A quadratically convergent Newton method for computing the polar decomposition of a full-rank matrix is presented and analysed. Acceleration parameters are introduced so as to enhance the initial rate of convergence and it is shown how reliable estimates of the optimal parameters may be computed in practice. To add to the known best approximation property of the unitary polar factor, the Hermit...

In this paper, we propose a new method for finding global optimum of continuous optimization problems, namely Level-Value Estimation algorithm(LVEM). First we define the variance function v(c) and the mean deviation function m(c) with respect to a single variable (the level value c), and both of these functions depend on the optimized function f(x). We verify these functions have some good prop...

Inverse square roots are used in several digital signal processing, multimedia, and scientific computing applications. This paper presents a high-speed method for computing inverse square roots. This method uses a table lookup, operand modification, and multiplication to obtain an initial approximation to the inverse square root. This is followed by a modified Newton-Raphson iteration, consisti...

This paper presents an undelayed solution to the bearing-only simultaneous localization and mapping problem (SLAM). We employ a square-root iterated Kalman filter for nonlinear state estimation. The proposed technique incorporates a modified Kalman update that is equivalent to a variable-step iterative Gauss-Newton method, and is numerically stable because it maintains a square-root decompositi...

This paper considers the extension of a number of passive multiplier theory based results, previously known only for linear time invariant scalar systems, to linear time varying (LTV) multivariable settings. The extensions obtained here have important applications to the stability of both adaptive systems and linear systems in general. We demonstratein this paper that at the heart of the extens...

Use of data assimilation techniques is becoming increasingly common across many application areas. The inverse Hessian (and its square root) plays an important role in several different aspects of these processes. In geophysical and engineering applications, the Hessian-vector product is typically defined by sequential solution of a tangent linear and adjoint problem; for the inverse Hessian, h...

Finding roots of nonlinear algebraic systems is paramount to many applications in applied mathematics because the physical laws of interest are usually nonlinear. If the functions comprising a system are complicated it is beneficial to reduce computing the functions as much as possible. The following describes a quasi-Newton root solver called Broyden Approximate Norm Descent, or ’BroydenAND’, ...

The paper presents (human-oriented) specification and (penand-paper) verification of the square root function. The function implements Newton method and uses a look-up table for initial approximations. Specification is done in terms of total correctness assertions with use of precise arithmetic and the mathematical square root √ . . ., algorithms are presented in pseudo-code with explicit disti...

This paper describes a program, called NEWTON, that finds approximate symbolic solutions to parameterized equations in one variable. N E W T O N derives an ini t ial approximat ion by solving for the dominant term in the equation, or if this fails, by bisection. It refines this approximation by a symbolic version of Newton's method. It tests whether the first Newton iterate lies closer to the s...