Local convergence of the method of pseudolinear equations for quasilinear elliptic boundary-value problems

A variant of the method of pseudolinear equations, an iterative method of solving quasilinear partial differential equations, is described for quasilinear elliptic boundary-value problems of the type -[p₁(u_x)]_x - [p₂(u_y)]_y = f on a bounded simply connected two-dimensional domain D. A theorem on local convergence in C^{2, λ}(D) of this variant, which has constant coefficients, is proved. Three other method of solving quasilinear elliptic boundary-value problems, namely. Newton's method, the Ka?anov method and a variant of the method of successive approximations that has constant coefficients, are briefly discussed. Results of a series of numerical experiments in a finite-difference setting of solving quasilinear Dirichlet problems of the above-mentioned type by the method of pseudolinear equations and these three methods are given. These results show that Newton's method converges for stronger nonlinearities than do the other methods, which, in order thereafter, are the Ka?anov method, the method of pseudolinear equations and, last, the method of successive approximations, which converges only for relatively weak nonlinearities. From fastest to slowest, the methods are: the method of successive approximations, the method of pseudolinear equations, Newton's method, the Ka?anov method.     

The Principia's theory of the motion of the lunar apse

An analysis of Newton's theory of the lunar apsidal motion in the Principia shows an inadequacy for which he attempted to compensate by adjusting his numerical assumptions.     

A faster,more stable method for computing the pth roots of positive definite matrices

An accelerated, more stable generalization of Newton's method for finding matrix pth roots is developed in a form suitable for finding the positive definite pth root of a positive definite matrix. Numerical examples are given and compared with the corresponding Newton iterates.     

A stochastic calculus for continuous N-parameter strong martingales

Let M be a 4N-integrable, real-valued continuous N-parameter strong martingale. Burkholder's inequalities prove to be an adequate tool to control the quadratic oscillations of M and the integral processes associated with it (i.e. multiple 1-stochastic integrals with respect to M and its quadratic variation) such that a 1-stochastic calculus for M can be designed. As the main results of this calculus, several Ito-type formulas are established: one in terms of the integral processes associated with M, another one in terms of the so-called ‘variations’, i.e. stochastic measures which arise as the limits of straightforward and simple approximations by Taylor's formula; finally, a third one which is derived from the first by iterated application of a stochastic version of Green's formula and which may be the strong martingale form of a prototype for general martingales.     

Hermite interpolation and A-stable methods for stiff ordinary differential equations

Hermite interpolation in the form of Newton's divided difference expressions is employed to give a generating function for A-stable difference methods of order 2n. These methods can be used to solve the initial value ordinary differential equation y′=g(y,t), y(a)=η. The extension to higher dimensions is considered, and practical suggestions are given for step size changes and order changes.     

Numerical solution of the Monge–Ampère equation by a Newton's algorithm

We solve numerically the Monge–Ampère equation with periodic boundary condition using a Newton's algorithm. We prove convergence of the algorithm, and present some numerical examples, for which a good approximation is obtained in 10 iterations. To cite this article: G. Loeper, F. Rapetti, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Sur les distances mutuelles d'une chorégraphie à masses distinctes

We state some simple properties of a configuration of N bodies whose masses are not all equal, and whose motion is a ‘choreography’. In such a solution of Newton's equations, the bodies chase each other around the same curve, with the same phase shift between consecutive bodies. It follows from those properties that for any dimension of space, the masses of a choreography are the same for a logarithmic potential. A similar argument shows that the vorticities of a choreography are the same for N vortices which satisfy Helmholtz's equations (Philos. Mag. 33 (1858) 485–512). We prove a more general result for any potential. In particular, for a choreography with distinct masses, the ratio between the smallest and the largest mutual distances is bounded by a constant which does not depend on the masses. To cite this article: M. Celli, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Error structures and parameter estimation

The error calculus based on the theory of Dirichlet forms is an extension of Gauss' approach to error propagation. The aim of this paper is to derive error structures from measurements. The links with Fisher's information lay the foundations of a strong connection with experiment. Here we show that this connection behaves well towards changes of variables and is related to the theory of asymptotic statistics. Finally the study of products permits one to lay the foundation of an infinite dimensional empirical error calculus. To cite this article: N. Bouleau, C. Chorro, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the solution of nonlinear two-point boundary value problems on successively refined grids

The solution of nonlinear two-point boundary value problems by adaptive finite difference methods ordinarily proceeds from a coarse to a fine grid. Grid points are inserted in regions of high spatial activity and the coarse grid solution is then interpolated onto the finer mesh. The resulting nonlinear difference equations are often solved by Newton's method. As the size of the mesh spacing becomes small enough. Newton's method converges with only a few iterations. In this paper we derive an estimate that enables us to determine the size of the critical mesh spacing that assures us that the interpolated solution for a class of two-point boundary value problems will lie in the domain of convergence of Newton's method on the next finer grid. We apply the estimate in the solution of several model problems.     

