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Normal Equations (normal + equation)
Selected AbstractsA semi-analytical estimation of the effect of second-order ionospheric correction on the GPS positioningGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2005H. Munekane SUMMARY We developed a semi-analytical method to evaluate the effect of the second-order ionospheric correction on GPS positioning. This method is based on the semi-analytical positioning error simulation method developed by Geiger and Santerre in which, assuming the continuous distribution of the satellites, a normal equation is formed to estimate the positioning error taking all the contributions of the ranging error by the visible satellites into account. Our method successfully reproduced the averaged time-series of three IGS sites which is comparable to the rigorous simulation. We then evaluated the effect of the ionospheric error on the determination of the reference frame. We evaluated the additional Helmert parameters that are required for the ionospheric effect. We found that the ionospheric effect can lead to annual scale changes of 0.1 ppb, with an offset of 1.8 mm and a semi-annual oscillation of 1 mm in the z -direction. However, these values are too small to explain the current deviations between the GPS-derived reference frame and the ITRF reference frame. Next, we estimated the apparent scale changes due to the ionospheric error in the GEONET coordinate time-series in Japan. We could qualitatively reproduce the observed semi-annual scale changes peaking at the equinoxes and having asymmetrical amplitudes between the vernal and autumnal equinoxes. [source] An efficient linear programming solver for optimal filter synthesisNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2007Jihong Ren Abstract We consider the problem of l, optimal deconvolution arising in high data-rate communication between integrated circuits. The optimal deconvolver can be found by solving a linear program for which we use Mehrotra's interior-point approach. The critical step is solving the linear system for the normal equations in each iteration. We show that this linear system has a special block structure that can be exploited to obtain a fast solution technique whose overall computational cost depends mostly on the number of design variables, and only linearly on the number of constraints. Numerical experiments validate our findings and illustrate the merits of our approach. Copyright © 2007 John Wiley & Sons, Ltd. [source] Preconditioning CGNE iteration for inverse problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 3 2007H. Egger Abstract The conjugate gradient method applied to the normal equations (CGNE) is known as efficient method for the solution of non-symmetric linear equations. By stopping the iteration according to a discrepancy principle, CGNE can be turned into a regularization method, and thus can be applied to the solution of inverse, in particular, ill-posed problems. We show that CGNE for inverse problems can be further accelerated by preconditioning in Hilbert scales, derive (optimal) convergence rates with respect to data noise, and give tight bounds on the iteration numbers. The theoretical results are illustrated by numerical tests. Copyright © 2007 John Wiley & Sons, Ltd. [source] How to multiply a matrix of normal equations by an arbitrary vector using FFTACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2008Boris V. Strokopytov This paper describes a novel algorithm for multiplying a matrix of normal equations by an arbitrary real vector using the fast Fourier transform technique. The algorithm allows full-matrix least-squares refinement of macromolecular structures without explicit calculation of the normal matrix. The resulting equations have been implemented in a new computer program, FMLSQ. A preliminary version of the program has been tested on several protein structures. The consequences for crystallographic refinement of macromolecules are discussed in detail. [source] A robust formulation of the ensemble Kalman filter,THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 639 2009S. J. Thomas Abstract The ensemble Kalman filter (EnKF) can be interpreted in the more general context of linear regression theory. The recursive filter equations are equivalent to the normal equations for a weighted least-squares estimate that minimizes a quadratic functional. Solving the normal equations is numerically unreliable and subject to large errors when the problem is ill-conditioned. A numerically reliable and efficient algorithm is presented, based on the minimization of an alternative functional. The method relies on orthogonal rotations, is highly parallel and does not ,square' matrices in order to compute the analysis update. Computation of eigenvalue and singular-value decompositions is not required. The algorithm is formulated to process observations serially or in batches and therefore easily handles spatially correlated observation errors. Numerical results are presented for existing algorithms with a hierarchy of models characterized by chaotic dynamics. Under a range of conditions, which may include model error and sampling error, the new algorithm achieves the same or lower mean square errors as the serial Potter and ensemble adjustment Kalman filter (EAKF) algorithms. Published in 2009 by John Wiley and Sons, Ltd. [source] | ||||||||||