moreover, analytical solutions are useful for determining the efficiency of numerical methods and for a qualitative presentation of the role of various system parameters.

Nonparametric Bayesian approaches to density estimation (“Bayesian field theories”) have typically to be solved numerically on a lattice. This is often numerically quite expensive. The Paper wants to show that such numerical calculations are nowadays feasible for some interesting problem classes. In particular, Bayesian field theories are defined by 1. a likelihood model, being a probabilistic description of the measurement process of observational data. and 2. a prior model, determining the generalization behavior of the theory by implementing available a priori knowledge. In this Paper some variations of prior and likelihood models are discussed: First, the implementation of approximate symmetries with Gaussian process priors is demonstrated for approximate periodic and for approximate fractal functions. Second, besides a discussion of the classical likelihood models of general density estimation and regression, special emphasis is put on the likelihood model of quantum theory to treat the inverse probl...

This paper considers the problem of the numerical solution of continuous system equations when the excitation is white noise. Many signal generation models consist of a linear system excited by white noise, and in simulating the performance of these systems it is usually necessary to solve the system differential equations numerically. The simplest representation of white noise is to represent it by independent random samples at the discrete time instants used in the numerical process. However it is shown in this paper that this is not appropriate when the time step is further subdivided in the numerical integration process, and doing so can lead to significant errors in the solution. Some simple examples are included to illustrate the difficulties.

We focus on the numerical analysis of a coupled queueing system with Poisson arrivals and exponentially distributed service times. Such a system consists of multiple queues served by a single server. Service is synchronised meaning that there is a departure from every queue upon service completion and there is no service whenever one of the queues is empty. It was shown before that the terms in the Maclaurin series expansion of the steady-state distribution of this queueing system when the service rate is sent to 0 (overload) can be calculated efficiently. In the present paper we extend this approach to lower loads. We focus on a sequence of Taylor series expansions of the stationary distribution around increasing service rates. For each series expansion, we use Jacobi iteration to calculate the terms in the series expansion where the initial solution is the approximation found by the preceding series expansion. As the generator matrix of the queueing system at hand is sparse, the numerical complexity of a single Jacobi iteration is O(NMK), where N is the order of the series expansion, K is the number of queues and M is the size of the state space. Having a good initial solution reduces the number of Jacobi iterations considerably, meaning that we can find a sequence of good approximations of the steady state probabilities fast.

In this paper, we proposed a new scheme for simulating water flows under shallow water assumption. The method is an extension of traditional shallow water equations. In contrast to traditional methods, we design a dynamic coordinate system for modeling in order to efficiently simulate water flows. Within this system, we derive our specialized shallow water equations directly from the Navier-Stockes equation. At the same time, we develop an implicit mechanism for solving the advection term and a vector projection operator for solving the external forces acting on water. We also present a two-way coupling method for simulating the interaction between water and rigid solid. The experimental results show that the proposed scheme can achieve a more realistic and accurate water model compared with the traditional methods, especially when the solid surfaces are too steep. Also we demonstrate the efficiency of our method in several scenes, all run at least 50 frames per second on average which allows real-time si...

Maximum likelihood estimates in GLMM are often difficult to be obtained since the calculation involves high dimensional integrals. It is not easy to find analytical solutions for the integral so that the approximation approach is needed. In this paper, we discuss several approximation methods to solve high dimension integrals including the Laplace, Penalized Quasi likelihood (PQL) and Adaptive Gaussian Quadrature (AGQ) approximations. The performance of these methods was evaluated through simulation studies. The ‘true’ parameter in the simulation was set to be similar with parameter estimates obtained by analyzing a real data, particularly salamander data (McCullagh & Nelder, 1989). The simulation results showed that the Laplace approximation produced better estimates when compared to PQL and AGQ approximations in terms of their relative biases and mean square errors.

In this paper, we simulate some of the benchmark problems proposed by the World Federation of Nondestructive Evaluation Centers (WFNDEC) using a full wave simulation model based on accurate solutions to the boundary integral equations for ultrasonic scattering. Much of the previous work on modeling these problems relied on the Kirch-hoff approximation to find the scattered fields from defects. Here we instead use a numerical method, called the Nystrom method, for finding the scattered fields more accurately by solving the boundary integral equations of scattering. We compare our model’s predictions with both measurements and Kirchhoff approximation based models. We expect the presented results to serve as a validation of our model as well as a comparison between the Kirchhoff approximation and the Nystrom method.

