Team Coprin

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Inria / Raweb 2004
Project: Coprin

# 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 Aij, bk 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.

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