(Jl{. n Xn set of real n x n matrices, ßRn set of real vectors with n components) there sometimes occurs the problem of varying the input data A, b within certain tolerances and looking for the set S of the resulting solutions x*. Examples of this problem are Wilkinson's backward analysis when solving linear systems on a computer (Wilkinson 1963) and an input-output model in economics which is ...

Techniques of reliable computing, like interval arithmetic, can be used to guarantee reliable solutions even in the presence of numerical round-off errors. The need to trace bounds for the error function separately can be eliminated using these techniques. In this extended abstract, we focus on presenting how the techniques and algorithms of reliable computing can be applied to the construction...

The hypotenuse function, z = √ x2 + y2, is sometimes included in math library packages. Assuming that it is being computed by a straightforward algorithm, in a binary floating point environment, with round to nearest rounding mode, a sharp roundoff error bound is derived, for arbitrary precision. For IEEE single precision, or higher, the bound implies that |z − z| < 1.222 ulp(z) and |z − z| < 1...

Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth periodic on the angle perturbations of the twist map, bounding the discretized map from above and from below. Using the well known Moser’s theorem we prove t...

In Part I and this Part II of our paper we investigate how extra-precise evaluation of dot products can be used to solve ill-conditioned linear systems rigorously and accurately. In Part I only rounding to nearest is used. In this Part II we improve the results significantly by permitting directed rounding. Linear systems with tolerances in the data are treated, and a comfortable way is describ...

The article presents the general notions and algorithm about the Gauss-Jordan method. An eloquent example is given and the Turbo C program illustrated this method. We conclude that we can obtain by this method the determinant, by simple calculations and reducing the rounding errors Cuvinte cheie: metoda Gauss-Jordan, matricea inversă, limbaj de programare Turbo C, pivot. 1. Noţiuni generale Met...

The CADNA library enables one to estimate round-off error propagation using a probabilistic approach. With CADNA the numerical quality of any simulation program can be controlled. Furthermore by detecting all the instabilities which may occur at run time, a numerical debugging of the user code can be performed. CADNA provides new numerical types on which round-off errors can be estimated. Sligh...

Interval arithmetic was original developed to estimate rounding errors on floating-point computations but it is used in a wide variety of applications from constraint solvers and global optimizers to power and timing analysis of software processes. The objective of interval analysis is to determine the output ranges (or interval) of a computation set. The main problem of classical interval anal...

1. Review of published ecological studies using path analysis suggests that path analysis is often misused and collinearity is a moderate to severe problem in a majority of the analyses. 2. Multiple regression is shown to be related to path analysis, and thus collinearity, which is a well-known problem in regression analysis, can also be a problem in path analysis. Collinearity occurs when inde...

Finite precision computations may affect the stability of algorithms and the accuracy of computed solutions. In this paper we first obtain a relation for computing the number of common significant digits between the exact solution and a computed solution of a one-dimensional initial-value problem obtained by using a single-step or multi-step method. In fact, by using the approximate solutions o...