Short time series prediction technique based on Bernstein polynomials is presented in this paper. Firstly, the straightforward Bernstein polynomial extrapolation scheme is improved by extending the interval of approximation. Secondly, the forecasting scheme is designed in the evolutionary computational setup which is based on the conciliation between the coarseness of the algebraic prediction and the smoothness of the time average prediction. Computational experiments with the test time series suggest that this time series prediction technique could be applicable for various forecasting applications.

In this paper, a quadratic spline is presented. The boundary conditions are adjusted to get new parameters that are used to get better approximation. Some numerical results are given to demonstrate the advantages and efficiency of the method. The proposed method is more accurate than the traditional quadratic method.

The residual cutting (RC) method has been reported to have superior converging characteristics in numerically solving elliptic partial differential equations. However, its application is limited to linear problems with diagonal-dominant matrices in general, for which convergence of a relaxation method such as SOR is guaranteed. In this study, we propose the generalized residual cutting (GRC) method, which is based on the Krylov subspace and applicable to general unsymmetric linear problems. Also, we perform numerical experiments with various coefficient matrices, and show that the GRC method has some desirable properties such as convergence characteristics and memory usage, in comparison to the conventional RC, BiCGSTAB and GMRES methods.At the request of the author of this paper, a corrigendum was issued on 22 June 2016 to correct an error in Eq. (2) and Eq. (3).

Quasi-Newton method has been widely used in solving unconstrained optimization problems. The popularity of this method is due to the fact that only the gradient of the objective function is required at each iterate. Since second derivatives (Hessian) are not required, quasi-Newton method is sometimes more efficient than the Newton method, especially when the computation of Hessian is expensive. On the other hand, standard quasi-Newton methods required full matrix storage that approximates the (inverse) Hessian. Hence, they may not be suitable to handle problems of large-scale. In this paper, we develop quasi-Newton updating formula diagonally using log-determinant norm such that it satisfies the weaker secant equation. The Lagrange multiplier is approximated using the Newton-Raphson method that is associated with weaker secant relation. An executable code is developed to test the efficiency of the proposed method with some standard conjugate-gradient methods. Numerical results show that the proposed method performs better than the conjugate gradient method.

Two-step method is introduced as an alternative method to classical methods in estimating the drift and diffusion parameters of the Stochastic Differential Equations (SDEs) models. Previous studies indicated that this method provides high percentage of accuracy of the estimated diffusion parameter of Lotka-Volterra model with simulated data. In this paper, a new improvement of two-step method is acquired to avoid the chosen of knots by applying Nadaraya-Watson kernel regression estimator in the first step of this method as a replacement of regression spline with truncated power series basis. The estimated parameters of Bachelier model by using modified two-step method are presented, including comparisons between two different kernel bandwidth methods, namely Asymptotic Mean Integrated Square Error (AMISE) for optimal bandwidth and Maximum Likelihood Cross-Validation (MLCV) technique. The performance of the new proposed method is evaluated with different number of sample sizes by using simulated data. Results indicate high percentage of accuracy of the estimated drift and estimated diffusion parameters of Bachelier model when AMISE for optimal bandwidth is applied compared to MLCV technique.

For Neill mapping function,there are 26 mathematical operations for hydrostatic component and 11 operations for wet components need to be carried out in order to get the mapping function value. The Neill mapping function model for both hydrostatics and wet component are given in a form of continued fraction. In order to obtain fast results, the mapping function model need to be simplified. In this paper, the nonlinear curve fitting is used to simplify Neill Mapping Function model. Selection of model to be simplified is based on the model ability to reach mapping function value as low as 3 degree of elevation angle. To sum of error shows how close the simplified model result with the original model. The number of mathematical operation can be reduced after the process of simplification of the model. The percentage of reduction of operation for simplified model, especially for hydrostatic component is more than 80 percent.

Conference of 42nd Annual Review of Progress in Quantitative Nondestructive Evaluation, QNDE 2015, Incorporating the 6th European-American Workshop on Reliability of NDE ; Conference Date: 26 July 2015 Through 31 July 2015; Conference Code:119404; International audience; In this paper, a numerical solution of WFNDEC 2015 eddy current benchmark is presented. In particular, the Surface Integral Equation (SIE) method has been employed for numerically solving the benchmark problem. The SIE method represent an effective and efficient alternative to standard numerical solver like Finite Element Method (FEM) when electromagnetic fields need to be calculated in problems involving homogeneous media. The formulation of SIE method allows to properly solve the electromagnetic problem by meshing the surface of the media instead to the complete media volume as done in FEM. The surface meshing enables to describe the problem with a smaller number of unknowns with respect to FEM. This property is directly translated in an obvious gain in terms of CPU time efficiency.

