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Historical development of the Newton-Raphson method

This expository paper traces the development of the Newton-Raphson method for solving nonlinear algebraic equations through the extant notes, letters, and publications of Isaac Newton, Joseph Raphson, and Thomas Simpson. It is shown how Newtons formulation differed from the iterative process of Raphson, and that Simpson was the first to give a general formulation, in terms of fluxional calculus, applicable to nonpolynomial equations. Simpsons extension of the method to systems of equations is exhibited.

Solving N + m nonlinear equations with only m nonlinear variables

We derive a method for solvingN+m nonlinear algebraic equations inN+m unknownsy≠Rm andz≠RN of the formA(y)z+b(y)=0, where the(N+m) × N matrixA(y) and vectorb(y) are continuously differentiable functions ofy alone. By exploiting properties of an orthonormal basis for null(AT(y)) the problem is reduced to solvingm nonlinear equations iny only. These equations are solved by Newtons method inm variables. Details of computational implementation and results are provided.ZusammenfassungEine Methode zur Lösung vonN+m nichtlinearen algebraischen Gleichungen mitN+m Variableny≠Rm undz≠RN non TypA(y)z+b(y)=0, in welchem die(N+m) × m MatrixA(y) und der Vektorb(y) stetig differenzierbare Funktionen der Variableny sind, wird hergeleitet. Durch Verwendung einer orthonormalen Basis im Nullraum vonAT(y) wird das Problem aufm nichtlineare Gleichungen allein mit der Variableny reduziert. Diese Gleichungen werden durch das Newton Verfahren (m Variable) gelöst. Einzelheiten der numerischen Rechnung werden beschrieben.

Relationships between order and efficiency of a class of methods for multiple zeros of polynomials

The behavior of a class of high order methods for solving polynomial equations is examined. It is shown that the number of iterations for local convergence to a multiple zero, to the limits of attainable accuracy, is bounded independent of the multiplicity of the zero, and decreases as the order of the method increases. For the higher order methods, the number of iterations decreases as the multiplicity increases. Computational efficiency as a function of degree, order and multiplicity is investigated, and an effective choice of order is recommended. Numerical examples are provided.

Homeopathic Potentization Based on Nanoscale Domains

OBJECTIVES The objectives of this study were to present a simple descriptive and quantitative model of how high potencies in homeopathy arise. DESIGN The model begins with the mechanochemical production of hydrogen and hydroxyl radicals from water and the electronic stabilization of the resulting nanodomains of water molecules. The life of these domains is initially limited to a few days, but may extend to years when the electromagnetic characteristic of a homeopathic agent is copied onto the domains. This information is transferred between the original agent and the nanodomains, and also between previously imprinted nanodomains and new ones. The differential equations previously used to describe these processes are replaced here by exponential expressions, corresponding to simplified model mechanisms. Magnetic stabilization is also involved, since these long-lived domains apparently require the presence of the geomagnetic field. Our model incorporates this factor in the formation of the long-lived compound. RESULTS Numerical simulation and graphs show that the potentization mechanism can be described quantitatively by a very simplified mechanism. The omitted factors affect only the fine structure of the kinetics. Measurements of pH changes upon absorption of different electromagnetic frequencies indicate that about 400 nanodomains polymerize to form one cooperating unit. Singlet excited states of some compounds lead to dramatic changes in their hydrogen ion dissociation constant, explaining this pH effect and suggesting that homeopathic information is imprinted as higher singlet excited states. CONCLUSIONS A simple description is provided of the process of potentization in homeopathic dilutions. With the exception of minor details, this simple model replicates the results previously obtained from a more complex model. While excited states are short lived in isolated molecules, they become long lived in nanodomains that form coherent cooperative aggregates controlled by the geomagnetic field. These domains either slowly emit biophotons or perform specific biochemical work at their target.

Empirical versus asymptotic rate of convergence of a class of methods for solving a polynomial equation

Abstract Given alternative methods with identical order of convergence for solving the polynomial equation -(z) = 0, the method with the smaller asymptotic error constant might be assumed to be superior in terms of the number of iterations required for convergence. We present empirical evidence for a parameterized class of methods of second order showing that a parameter choice which does not correspond to the minimal asymptotic error constant may nevertheless be superior in practice.

