Local Stability of Euler's and Kahan's Methods

Two discretization methods, the forward Euler's method and the Kahan's reflexive method, are compared by looking at the local stabilities of fixed points of a system of differential equations. We explain why forward Euler's method is not as good from the viewpoint of complex analysis. Conformal mappings are used to relate the eigenvalues of the Jacobian matrices of the differential equations system and the resulting difference equations system. The Euler's method will not preserve Hopf bifurcation. The Kahan's method preserves the local stability of the fixed points of the differential equations.     

Some remarks on the Jacobian question

This revised version of Abhyankar's old lecture notes contains the original proof of the Galois case of then-variable Jacobian problem. They also contain proofs for some cases of the 2-variable Jacobian, including the two characteristic pairs case. In addition, proofs of some of the well-known formulas enunciated by Abhyankar are actually written down. These include the Taylor Resultant Formula and the Semigroup Conductor formula for plane curves. The notes are also meant to provide inspiration for applying the expansion theoretic techniques to the Jacobian problem.     

Efficient similarity factorization of rank-1 modifications

For the solution of stiff ODEs by implicit methods one has to solve sequences of algebraic systems, whose Jacobian is the identity minus a multiple of the 'system' Jacobian of the right hand side. The latter can be successively approximated by a new 'two-sided rank-1' formula yielding only a moderate increase in the number of iterations compared to Newton's method. However, so far we have only been able to achieve the reduction in the linear algebra effort per integration step from O (n 3) to O (n 2) as long as the size is kept constant. To obtain an effort of O (n 2) seems to require an update of a similarity reduction of the system Jacobian to Hessenberg form or even a band matrix. As we have found no O (n 2) algorithms for updating such factorizations in literature, we discuss an idea to approximate the update with enough accuracy to ensure rapid convergence of the underlying implicit equation solver. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some thoughts on the Jacobian Conjecture,Part I

In Abhyankar's Purdue lectures of 1971, the bivariate Jacobian Conjecture was settled for the case of two plus epsilon characteristic pairs. In the published version, the epsilon part got left out. Here that omission is taken care of by proving a sharper result.     

An Acceleration Method in the Homotopy Newton's Continuation for Nonlinear Singular Problems

The nonlinear singular problem $f(u)=0$ is considered. Here $f$ is a $C^3$ mapping from $E^n$ to $E^n$. The Jacobian matrix $f'(u)$ is singular at the solution $u^*$ of $f(u)=0$. A new acceleration method in the homotopy Newton's continuation is proposed. The quadratic convergence of the new algorithm is proved. A numerical example is given.     

Estimates for the inverse of tridiagonal matrices arising in boundary-value problems

By considering tridiagonal matrices as three-term recurrence relations with Dirichlet boundary conditions, one can formulate their inverses in terms of Green's functions. This analysis is applied to three-point difference schemes for 1-D problems, and five-point difference schemes for 2-D problems. We derive either an explicit inverse of the Jacobian or a sharp estimate for both uniform and nonuniform grids.     

Newton's method and a surface with Jacobian zero

Suppose a system of nonlinear real equations P(x)=0, where P and x are n-dimensional vectors, is solved by means of the continuous analog of Newton's method. We study the behavior of the method near the surface S with Jacobian zero: S={x¦det P′(x)=0}. A computational strategy is suggested in the case where the method diverges.     

Characterization of Quasiconvexity for Locally Lipschitz Vector-Valued Functions

The aim of the article is to characterize the locally Lipschitz vector-valued functions which are K -quasiconvex with respect to a closed convex cone K in the sense that the sublevel sets are convex. Our criteria are written in terms of a K -quasimonotonicity notion of the generalized directional derivative and of Clarke's generalized Jacobian. This work could be compared to Sach's one in which the author gives necessary and sufficient conditions for a locally Lipschitz map f between two Euclidean spaces to be scalarly K -quasiconvex in the sense that, for any continuous linear form of the nonnegative polar cone K + , the composite function f is quasiconvex.     