Considering the growth of interest in understanding flow phenomena in rotational machines, computational fluid dynamics (CFD) is a powerful tool to reach this goal. Especially unsteady simulations are becoming a focus of interest. Nevertheless, unsteady simulations require huge computational times and ressources, thus it is necessary to investigate other methods to find more appropriate approaches to model time-periodic cases.For time-periodic flows the time spectral method (TSM) presents an interesting alternative to the regular time marching solvers. The TSM is well-known for computation of compressible time-periodic flows, but applications to incompressible cases are limited. This paper presents an extension of the TSM to incompressible flows. While there have been previous implementations using pressure correction method with an explicit treatment of time coupling, here an implicit treatment is chosen. To increase efficiency and employ a more robust coupling of the individual time instances the moment...

Drying is often related to the food industry as a process of shifting heat and mass inside food, which helps in preserving food. Previous research using a mass transfer equation showed that the results were mostly concerned with the comparison between the simulation model and the experimental data. In this paper, the finite difference method was used to solve a mass equation during drying using different kinds of boundary condition, which are equilibrium and convective boundary conditions. The results of these two models provide a comparison between the analytical and the numerical solution. The result shows a close match between the two solution curves. It is concluded that the two proposed models produce an accurate solution to describe the moisture distribution content during the drying process. This analysis indicates that we have confidence in the behaviour of moisture in the numerical simulation. This result demonstrated that a combined analytical and numerical approach prove that the system is behaving physically. Based on this assumption, the model of mass transfer was extended to include the temperature transfer, and the result shows a similar trend to those presented in the simpler case.

Matter of this paper is the study of the flow and the corresponding heat transfer in a U-shaped heat exchanger. We propose a mathematical model that is formulated as a forced convection problem for incompressible and Newtonian fluids and results in the unsteady Navier-Stokes problem. In order to get a solution, we discretise the equations with both the Finite Elements Method and the Finite Volumes Method. These procedures give rise to a non-symmetric indefinite quadratic system of equations. Thus, three regularisation techniques are proposed to make approximations effective and ideas to compare their results are provided.

The Euler equations have been used to model gas dynamics for decades. They consist of mathematical equations for the conservation of mass, momentum, and energy of the gas. For a large time value, the solution may contain discontinuities, even when the initial condition is smooth. A standard finite volume numerical method is not able to give accurate solutions to the Euler equations around discontinuities. Therefore we solve the Euler equations using an adaptive mesh finite volume method. In this paper, we present a new construction of the adaptive mesh finite volume method with an efficient computation of the refinement indicator. The adaptive method takes action automatically at around places having inaccurate solutions. Inaccurate solutions are reconstructed to reduce the error by refining the mesh locally up to a certain level. On the other hand, if the solution is already accurate, then the mesh is coarsened up to another certain level to minimize computational efforts. We implement the numerical entropy production as the mesh refinement indicator. As a test problem, we take the Sod shock tube problem. Numerical results show that the adaptive method is more promising than the standard one in solving the Euler equations of gas dynamics.

This paper will introduce the maximum likelihood method of estimation for delay differential equation model governed by unknown delay and other parameters of interest followed by a numerical solver approach. As an example we consider the delayed logistic differential equation. A grid based estimation framework is proposed. Our methodology estimates correctly the delay parameter as well as the initial starting value of the dynamical system based on simulation data. The computations have been carried out with help of mathematical software: MATLAB® 8.0 R2012b.

Shallow water flows can be modeled mathematically. One available model is the shallow water equations. Accurate solutions to the shallow water equations are desired. In this paper we solve the one-dimensional shallow water equations for flat topography. We present the performance of three numerical methods. The first is the implicit collocated finite difference method proposed by Crowhurst and Li (the Crowhurst-Li FDM). The second is the explicit collocated Lax-Friedrichs finite volume method (the Lax-Friedrichs FVM). The third is the explicit staggered finite volume method proposed by Stelling and Duinmeijer (the Stelling-Duinmeijer FVM). All three methods are stable. We find that the Lax-Friedrichs FVM is simplest in terms of computation. We also find that the Stelling-Duinmeijer FVM is more accurate in solving dam break problems than the other two methods.

