24 research outputs found
Improved Computational Performance of a Modified Conjugate Gradient Coefficient for Solving Nonlinear System of Equations
No full textAbubakar Dauda. — “They love us because we give them Zakāt.” The Distribution of Wealth and the Making of Social Relations in Northern Nigeria
No full textThis book is based on a dissertation submitted in partial fulfillment of the requirements for the Ph.D at the Graduate School “Muslim Cultures and Societies” of the Freie Universität of Berlin. It is published as the twenty-third volume of the “Islam in Africa” series inititated by E. J. Brill in 2003. It is a good addition to this series, which had already made a considerable contribution to the understanding of Islam and Muslim societies in Africa. Its author, Dauda Abubakar, analyzes the p..
Finite element modelling of hot rolling of Al-3%Mg and the kinetics of static recrystallisation
No full textThe principal objectives of this work were (i) to investigate the effect of geometry on the through-thickness gradient in microstructure evolved during post-deformation annealing treatment of rolled AI-3%Mg slabs and (ii) to employ the finite element method and empirical equations characterising the rate of static recrystallisation to predict the gradient in the volume fraction recrystallised through the slab thickness. Geometry was characterised by the aspect ratio of the deformation zone. The finite element method was used principally to simulate the evolution and spatial distribution of process parameters namely the strain, strain rate and temperature of deformation through the rolled slab thickness. Standard metallographic techniques and quantitative metallography combined with optical microscopy under polarised light were employed to measure the volume fraction recrystallised. The geometric orientation of microbands developed due to hot deformation was also characterised. Rate-dependent, thermomechanical material constitutive data based on the hyperbolic sine and Voce type of flow stress/strain relationships were used as input in the two-dimensional finite element model together with a friction model based on the penalty method. Published empirical equations linking the rate of static recrystallisation and the process parameters were then employed to predict the measured volume recrystallised. Results indicated that gradients in microstructure occurred through the rolled slab thickness for all the slab/geometry and rolling conditions considered and that the orientation of microbands developed independent of the rolling reduction. The main conclusions drawn were that (i) geometry had a profound effect on the evolution of microstructure through the rolled slab thickness; (ii) the finite element method can be used as an effective tool in the prediction of through-thickness gradient in microstructure evolved in the post-deformation annealing treatment; and (iii) microband development was an important microstructural feature during hot rolling of AI-3%Mg, acting as potential nucleation sites for subsequent static microstructural transformation processes. (author)Available from British Library Document Supply Centre-DSC:DXN045785 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo
The trade-off between the PLSR and PCR methods for modeling data with collinear structure
No full textThis paper investigates the partial least squares regression (PLSR) and principal component regression (PCR) methods as versatile alternative regression techniques when the use of the ordinary least squares method breaks down. Emphasis is more on the situation where the predictor variables are evidently correlated. Data sets with Gaussian non-orthogonal predictor variables were simulated at different sample sizes ranging from 20 to 1000 to examine the performance of the two regression types under varying situations. The data were randomly partitioned into training and test sets with both PLSR and PCR models constructed on the training sets while their performances were evaluated on the test sets using themean square error of predictions and other indices. At each fit of the models, the leave-one-out cross-validation technique was employed to enhance the efficiency and stability of the fitted models. Results from the simulation studies revealed the goodness of the two regression methods but at varying degrees of accuracy. More importantly, it is evident from the results that though, both the PLSR and PCR techniques yielded good regression models, the PLSRtechniqueis consistently more efficient on the test datain terms of good predictions than the PCR method irrespective of sample sizes. Also in terms of model parsimony, the PLSR technique yielded efficient regression models with relatively fewer latent components than the PCR method. Data sets on the performance of M.Sc. graduates from the Department of Statistics, University of Ilorin, Nigeria during the 2012 academic session were used to validate the results from the Monte Carlo studies
Analyzing the transmission dynamics of tuberculosis in Kaduna Metropolis, Nigeria
No full textA mathematical model for the transmission dynamics of tuberculosis in Kaduna metropolis, is formulated and analysed. For the prevalence of the disease, the model was considered in proportions of susceptible, exposed, infectious and recovered compartments. The disease-free equilibrium (DFE) and Endemic Equilibrium (EE) states of the model in proportions were obtained and DFE state was used to compute the basic reproduction number 0, as important threshold whose values allow to establish whether an infection will spread in a population or not. The stability analysis shows that the disease-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number is less than unity using Routh – Hurwitz stability criterion and Lyapunov function respectively. It is further proved using Routh-Hurwitz that the endemic equilibrium state is locally asymptotically stable whenever the basic reproduction number is greater than unity. The computed results of the basic reproduction number 0 estimated to be 1.0623, as well as the stability analysis revealed that tuberculosis infection will remain endemic (persist) in Kaduna metropolis. Furthermore, effective control measures such as expanded and regular immunization campaign will decrease the infection burden
Comparison of change-points in multivariate statistical process control using the performance of Lapage-type (nonparametric)
No full textThe inability of the Shewhart‟s, the EWMA, and the CUSUM, Hotelling‟s T2 and many other control charts to indicate the time of shift poses great problems in production, Medicine, etc. To overcome the problems the need to identify the period of change (shift) in the process becomes inevitable. The study used Lapage-type Change-point (LCP) to detect the simultaneous shift in both mean and variance. In the study we compare the performance of generalized likelihood ratio change-point (GLRCP) a parametric-base with our proposed method (LCP) at different varying start-ups using real life data. We run the data on Normal, Laplace and Lognormal distributions and also Average Run Length (ARL0) to assess the performance of the methods. Evaluating in-control ARLs (IC-ARLs) for each of the methods at change-point 250 and ARL0 500 indicates the same performance irrespective of the start-up value; LCP and GLR methods have rather a similar performance IC-ARLs at change-point 50 and change-point 100 under the normality assumptions, but under non-normal distributions, LCP has substantially higher IC-ARLs compared to GLRCP at 20. The LCP outperformed the GLRCP when applied to children bronchial pneumonia status. We therefore recommend that new method be used in short-run situations and also when underlying distributions are usually unknown
