### On the Use of Interval Arithmetic to Study Error Propagions in the Artificial

#### Abstract

Two of the most important numerical methods that are widely used in the artificial satellite orbit determination are the least squares procedures to process the tracking station observations and the numerical integration of ordinary differential equations. The propagation of errors in these methods is a very important problem in the area of orbital mechanics. The recently developed Interval Analysis is a technique to provide lower and upper bounds for the exact values of this type of errors and can be used in these error propagations. Interval Arithmetic replaces the computation on real numbers by a computation on pairs (a,b) where the a and b are real numbers: a is the lower bound and b the upper bound. This is thus a

convenient way of representing the error that may be associated with a number.

The number is supposed to be anywhere (uniformly) between these two limits a and b. With the idea of using this technique in this problem, our work aims at comparing two different methods of least squares, using numerical calculations in Interval Arithmetic: the standard method with inversion of A-transpose-A and the

orthogonalization method with the Gram-Schmidt decomposition. The basic problem in question, the two methods of least squares being compared, and the

preliminary computer program listings using interval arithmetic alongwith the results and comments are given here. A comparison of two numerical integration methods -Runge-Kutta methods of orders 4 and S - is proposed to be done in a later work.

convenient way of representing the error that may be associated with a number.

The number is supposed to be anywhere (uniformly) between these two limits a and b. With the idea of using this technique in this problem, our work aims at comparing two different methods of least squares, using numerical calculations in Interval Arithmetic: the standard method with inversion of A-transpose-A and the

orthogonalization method with the Gram-Schmidt decomposition. The basic problem in question, the two methods of least squares being compared, and the

preliminary computer program listings using interval arithmetic alongwith the results and comments are given here. A comparison of two numerical integration methods -Runge-Kutta methods of orders 4 and S - is proposed to be done in a later work.

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Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

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**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**