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Implementation

The simplest Gaussian elimination scheme is implemented as:

 
int Gauss_Elimination(INTERVAL_MATRIX &Ain,
                     INTERVAL_VECTOR &b,INTERVAL_VECTOR &bout)
where The procedure returns 1 if it has succeeded in finding the enclosure, 0 otherwise (one of the interval pivot in the Gaussian scheme includes 0).

This computation for the initial system and a pre-conditioned system has been implemented in the procedure:

 
int Gauss_Elimination_Derivative(MATRIX &Cond,INTERVAL_MATRIX &Ain,
              INTERVAL_MATRIX &ACondin,
               const INTERVAL_VECTOR bin, 
               const INTERVAL_VECTOR bCondin, 
              INTERVAL_VECTOR &bout, 
              INTERVAL_VECTOR & Param,
              INTERVAL_VECTOR (* Func)(int l1, int l2, INTERVAL_VECTOR & v_IS),
              INTERVAL_MATRIX (* JFunc)(int l1, int l2, INTERVAL_VECTOR & v_IS),
              INTERVAL_VECTOR (* bFunc)(int l1, int l2, INTERVAL_VECTOR & v_IS),
              INTERVAL_MATRIX (* JbFunc)(int l1, int l2, INTERVAL_VECTOR & v_IS))
where

This procedure will return 1 if the Gauss elimination scheme has been completed, 0 otherwise. It will return in general a much more better enclosure than the classical interval Gauss elimination scheme.

For example consider the system

\begin{displaymath}
\left[ \begin{array}{cc}
x & y \\
x & x
\end{array}
\ri...
... Y =
\left[
\begin{array}{c}
x \\
x
\end{array}
\right]
\end{displaymath}

for $x$ in [3,4] and $y$ in [1,2]. Using the classical Gauss elimination scheme the enclosure of the solution is:

\begin{displaymath}
Y_1=[0.76923,10]~~~~Y_2=[-13,-0.0769]
\end{displaymath}

while if we use the derivatives we get

\begin{displaymath}
Y_1=[0.9166,2.6666]~~~~Y_2=[-2,-.66666]
\end{displaymath}

while the solutions are $Y_1=(y+x)/x$ (which has an enclosure of [1.25,1.666]), $Y_2=-1$.

The ALIAS-Maple procedure LinearBound implements this calculation.


next up previous contents
Next: Regularity of parametric interval Up: Solving systems of linear Previous: Mathematical background   Contents
Jean-Pierre Merlet 2012-12-20