Differentiability for Green's operators of variational inequalities and applications to the determination of bifurcation points

We study Fréchet differentiability at the origin, in the Hilbert space $\text{L}{}^2\text{(Ω)}$, for the Green's operator P and we apply these results to the calculus of bifurcation points.     

Intersections among Steiner systems

A Steiner system S(l, m, n) is a system of subsets of size m (called blocks) from an n-set S, such that each d-subset from S is contained in precisely one block. Two Steiner systems have intersection k if they share exactly k blocks. The possible intersections among S(5, 6, 12)'s, among S(4, 5, 11)'s, among S(3, 4, 10)'s, and among S(2, 3, 9)'s are determined, together with associated orbits under the action of the automorphism group of an initial Steiner system. The following are results: (i) the maximal number of mutually disjoint S(5, 6, 12)'s is two and any two such pairs are isomorphic; (ii) the maximal number of mutually disjoint S(4, 5, 11)'s is two and any two such pairs are isomorphic; (iii) the maximal number of mutually disjoint S(3, 4, 10)'s is five and any two such sets of five are isomorphic; (iv) a result due to Bays in 1917 that there are exactly two non-isomorphic ways to partition all 3-subsets of a 9-set into seven mutually disjoint S(2, 3, 9)'s.     

Chebyshev-type methods and preconditioning techniques

Recently, a Newton’s iterative method is attracting more and more attention from various fields of science and engineering. This method is generally quadratically convergent. In this paper, some Chebyshev-type methods with the third order convergence are analyzed in detail and used to compute approximate inverse preconditioners for solving the linear system Ax = b. Theoretic analysis and numerical experiments show that Chebyshev’s method is more effective than Newton’s one in the case of constructing approximate inverse preconditioners.     

“Dart calculus” of induced subsets

The paper contains some basic results from “dart calculus” of induced subsets. We obtain as their consequence a negative answer to Hajnal's hypothesis: (m, +1∑i=0n?1(^m_i))→(m?1, 1+∑i?0n?1(^m?1_i)).     

Pelikán's conjecture and cyclotomic cosets

The following conjecture was recently made by J. Pelikán. Let a₀ ,…, a_n be an (n + 1)-tuple of 0's and 1's; let A_k = ?_i=0^n?ka_ia_i+k for k = 0,…, n. Then if n ? 4 some A_k is even.This paper shows that Pelikán's conjecture is false for infinitely many values of n. On the other hand it is also shown that the conjecture is true for most values of n, and a characterization is given of those values of n for which it fails.     

American attempts to solve Newton's pasturage problem

Frederick Emerson's North American Arithmetic contained a “pasturage problem” which baffled his compatriots. Actually, as the Americans discovered forty-two years later, this problem was taken from Isaac Newton's Arithmetica Universalis. The history of this problem illuminates the tradition of standard artificial exercises, the isolation of American mathematics, a chain of mathematical and historical plagiarisms, and changing patterns of arithmetical reasoning.     

On iterative methods for the quadratic matrix equation with M-matrix

In this paper, we consider Newton’s method and Bernoulli’s method for a quadratic matrix equation arising from an overdamped vibrating system. By introducing M-matrix to this equation, we provide a sufficient condition for the existence of the primary solution. Moreover, we show that Newton’s method and Bernoulli’s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition.     

A penalty/Newton/conjugate gradient method for the solution of obstacle problems

Motivated by the search for non-negative solutions of a system of Eikonal equations with Dirichlet boundary conditions, we discuss in this Note a method for the numerical solution of parabolic variational inequality problems for convex sets such as $\text{K={v∣v∈H}{}_0{}^1\text{(Ω)}$, v?ψ a.e. on $\text{Ω}.}$ The numerical methodology combines penalty and Newton's method, the linearized problems being solved by a conjugate gradient algorithm requiring at each iteration the solution of a linear problem for a discrete analogue of the elliptic operator I?μΔ. Numerical experiments show that the resulting method has good convergence properties, even for small values of the penalty parameter. To cite this article: R. Glowinski et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.