# Project : coprin

## Section: New Results

### Interval linear systems

#### Library for solving interval system of linear equations

Keywords : interval linear equations, interval analysis.

Participants : David Daney, Jean-Pierre Merlet.

Solving an interval linear system of equations is crucial for two reasons:

a module solving this problem is a basic component of numerous interval analysis algorithms (such as interval Newton)

this problem appears in many applications (see the example in the robotics section)

We have implemented a C++ package which collects the classical
methods known in the field of interval analysis (Gauss-Seidel,
Gauss-elimination, Krawczyk's method with or without pre-conditioning) and
also additional methods based on the constraint programming approach
and linear programming. This package is connected to the libraries
`ALIAS` and `ICOSALIAS` and is used within a specific solver
of non-linear over-constrained systems of equations.

#### Improved algorithm for parametrized linear interval system

Keywords : interval linear equations, interval analysis.

Participants : David Daney, Jean-Pierre Merlet.

Solving linear interval systems of type AX = b consists
in determining a box that includes all the
solutions in X of such set of linear systems. But most of the time
the real problem
may be written as where is a set of parameters with interval values.
Classical interval
analysis method (such as Gauss elimination) usually overestimate
largely the box including all the solutions in X as the dependency
in
between the coefficients A_{ij}, b_{k} are not taken into
account.

We have shown that using the monotonicity
for improving the interval evaluations of the expressions used in
the Gauss elimination scheme (by
considering their derivatives with respect to )
may allow to drastically improve the box
for X. This algorithm has been incorporated into `ALIAS` and
has been used for a robotics application.