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### Improvement of the function evaluation and of the Jacobian

An improved value of the Jacobian is obtained by taking account its derivative in the procedure:

```
int Dimension_Eq,
INTERVAL_MATRIX (* Gradient)(int, int, INTERVAL_VECTOR &),
INTERVAL_MATRIX (* Hessian)(int, int, INTERVAL_VECTOR &),
INTERVAL_VECTOR &Input,
```
where
• Exact: if 1 the calculation for one element of the Jacobian will stop as soon as the method has found that the interval evaluation of the element will not have a constant sign. If 0 the best interval evaluation will be computed
• InGrad: if this matrix is not the zero matrix we will assume that the non zero elements of this matrix are the interval evaluation of the Jacobian

To compute only the best value of the jacobian element at l-th row nad j-th column you may use:

```
int Dimension_Eq,
INTERVAL_MATRIX (* Gradient)(int, int, INTERVAL_VECTOR &),
INTERVAL_MATRIX (* Hessian)(int, int, INTERVAL_VECTOR &),
INTERVAL_VECTOR &Input,int Exact)
```

We may also obtain the best interval evaluation of the equations through the procedure

```
int Dimension_Eq,
INTERVAL_VECTOR (* TheIntervalFunction)(int,int,INTERVAL_VECTOR &),
INTERVAL_MATRIX (* Gradient)(int, int, INTERVAL_VECTOR &),
INTERVAL_MATRIX (* Hessian)(int, int, INTERVAL_VECTOR &),
INTERVAL_VECTOR &Input, int Exact)
```

Jean-Pierre Merlet 2012-12-20