الفهرس 
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Abstract
Many computer applications and real-life problems can be modelled by systems of linear equations or safely transformed to the linear case. when uncertain models parametrs are introduced by intervals, then a parametric interval linear system must proparly be solved be solved to meet all possible scenarios and yield useful results.
In this work we solved parametric linear systems of equations whose
coefficients are, in the general case , nonlinear functions of interval parameters.
Here solution means that we enclose the set of all solutions, the so-called
parametric solution set, obtained when all parameters are allowed to vary
within their intervals. This task appears in many scientific and engineering
problems involving uncertainties.
A C-XSC implementation of a parametric fixed-point iteration method for
computing an outer enclosure for the solution set is proposed in this work. This
method requires to bound the range of a multivariate function over a given box
and often delivers intervals which are too wide for practical applications. We
computed tight enclosures of the parametric solution set by using the new
extended generalized interval arithmetic which is an arithmetic for intervals
(which are representing uncertainties). The most important property of this
method is to reduce the effect of the dependency problem which is inherent in
the computation with standard interval arithmetic. We used the new arithmetic
to tightly bound the range of a multivariate nonlinear function over a box,
a task to which many problems in mathematics and its applications can
be reduced.
We applied the new bounding technique to improve the efficiency of the
solution for parametric systems. Numerical examples illustrating the
applicability of the proposed method are solved, and the proposed method is
compared with other methods.
Interval analysis is an enormously valuable tool to solve the problems occur when
we use computer to carry out mathematical computation using floating point
arithmetic which real numbers are approximated by machine numbers. Because of
this representation two types of errors are generated. The first type of error occurs
when a real valued input data is approximated by a machine numbers. The second
type of error is caused by intermediate results being approximated by machine
numbers. Therefore, the results of the computations performed will usually be
affected by rounding errors and in the worst cases lead to completely wrong
results. This problem is getting even worse since computers are becoming faster,
and it is possible to execute more and more computations within a fixed time. It is
possible to verify the accuracy of the results generated by some complicated
programs using other tools.
By Interval analysis we can estimate and control the errors (which occur on the
computers) automatically. Instead of approximating a real value by a machine
number, the real value is approximated by an interval [] that includes a
machine number. The upper and lower boundaries of this interval contain the
usually unknown value . The width of this interval may be used as a measure for
the quality of the approximation. It is desirable to make interval bounds as narrow
as possible. A major focus of interval analysts is developing interval Algorithms
that produce sharp or nearly sharp bounds on the solution of numerical computing
problems.A C-XSC implementation of a parametric fixed-point iteration method for
computing an outer enclosure for the solution set is proposed in this work. This
method requires to bound the range of a multivariate function over a given box
and often delivers intervals which are too wide for practical applications. We
computed tight enclosures of the parametric solution set by using the new
extended generalized interval arithmetic which is an arithmetic for intervals
(which are representing uncertainties). The most important property of this
method is to reduce the effect of the dependency problem which is inherent in
the computation with standard interval arithmetic. We used the new arithmetic
to tightly bound the range of a multivariate nonlinear function over a box,
a task to which many problems in mathematics and its applications can
be reduced.
We applied the new bounding technique to improve the efficiency of the
solution for parametric systems. Numerical examples illustrating the
applicability of the proposed method are solved, and the proposed method is
compared with other methods.