New two-step predictor-corrector method with ninth order convergence for solving nonlinear equations

In this paper, we suggest and analyze a new two-step predictor-corrector type iterative method for solving nonlinear equations of the type. This method based on a Halley and Householder iterative method and using predictor corrector technique. The convergence analysis of our method is discussed. It is established that the new method has convergence order nine. Numerical tests show that the new methods are comparable with the well known existing methods and gives better results

A Gradient Algorithm Locally Equivalent to the Em Algorithm

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/146826/1/rssb02037.pd

Globally Convergent Algorithms for Maximum a Posteriori Transmission Tomography

This paper reviews and compares three maximum likelihood algorithms for transmission tomography. One of these algorithms is the EM algorithm, one is based on a convexity argument devised by De Pierro in the context of emission tomography, and one is an ad hoc gradient algorithm. The algorithms enjoy desirable local and global convergence properties and combine gracefully with Bayesian smoothing priors. Preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the EM algorithm. This superiority stems from the larger number of exponentiations required by the EM algorithm. The convex and gradient algorithms are well adapted to parallel computing.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86016/1/Fessler101.pd

Sensitivity analysis and approximation methods for general eigenvalue problems

Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought

Continuation methods for nonlinear equations

Feasibility of overcoming local convergent nature of continuation methods for nonlinear equation

Matrices with non-negative elements

Chapter 1 is a short introductory chapter dealing with some definitions and basic properties of matrices and vectors.In Chapter 2 we introduce a partial order between matrices with real elements. For matrices with non-negative elements our notation' and terminology differ from the usual ones. The casual reader is advised to read section 2.3 before glancing further. The term "Positive Matrix" will be used from now on in the sense of 2.3.In Chapter 3 we consider the "normal form" of a reducible matrix, and some associated sets. We define the "R-functions", a principal tool of investigation in later chapters.Chapter 4 contains some consequences of the partial order "between matrices. Some analogues of the properties of positive matrices and positive numbers are developed.In Chapter 5 we consider "chains of elements" and powers of positive matrices.Chapter 6 contains a resumé of the chief algebraic properties of a matrix that are required in the rest of the thesis. These concern latent roots and latent vectors, sets of "generalized latent vectors", classical canonical submatrices, and principal idempotent and nilpotent elements.In Chapter 7 we describe a method of proving the fundamental properties of positive matrices, which is based on some work by Probenius.In Chapter 8 we review a method due to Wielandt (1950) of proving the basic results for irreducible positive matrices. We give a variant of our own. Lower and upper bounds are found in terms of the elements of the matrix, for the ratios of elements of the strictly positive latent column vector associated with the largest positive latent root of an irreducible positive matrix.In Chapter 9 we deal with "P-matrices". We deduce a large number of algebraic results purely by inspection of positive elements, finally we examine "sets" of latent row vectors.Chapter 10 is the longest chapter. In It we consider the singular matrix A = ρ I - P , where P is a positive matrix and ρ its largest positive latent root. We examine the number of linearly independent latent vectors associated with the latent root 0 (10. C1 ), sets of positive generalized latent vectors (10. 16), and, when the multiplicity of 0 does not exceed three, the classical canonical submatrices associated with 0. Provided we know which are singular when A is in normal form, these questions may be answered by an inspection of the positions of non-zero elements. In general the orders of the classical canonical submatrices associated with 0 can not be settled in this way, though such methods suffice to determine whether they are all equal to 1, (10. 31 ). Finally we consider the principal idempotent and nilpotent elements of A associated with 0, in some special cases.The Bibliography then follows. In the text we give as reference the author's name and the date of publication, thus: Probenius (1912).The Appendix consists of a paper accepted by the Journal of the London Mathematical Society, "An inequality for latent roots applied to determinants with dominant principal diagonal". The theory of matrices with dominant principal diagonal is closely connected with that for positive matrices. In this paper we "rejected" notation and terminology of 2.3

Numerical modelling of dynamical systems in isothermal chemical reactions and morphogenesis

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Mathematical models of isothermal chemical systems in reactor problems and Turing's theory of morphogenesis with an application in sea-shell patterning are studied. The reaction-diffusion systems describing these models are solved numerically. First- and second-order difference schemes are developed, which are economical and reliable in comparison to classical numerical methods. The linearization process decouples the reaction-diffusion equations thereby allowing the use of different time steps for each differential equation, which may be large due to the excellent stability properties of the methods. The methods avoid having to solve a non-linear algebraic system at each time step. The schemes are suitable for implementation on a parallel machine.This study is funded by the University of Dokuz Eylul

Hydrodynamic models of a cepheid atmosphere

A method for including the solution of the transfer equation in a standard Henyey type hydrodynamic code was developed. This modified Henyey method was used in an implicit hydrodynamic code to compute deep envelope models of a classical Cepheid with a period of 12(d) including radiative transfer effects in the optically thin zones. It was found that the velocity gradients in the atmosphere are not responsible for the large microturbulent velocities observed in Cepheids but may be responsible for the occurrence of supersonic microturbulence. It was found that the splitting of the cores of the strong lines is due to shock induced temperature inversions in the line forming region. The adopted light, color, and velocity curves were used to study three methods frequently used to determine the mean radii of Cepheids. It is concluded that an accuracy of 10% is possible only if high quality observations are used