Solving separable nonlinear equations with jacobians of rank deficiency one

Nonlinear systems of equations of the separable form A(y)z + b(y) = 0, with only one nonlinear variable y∈ℝ, can be reduced to a single nonlinear equation in y. We develop a technique for the case in which A(y) has rank deficiency one. The method requires only one LU factorization per iteration and is quadratically convergent. Numerical examples and applications are provided.

Action of Excited State Molecular Networks

Nanodomains are groups of water molecules held together by an electron in an excited state. We investigate the interaction of nanodomains with living matter through acceleration of an enzyme cycle. We formulate a mechanistic model with four enzyme forms in a cycle and three successive phases. In Phase 1 a slowly catalyzing reaction approaches steady state. In Phase 2 the enzyme forms convert to their excited states using nanodomain energy, and a new stationary state is reached. The high rate of excited state energy movement in living systems leads to rapid conversion to the excited state, and the excitation energy needs to be supplied for only a short period. The excited state produces a very fast cycle, which is stable for a much smaller enzyme concentration than needed for the slow cycle. In Phase 3 the excited states decay. These phases are simulated by solving differential equations numerically.

Short-lived intermediates in aspartate aminotransferase systems

The kinetics of the reaction of aspartate aminotransferase with erythro-beta-hydroxy-aspartate, in which rapid mixing is followed (upon reaching a suitable stationary state) by a very fast temperature jump, is numerically simulated. Values for rate constants are used to the extent known, otherwise estimated. It is shown that reaction steps not resolvable by rapid mixing can be resolved by subsequent chemical relaxation. Since several absorption spectra of enzyme complexes overlap, use of a pH-indicator is investigated. When the pH-indicator is coupled to the protonic dissociation of free enzyme, the fast steps are easily detected in the chemical relaxation portion of the simulation. When the pH-indicator is coupled to the protonic dissociation of the (short-lived) quinoid intermediate, protonic dissociation is easily detectable in the stopped flow phase and in the chemical relaxation phase. Such transient protonic dissociation has not been detected experimentally, but is predicted by the simulation. When natural substrates are used, the magnitude of the rate constants makes it unlikely that transient proton dissociation can be detected by stopped flow alone, but a combination of stopped flow with very fast temperature perturbation allows detection of the transient proton through use of a suitable nonbinding pH-indicator. This is demonstrated by simulation for a specific case. Finally, an alternate mechanism is introduced and distinction of its kinetics from that of the original mechanism is demonstrated.

A saxpy formulation for plane rotations

An algorithm for pre- or post-multiplication of a matrix by a plane rotation, using only three vector saxpy operations instead of the four vector operations usually considered necessary, is described. No auxiliary storage for overwriting is required. The method is shown to be numerically stable.

Solving separable nonlinear least squares problems using the QR factorization

Abstract We present a method for solving the separable nonlinear least squares problem min y , z ‖ F ( y , z ) ‖ , where F ( y , z ) ≡ A ( y ) z + b ( y ) with a full rank matrix A ( y ) ∈ R ( N + l ) × N , y ∈ R n , z ∈ R N and the vector b ( y ) ∈ R N + l , with small l ≥ n . We show how this problem can be reduced to a smaller equivalent problem min y ‖ f ( y ) ‖ where the function f has only l components. The reduction technique is based on the existence of a locally differentiable orthonormal basis for the nullspace of A T ( y ) . We use Newton’s method to solve the reduced problem. We show that successive iteration points are independent of the nullspace basis used at any particular iteration point; thus the QR factorization can be used to provide a local basis at each iteration. We show that the first and second derivative terms that arise are easily computed, so quadratic convergence is obtainable even for nonzero residual problems. For the class of problems with N much greater than n and l the main cost per iteration of the method is one QR factorization of A ( y ) . We provide a detailed algorithm and some numerical examples to illustrate the technique.