Newton System Research Articles

Primal-Dual Path-Following Algorithms for Determinant Maximization Problems With Linear Matrix Inequalities

Primal-dual path-following algorithms are considered for determinant maximization problem (maxdet-problem). These algorithms apply Newton‘s method to a primal-dual central path equation similar to that in semidefinite programming (SDP) to obtain a Newton system which is then symmetrized to avoid nonsymmetric search direction. Computational aspects of the algorithms are discussed, including Mehrotra-type predictor-corrector variants. Focusing on three different symmetrizations, which leads to what are known as the AHO, H..K..M and NT directions in SDP, numerical results for various classes of maxdet-problem are given. The computational results show that the proposed algorithms are efficient, robust and accurate.

The use of approximate factorization in stiff ODE solvers

We consider implicit integration methods for the numerical solution of stiff initial-value problems. In applying such methods, the implicit relations are usually solved by Newton iteration. However, it often happens that in subintervals of the integration interval the problem is nonstiff or mildly stiff with respect to the stepsize. In these nonstiff subintervals, we do not need the (expensive) Newton iteration process. This motivated us to look for an iteration process that converges in mildly stiff situations and is less costly than Newton iteration. The process we have in mind uses modified Newton iteration as the outer iteration process and a linear solver for solving the linear Newton systems as an inner iteration process. This linear solver is based on an approximate factorization of the Newton system matrix by splitting this matrix into its lower and upper triangular part. The purpose of this paper is to combine fixed point iteration, approximate factorization iteration and Newton iteration into one iteration process for use in initial-value problems where the degree of stiffness is changing during the integration.

Quantum computation, entanglement and state reduction

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results

Primal-dual interior-point path-following methods for semidefinite programming are considered. Several variants are discussed, based on Newton's method applied to three equations: primal feasibility, dual feasibility, and some form of centering condition. The focus is on three such algorithms, called the XZ, XZ+ZX, and Q methods. For the XZ+ZX and Q algorithms, the Newton system is well defined and its Jacobian is nonsingular at the solution, under nondegeneracy assumptions. The associated Schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption. Practical aspects are discussed, including Mehrotra predictor-corrector variants and issues of numerical stability. Compared to the other methods considered, the XZ+ZX method is more robust with respect to its ability to step close to the boundary, converges more rapidly, and achieves higher accuracy.

Solution of nonlinear Poisson-type equations

We consider approximate and truncated Newton methods for solving nonlinear Poisson-type systems of equations with emphasis on vector computers. We have selected two model differential equations for our analysis. The first, after finite difference discretization, yields a nonlinear system with a symmetric positive definite Jacobian matrix, while the second yields a nonsymmetric Jacobian matrix which is not, in general, positive definite. In the case of a symmetric positive definite Jacobian matrix, the linearized Newton systems are solved by (preconditioned) conjugate gradient methods. In particular, we consider incomplete Cholesky factorization in conjunction with the red-black ordering scheme and symmetric diagonal scaling as preconditioners. In the case of a nonsymmetric Jacobian matrix, we construct symmetric positive definite approximations and analyze the effect of these approximations. We develop, and present theory for, a truncated approximate Newton method. This method uses an approximate Jacobian matrix in addition to truncation in the solution of the Newton system. Convergence properties and work reduction of this new method have been found to be very competitive.

A Newton's method scheme for solving free‐surface flow problems

AbstractA Newton's method scheme is described for solving the system of non‐linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy‐driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.

Resolution of Loschmidt's paradox: The origin of irreversible behavior in reversible atomistic dynamics.

We show that Nosroman\ifmmode \bar{e}\else \={e}\fi{} mechanics provides a link between computer simulations of nonequilibrium processes and real-world experiments. Reversible Nose\ifmmode\bar\else\textasciimacron\fi{} equations of motion, when used to constrain non- equilibrium boundary regions, generate stable dissipative behavior within an adjoining bulk sample governed by Newton's equations of motion. Thus, irreversible behavior consistent with the second law of thermodynamics arises from completely reversible microscopic motion. Loschmidt's reversibility paradox is surmounted by this Nose\ifmmode\bar\else\textasciimacron\fi{}-Newton system, because the steady-state nonequilibrium probability density in the many-body phase space is confined to a zero-volume attractor.

A quadratically convergent method for minimizing a sum of euclidean norms

We consider the problem of minimizing a sum of Euclidean norms. $$F(x) = \sum\nolimits_{i = 1}^m {||r_i } (x)||$$ here the residuals {r i(x)} are affine functions fromR n toR 1 (n≥1≥2,m>-2). This arises in a number of applications, including single-and multi-facility location problems. The functionF is, in general, not differentiable atx if at least oner i (x) is zero. Computational methods described in the literature converge quite slowly if the solution is at such a point. We present a new method which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along whichF is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method recently described by Calamai and Conn. The new method has quadratic convergence to a solutionx under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at* x, thenF is differentiable at* x and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, sayr i * x), is zero, then, as the iteration proceeds, the Hessian ofF becomes extremely ill-conditioned. It is proved that this illconditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold (xℛr i (x)=0}. If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence for Newton's method applies. If several residuals are zero at* x, several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results.