Zolotarev's conformal mapping and Chebotarev's problem

Consider the Chebotarev problem of finding a continuum S in the complex plane including some given points such that the logarithmic capacity of S is minimal. In this paper, we give a complete solution of this problem for the case of three given points with the help of Zolotarev's conformal mapping using Jacobian elliptic and theta functions. Moreover, for four given points, some special cases can be treated.     

Modified Newton’s method for systems of nonlinear equations with singular Jacobian

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.     

Error estimate for the modified Newton method with applications to the solution of nonlinear,two-point boundary-value problems

The solution of nonlinear, two-point boundary value problems by Newton's method requires the formation and factorization of a Jacobian matrix at every iteration. For problems in which the cost of performing these operations is a significant part of the cost of the total calculation, it is natural to consider using the modified Newton method. In this paper, we derive an error estimate which enables us to determine an upper bound for the size of the sequence of modified Newton iterates, assuming that the Kantorovich hypotheses are satisfied. As a result, we can efficiently determine when to form a new Jacobian and when to continue the modified Newton algorithm. We apply the result to the solution of several nonlinear, two-point boundary value problems occurring in combustion.     

Modified deflation algorithm for the solution of singular problems. II. Nonlinear multipoint boundary value problems

Various kinds of iterative methods have been proposed for the solution of nonlinear multipoint boundary-value problems MPBVP's. However, it is necessary for these methods that the adjusting matrix, which corresponds to the Jacobian of nonlinear equations, is nonsingular at the solution. In this paper an algorithm for the singular solution of nonlinear MPBVP's, which is an extension of the modified deflation algorithm for the singular root of nonlinear algebraic equations developed by the author is presented. According to the present method, the singular solution can ultimately be reduced to the usual simple solution and both convergency and accuracy can greatly be improved. The effectiveness of the present method is shown by solving two illustrative examples.     

Two classes of multisecant methods for nonlinear acceleration

Many applications in science and engineering lead to models that require solving large‐scale fixed point problems, or equivalently, systems of nonlinear equations. Several successful techniques for handling such problems are based on quasi‐Newton methods that implicitly update the approximate Jacobian or inverse Jacobian to satisfy a certain secant condition. We present two classes of multisecant methods which allow to take into account a variable number of secant equations at each iteration. The first is the Broyden‐like class, of which Broyden's family is a subclass, and Anderson mixing is a particular member. The second class is that of the nonlinear Eirola–Nevanlinna‐type methods. This work was motivated by a problem in electronic structure calculations, whereby a fixed point iteration, known as the self‐consistent field (SCF) iteration, is accelerated by various strategies termed ‘mixing’. Copyright © 2008 John Wiley & Sons, Ltd.     

Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions

Jacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.     

Jacobian smoothing Brown’s method for NCP

Jacobian smoothing Brown’s method for nonlinear complementarity problems (NCP) is studied in this paper. This method is a generalization of classical Brown’s method. It belongs to the class of Jacobian smoothing methods for solving semismooth equations. Local convergence of the proposed method is proved in the case of a strictly complementary solution of NCP. Furthermore, a locally convergent hybrid method for general NCP is introduced. Some numerical experiments are also presented.     

A special newton-type optimization method

The Kuhn–Tucker conditions of an optimization problem with inequality constraints are transformed equivalently into a special nonlinear system of equations T ₀(z) = 0. It is shown that Newton's method for solving this system combines two valuable properties: The local Q-quadratic convergence without assuming the strict complementary slackness condition and the regularity of the Jacobian of T ₀ at a point z under reasonable conditions, so that Newton’s method can be used also far from a Kuhn–Tucker point     

Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L² and H¹ norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes

This a first step to develop a theory of smooth, étale, and unramified morphisms between Noetherian formal schemes. Our main tool is the complete module of differentials, which is, a coherent sheaf whenever the map of formal schemes is of pseudofinite type. Among our results, we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of Noetherian formal schemes is flat and its module of differentials is locally free. The article closes with a version of Zariski's Jacobian criterion.