Applications of the Newton-Raphson optimization technique to molecular mechanics calculations are discussed. The theory for applying this method is developed in terms of the spectroscopic Wilson S vectors for stretching, bending and torsional internal coordinates. An exact method for calculating the geometries of transition states for interconversion processes is shown to follow directly from t...

We generalize Vieta formula for the product of roots of a polynomial to the multidimensional case. We compute in the group (C∗)n the product of all roots of a system of n polynomial equations with sufficiently general Newton polyhedra. We present two different formulas for this product. In the first formula we use the so called Parshin symbols, in the second formula we use derivatives of the mi...

In this paper, a recursive Gauss-Newton (RGN) algorithm is first developed for adaptive tracking of the amplitude, frequency and phase of a real sinusoid signal in additive white noise. The derived algorithm is then simplified for computational complexity reduction as well as improved with the use of multiple forgetting factor (MFF) technique to provide a flexible way of keeping track of the pa...

Newton-Krylov methods are standard tools for solving nonlinear problems. A common approach is to “lag” the Jacobian when assembly or preconditioner setup is computationally expensive, in exchange for some degradation in the convergence rate and robustness. We show that this degradation may be partially mitigated by using the lagged Jacobian as an initial operator in a quasi-Newton method, which...

This paper presents an overlapped block-parallel Newton method for solving large nonlinear systems. The graph partitioning algorithms are first used to partition the Jacobian into weakly coupled overlapping blocks. Then the simplified Newton iteration is directly performed, with the diagonal blocks and the overlapping solutions assembled in a weighted average way at each iteration. In the algor...

We present MH-MGT, a multivariate technique for sampling from twice-differentiable, log-concave probability density functions. MHMGT is Metropolis-Hastings sampling using asymmetric, multivariate Gaussian proposal functions constructed from Taylor-series expansion of the log-density function. The mean of the Gaussian proposal function represents the full Newton step, and thus MH-MGT is the stoc...

In this paper, we propose two techniques to solve the nonlinear constrained optimization problem in large scale mesh-interconnected system. The first one is a diagram-method-based decomposition technique which decomposes the large scale system into some small subsystems. The second technique is a projected-Jacobi-based parallel dual-type method which can solve the optimization problems in the d...

Methods of interval arithmetic can be used to reliably find with certainty all solutions to nonlinear systems of equations. In such methods, the system is transformed into a linear interval system and a preconditioned interval Gauss-Seidel method may then be used to compute such solution bounds. In this work, a new heuristic for solving polynomial systems is presented, called reordering techniq...

Many problems in image processing are addressed via the minimization of a cost functional. The most prominent optimization technique is the gradient descent, often used due to its simplicity and applicability where other techniques, e.g., those coming from discrete optimization, can not be used. Yet, gradient descent suffers from a slow convergence, and often to just local minima which highly d...