Kommentar zu Hegels Logik in seiner "Enzyklopädie" von 1830 (review)

BOOK REVIEWS 317 with scriptural revelation. With most men of letters in the early eighteenth century, Whiston considers Newton's system as delineated in the Principia to be certain, and he utilizes the results of Newton's scientific investigations to corroborate the merely probable testimony of historical revelation. He delineates this view in a book dedicated to his mentor, Newton, and to the rest of the members of the Royal Society-Astronomical Principles of Religion, Natural and Reveal'd (1717). Attfield is correct in his view that secular science (or secular morality or belief in God independently of scripture on purely rational grounds) is compatible with Judeo -Christian theism. But in arguing historically he prises his "secular" authors out of their historical context for the purpose of his own argument and inevitably distorts their positions in the process. He uses his historical characters as if all they represent is a prospectus of positions, a grab-bag of technical concepts and arguments independent of all historical circumstance and particularity. Such semantic atomism is useful to a person making a case for his own philosophical position, which is the best part of what Attfield is doing, but it is dangerous for an accurate historical understanding. Attfield's conclusion about the compatibility of theism with independent reason would stand on its own, and he ought to have written the book as a straightforward essay in the philosophy of religion. But in insisting that he must delineate "in some measure the theories, attitudes and the spirit of the secularisers" he does violence to them and to the way they link reason to their own religion beliefs. By calling them "secularizers," by his definition, he simultaneously distorts their philosophies and begs the question. JAMES E. FORCE University of Kentucky Bernhard Lakebrink. Kommentar zu Hegels Logik in seiner "Enzyklopi~die" yon 183o. Vol. 1, Sein und Wesen. Freiburg and Munich: Karl Alber, 1979. Pp. 453. DM 48.oo. Kant's hermeneutical claim of understanding other philosophical authors "better" than they would understand themselves seems to fail when applied to that philosopher who tried to reverse by his understanding the entire philosophical tradition-Hegel . There are, of course, various interpretations, explanations, or transformations of Hegel's Logic. However, it seems that nearly all interpreters' managed to avoid the most simple requirement of a commentary, to "spell out" (buchstabieren) the text itself. Lakebrink's book, ~which is the first German commentary on Hegel's Logic, tries to fill this gap. "Nirgendwo verlangt zudem ein vorliegender Text ein solches Ausmass an Ann~iherung und Angleichung wie eben hier" (p. 23). This pretension has to justify its own validity with regard to a hermeneutical tradition going back to Hegel's own time. ' The only exception is A. L6onard, Commentairelittoralde la LogiquedeHegel(Paris-Louvain, 1974); see the review in Journal of the Historyof Philosophy 16 (1978):125-28. 2 See B. Lakebrink, HegelsdialektischeOntologieund die Thom.~itischeAnalektik(Cologne, 1955); Die EuropgdscheIdee der Freiheit,vol. 1, HegelsLogik und die Traditionder Selbstbestimmung(Leiden, 1968); and Studien zur MetaphysikHegels(Freiburg, 1969). 318 HISTORY OF PHILOSOPHY The text in question is the Encyclopedia Logic, of 183o, and--at least to a certain extent--the corresponding passages of the Great Logic of 1812-13. The first part of the commentary covers the sections on "Being" (Sein) and "Essence" (Wesen); it is announced that a second volume, on the logic of "Concept" (Begrif]), is to follow. The book begins with a systematic introduction (pp. I 1-96) discussing the current tendencies of topics such as hermeneutics (p. 11), language and thought (p. 2 t), universality and singularity (p. 26), spirit and history (p. 29), the finite and the infinite (p. 46), logic and eschatalogy (p. 56), and the problem of metaphysics with reference to idealism and realism (p. 76). These introductory chapters try to place Hegel's Logic in the tradition of the leading themes of classical philosophy. In this respect, the commentary follows a retrospective line by constructing a possible dialogue between Aristotle, Thomas Aquinas, and Hegel. On the other hand, one may regret that there are only a few points of contact that might facilitate a comparison between Hegel's dialectics and the conceptions of modern...

Introduction des théories de Newton en France au XVIIe siècle avant 1738

THE eighteenth century witnessed the bitter controversies between the Cartesian and the Newtonian conception of the physical world. For decades, the Cartesians tried all the subtleties of logic and science to defend the vortex theory against the views based on universal attraction. But in the end, they had to give way; and the popular defence of Newton's philosophy by Voltaire marked the turning of the tide. It is the epic of these controversies which Prof. Brunet describes for us with a wealth of details giving a scholarly interest to his exposition. The opposition of the Cartesians can be explained by the fact that their master's theory was the first universal explanation of the world, independent of the occult forces which were in favour during previous centuries. The scientific atmosphere of the time was quite at ease with the mechanist conception of Descartes, and loathed any system which had even a vague resemblance to occult qualities. No doubt these controversies help to clear the implications of Newton's system of the dogmatic blemishes which had to be ultimately recognised in Descartes' cosmology. In bringing to light such and other important points, Prof. Brunet has rendered a great service to the history of science. Introduction des théories de Newton en France au XVIIe siècle avant 1738. Prof. Pierre Brunet Par. Pp. vii + 355. (Paris: Albert Blan-chard, 1931.) 55 francs.