This paper considers the problem of model order reduction of nonlinear affine systems by using Krylov subspace projection. The nonlinear affine system is described with the concepts of differential geometry. The main properties of such systems are shown and the tools for reachability and observability analysis are provided. The main features of the Krylov subspace projection methods are presented and the extension of these methods in the nonlinear case are developed. The presented method uses the concept of nonlinear reachability and observability distributions and implements the Gram-Schmidt orthogonalization procedure on the members of these distributions. The main difference with the linear methods is that the orthogonalization procedure is im-plemented directly on the distribution elements, while in the linear case the orthogonalization is performed iteratively by using the system matrices. The performance of the presented method is demonstrated by numerical examples.

In this paper the reliability function is defined by the stochastic failure rate process with a non negative and right continuous trajectories. Equations for the conditional reliability functions of an object, under assumption that the failure rate is a semi-Markov process with an at most countable state space are derived. A proper theorem is presented. The linear systems of equations for the appropriate Laplace transforms allow to find the reliability functions for the alternating, the Poisson and the Furry-Yule failure rate processes.

Conjugate gradient (CG) methods are one of the tools in optimization. Due to its low computational memory requirement, this method is used in solving several of nonlinear unconstrained optimization problems from designs, economics, physics and engineering. In this paper, a new modification of CG family coefficient (βk) is proposed and posses global convergence under exact line search direction. Numerical experimental results based on the number of iterations and central processing unit (CPU) time show that the new βk performs better than some other well known CG methods under some standard test functions.

Modelling the transport processes in a vadose zone, e.g. modelling contaminant transport or the effect of the soil water regime on changes in soil structure and composition, plays an important role in predicting the reactions of soil biotopes to anthropogenic activity. Water flow is governed by the quasilinear Richards equation. The paper concerns the implementation of a multi-time-step approach for solving a nonlinear Richards equation. When modelling porous media flow with a Richards equation, due to a possible convection dominance and a convergence of a nonlinear solver, a stable finite element approximation requires accurate temporal and spatial integration. The method presented here enables adaptive domain decomposition algorithm together with a multi-time-step treatment of actively changing subdomains.

Semiparametric analysis and rank-based inference for the accelerated failure time model are complicated in the presence of interval censored data. The main difficulty with the existing rank-based methods is that they involve estimating functions with the possibility of multiple roots. In this paper a class of asymptotically normal rank estimators is developed which can be acquired via linear programming for estimating the parameters of the model, and a two-step iterative algorithm is introduced for solving the estimating equations. The proposed inference procedures are assessed through a real example. The results of applying the proposed methodology on the breast cancer data show that the algorithm converges after three iterations, and the estimations of model parameter based on Log-rank and Gehan weight functions are fairly close with small standard errors.

UNBabc mapping function for wet component needs 11 operations to be carried out in order to obtain the mapping function value. This wet component is provided in a form of continued fraction. To obtain faster result faster calculation is required. Therefore, there is a need to simplify the mapping function models. In this paper the method used to simplify non hydrostatic UNBabc mapping function model is nonlinear curve fitting method. This model has been selected to be simplified, due to its ability to achieve mapping function value, down to 2 degree of elevation angle. To show that the simplified model result approach the original model the sum of error are used. The comparison between the original models with the simplified models can be shown by calculating the sum of error. The number of mathematical operation can be reduced from 11 operations to 2 operations after the process of simplification of the model.

The iterative Induced Dimension Reduction (IDR) methods are considered for solving large systems of linear algebraic equations (SLAEs) with nonsingular nonsymmetric matrices. These approaches are investigated by many authors and are charachterized sometimes as the alternative to the classical processes of Krylov type. The key moments of the IDR algorithms consist in the construction of the embedded Sonneveld subspaces, which have the decreasing dimensions and use the orthogonalization to some fixed subspace.Other independent approaches for research and optimization of the iterations are based on the augmented and modified Krylov subspaces by using the aggregation and deflation procedures with present various low rank approximations of the original matrices. The goal of this paper is to show, that IDR method in Sonneveld subspaces present an original interpretation of the modified algorithms in the Krylov subspaces. In particular, such description is given for the multi-preconditioned semi-conjugate direct...