The simplest Gaussian elimination scheme is implemented as:

int Gauss_Elimination(INTERVAL_MATRIX &Ain, INTERVAL_VECTOR &b,INTERVAL_VECTOR &bout)where

`Ain`: the**A**interval matrix`b`: the**b**interval vector`bout`: the enclosure of the set of solutions

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

`Cond`: the pre-conditioning matrix. If all the elements of`Cond`are 0 the procedure will implement the Gauss elimination scheme only on the initial system`Ain`: the interval evaluation of the matrix`ACondin`: the interval evaluation of the`Cond`matrix`bin`: the interval evaluation of`bCondin`: the interval evaluation of`Cond``bout`: the enclosure of the interval linear system`Param`: the ranges for`Func`: a procedure that computes the interval evaluation of the elements of . These elements are stored rows by rows in an interval vector (see the note 2.3.4.3)`Jfunc`: a procedure that computes the interval evaluation of the derivatives of the elements of , see the note 2.4.2.2`bFunc`:a procedure that computes the interval evaluation of the elements of , see the note 2.3.4.3`Jbfunc`: a procedure that computes the interval evaluation of the derivatives of the elements of , see the note 2.4.2.2

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

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

while if we use the derivatives we get

while the solutions are (which has an enclosure of [1.25,1.666]), .

The ALIAS-Maple procedure `LinearBound` implements this
calculation.