Improved Quasi-Newton method via PSB update for solving systems of nonlinear equations
No full textThe Newton method has some shortcomings which includes computation of the Jacobian matrix which may be difficult or even impossible to compute and solving the Newton system in every iteration. Also, the common setback with some quasi-Newton methods is that they need to compute and store an n × n matrix at each iteration, this is computationally costly for large scale problems. To overcome such drawbacks, an improved Method for solving systems of nonlinear equations via PSB (Powell-Symmetric-Broyden) update is proposed. In the proposed method, the approximate Jacobian inverse Hk of PSB is updated and its efficiency has improved thereby require low memory storage, hence the main aim of this paper. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations
No full textResearch from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear
equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the
years, there has been significant theoretical study to develop methods for solving such systems, despite these efforts,
unfortunately the methods developed do have deficiency. In a contribution to solve systems of the form F (x) = 0, x is
an element of R-n, a derivative free method via the classical Davidon-Fletcher-Powell (DFP) update is presented. This
is achieved by simply approximating the inverse Hessian matrix with Q(k+1)(-1) to theta I-k. The modified method
satisfied the descent condition and possess local superlinear convergence properties. Interestingly, without
computing any derivative, the proposed method never fail to converge throughout the numerical experiments. The
output is based on number of iterations and CPU time, different initial starting points were used on a solve 40
benchmark test problems. With the aid of the squared norm merit function and derivative-free line search technique,
the approach yield a method of solving symmetric systems of nonlinear equations that is capable of significantly
reducing the CPU time and number of iteration, as compared to its counterparts. A comparison between the proposed
method and classical DFP update were made and found that the proposed methodis the top performer and
outperformed the existing method in almost all the cases. In terms of number of iterations, out of the 40 problems
solved, the proposed method solved 38 successfully, (95%) while classical DFP solved 2 problems (i.e. 05%). In terms
of CPU time, the proposed method solved 29 out of the 40 problems given, (i.e. 72.5%) successfully whereas classical
DFP solves 11 (27.5%). The method is valid in terms of derivation, reliable in terms of number of iterations and
accurate in terms of CPU time. Thus, suitable and achived the objective
Derivative Free Conjugate Gradient Method via Broyden's Update for solving symmetric systems of nonlinear equations
No full textThe applications of mathematics in many areas of computing, scientific and engineering research mostly give rise to a
systems of nonlinear equations. Various iterative methods have been developed to solve such equations, this includes
Newton method, Quasi-Newton's etc. Over the years, there has been significant theoretical study on quasi-Newton
methods for solving such systems, but unfortunately the methods suffers setback. To overcome such problems, a
Derivative free Method for Solving Symmetric Systems of Nonlinear Equations Using Broyden's Update is presented.
The modification is achieved by simply approximating the inverse Hessian matrix to with (δ and I represents
acceleration parameter and an identity matrix respectively) without computing any derivative. The method uses the
symmetric structure of the system sufficiently and the generalized classical Broyden's update method for
unconstrained optimization problems. The squared norm merit function is used, both the direction and the line
search technique are derivative-free, this attractive feature of the proposed method makes it to have a very low storage
requirement thereby solving large scale problems successfully. In an effort to solve nonlinear problems of the form F(x)
= 0, 0, x ∈ R different initial starting points were used on a set of benchmark test problems, the output is based on
number of iterations and CPU time. A comparison between the proposed method and the classical methods were
made and found that the proposed method is efficient, robust and outperformed the existing method
Derivative free Davidon-Fletcher-Powell (DFP) for solving symmetric systems of nonlinear equations
No full textResearch from the work of engineers, economist, modelling, industry, computing, and scientist are mostly nonlinear
equations in nature. Numerical solution to such systems is widely applied in those areas of mathematics. Over the
years, there has been significant theoretical study to develop methods for solving such systems, despite these efforts,
unfortunately the methods developed do have deficiency. In a contribution to solve systems of the form F (x) = 0, x is
an element of R-n, a derivative free method via the classical Davidon-Fletcher-Powell (DFP) update is presented. This
is achieved by simply approximating the inverse Hessian matrix with Q(k+1)(-1) to theta I-k. The modified method
satisfied the descent condition and possess local superlinear convergence properties. Interestingly, without
computing any derivative, the proposed method never fail to converge throughout the numerical experiments. The
output is based on number of iterations and CPU time, different initial starting points were used on a solve 40
benchmark test problems. With the aid of the squared norm merit function and derivative-free line search technique,
the approach yield a method of solving symmetric systems of nonlinear equations that is capable of significantly
reducing the CPU time and number of iteration, as compared to its counterparts. A comparison between the proposed
method and classical DFP update were made and found that the proposed methodis the top performer and
outperformed the existing method in almost all the cases. In terms of number of iterations, out of the 40 problems
solved, the proposed method solved 38 successfully, (95%) while classical DFP solved 2 problems (i.e. 05%). In terms
of CPU time, the proposed method solved 29 out of the 40 problems given, (i.e. 72.5%) successfully whereas classical
DFP solves 11 (27.5%). The method is valid in terms of derivation, reliable in terms of number of iterations and
accurate in terms of CPU time. Thus, suitable and achived the objective
