Solution Of Ill-conditioned Systems Research Articles

Gauss method application to solution of ill-conditioned systems of linear algebraic equations

Gauss method application to solution of ill-conditioned systems of linear algebraic equations

Dynamical Newton-Like Methods for Solving Ill-Conditioned Systems of Nonlinear Equations with Applications to Boundary Value Problems

In this paper, a general dynamical method based on the construction of a scalar homotopy function to transform a vector function of Non-Linear Algebraic Equations (NAEs) into a time-dependent scalar function by introducing a fictitious time-like variable is proposed. With the introduction of a transformation matrix, the proposed general dynamical method can be transformed into several dynamical Newton-like methods including the Dynamical Newton Method (DNM), the Dy- namical Jacobian-Inverse Free Method (DJIFM), and the Manifold-Based Expo- nentially Convergent Algorithm (MBECA). From the general dynamical method, we can also derive the conventional Newton method using a certain fictitious time- like function. The formulation presented in this paper demonstrates a variety of flexibility with the use of different transformation matrices to create other possible dynamical methods for solving NAEs. These three dynamical Newton-like meth- ods are then adopted for the solution of ill-conditioned systems of nonlinear equa- tions and applied to boundary value problems. Results reveal that taking advantages of the general dynamical method the proposed three dynamical Newton-like meth- ods can improve the convergence and increase the numerical stability for solving NAEs, especially for the system of nonlinear problems involving ill-conditioned Jacobian or poor initial values which cause convergence problems.

Addendum - Least-Squares Solution of Ill-Conditioned Systems - Part Two

Addendum - Least-Squares Solution of Ill-Conditioned Systems - Part Two

Least squares solution of ill-conditioned systems

view Abstract Citations (4) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Least squares solution of ill-conditioned systems. Branham, R. L., Jr. Abstract In a rediscussion of previous results, observations of minor planets 6 (Hebe), 7 (Iris), 8 (Flora), 9 (Metis), and 15 (Eunomia) are used to determine corrections to a number of astronomical parameters. The methods used are similar to singular value analysis, but directly employ the normal equations previously calculated for a traditional least squares solution. Solutions by individual minor planets and by various subsets of the observational data show much less variation than solutions given by traditional least squares. But there are discrepancies between the determinations of certain unknowns, particularly the equinox correction, its secular variation, and the correction to the precession given by traditional least squares and the approaches outlined in this paper. External evidence indicates that the results from traditional least squares are superior. Publication: The Astronomical Journal Pub Date: October 1979 DOI: 10.1086/112586 Bibcode: 1979AJ.....84.1632B Keywords: Asteroids; Astrometry; Equinoxes; Least Squares Method; Orbit Calculation; Planet Ephemerides; Correlation; Eigenvectors; Error Analysis; Stellar Motions; Lunar and Planetary Exploration; Minor Planets full text sources ADS |