General Error Analysis Research Articles

Petrological and geochemical mass-balance equations: an algorithm for least-square fitting and general error analysis

Least-square fitting of petrological and geochemical overdetermined linear systems is carried out by means of a rapidly converging algorithm which allows a general treatment of coefficient uncertainties. Both constrained and unconstrained solutions are presented. The modal analysis of lunar rock 10022 and stoichiometric coefficients for two coronitic reactions are so determined, as an example, from application of the proposed procedure to available chemical analyses.

Error Analysis of Euler Angle Transformations

Error analyses of Euler angle transformations arise in the design of precision pointing systems, guidance systems and other systems containing gimbals. The generalized problem, including nonorthogonality of nominally orthogonal coordinate axes as well as errors in the Euler angles, is treated. The usual approach by tedious matrix techniques is simplified to yield a vector solution by application of a similarity transformation to the skew- symmetric error matrices. A vector solution is obtained directly by invoking the vector property of infinitesimal rotations. Piograms, symbolic representations of Euler angle transformations, are used to formulate the problem and to develop the vector solution. EQUENCES of angular rotations are universally used in the analysis of rigid-body dynamics. Examples can be widely found in analysis of precision pointing systems and aircraft, missiles, and space vehicles. As shown by Euler, a minimum of three rotations are required to specify the relative orientation of two orthogonal coordinate systems with arbitrary attitudes. Any physical realization of coordinate transformations, as with gimbals, for example, will introduce angular errors so that each coordinate axis is perturbed from its idealized position. The most general error analysis of an Euler angle sequence must include the effects of nonorthogonal ity of the nominally orthogonal coordinate axes as well as errors in the angles of rotation. An example problem is presented to illustrate the various approaches available for error analysis of an Euler angle sequence. The usual but tedious solution by matrix techniques is given, A vector solution is obtained by applying a similarity transformation to the matrix solution. In a second approach a vector solution to the problem is obtained directly by invoking the vector property of infinitesimal rotations. Piograms, symbolic representations of rotational coordinate transformations,1'2 are used to define the problem and are shown to give the vector solution in a particularly compact way. Three different Euler angle sequences are commonly used in the literature. One of these three is used as the example problem in this paper. The method of solution presented in the paper is applicable to any of the twelve possible three-angle Euler sequences. The method is applicable to rotational sequences of any length by an obvious extension of the technique.

Methods of Series Analysis. I. Comparison of Current Methods Used in the Theory of Critical Phenomena

We survey the principal types of methods of series analysis which have been used in the study of critical phenomena with a view to determining their accuracy and applicability to the treatment of the critical-point singularities. These methods include the ratio method and its variants, such as the Neville-table method; the Pad\'e-approximant procedures; and the procedures based on the generalized-Polya theorems of Thompson et al. We show that the actual procedures of Thompson et al. are mathematically equivalent to certain of the Pad\'e-approximant procedures. We give a general error analysis for the series-analysis procedures and derive a relation between the expected magnitude of the errors in the parameters of $A{(1\ensuremath{-}yx)}^{\ensuremath{-}\ensuremath{\gamma}}$, namely, $\ensuremath{\Delta}y:\ensuremath{\Delta}\ensuremath{\gamma}:\ensuremath{\Delta}A$ as $1:J:J \mathrm{ln}J$, where $J$ is the order of the last term of the series analyzed. This relation is briefly illustrated by numerical data. We further give procedures for establishing estimates of the magnitude of errors in the parameters that have the same type of validity as those commonly used to determine the accuracy of a truncated Taylor series. We discuss the commonly occurring but anomalous case of "defects" (errant close pole and zero) in Pad\'e-approximant procedures. We show them to be related to Pad\'e's block structure of the approximant table and emphasize the artificial nature of the apparently rapid convergence that they cause. By numerical investigation of many test functions, similar in structure to those believed to appear in problems of critical phenomena, we have illustrated the following conclusions. First, for series where there is only a simple algebraic singularity, closer to the origin and well separated from any other singularity, the ratio method, perhaps with Neville-table improvement, is the most effective procedure. Second, for series where there are interfering singularities close to the one considered, or where there are singularities either closer than or nearly at the same distance from the origin as the one considered, the Pad\'e procedures are best. Finally, for not exactly representable singularity structures of the type just decscribed, the convergence of even the Pad\'e-approximant procedures are significantly slowed. None of the general methods described does a very impressive job in computing the $\ensuremath{\gamma}$ value if the function is in fact of the form $A{(1\ensuremath{-}yx)}^{\ensuremath{-}\ensuremath{\gamma}}\mathrm{ln}|1\ensuremath{-}yx|$.

A Unified Approach to the Error Analysis of Augmented Dynamically Exact Inertial Navigation Systems

A model for pure inertial navigation systems is defined which describes a wide class of dynamically exact systems for terrestrial and space applications. Based on this model a standard error block diagram is described and it is shown how the conventional simplifying assumptions lead to the specific version used for slow moving terrestrial applications. To efficiently treat augmented inertial systems, a generalized approach to error analysis of estimation systems is introduced. This approach, discussed in an earlier paper, is amplified as it pertains to augmented inertial systems. It is shown how the standard error propagation block diagram can be used to place the augmented inertial navigation system error analysis in the canonical form necessary for a generalized error analysis. Augmentations considered are position, velocity and attitude fixes. Specific examples are given.