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Evaluation procedure using the Jacobian

A better evaluation of the function intervals than the `IntervalFunction`
can be obtained using the Jacobian matrix. A specific procedure
can be used to obtain this evaluation:

INTERVAL_VECTOR Compute_Interval_Function_Gradient(int m,int n,
INTEGER_VECTOR &Type_Eq,
INTERVAL_VECTOR (* IntervalFunction)(int,int,INTERVAL_VECTOR &),
INTERVAL_MATRIX (* IntervalGradient)(int, int, INTERVAL_VECTOR &),
INTERVAL_VECTOR & Input, int Exact,
INTEGER_VECTOR &AG,INTEGER_MATRIX &AR)

This procedure computes the function intervals for the box
`Input`.
`Type_Eq` is an integer array whose `Dimension_Eq` elements
indicates the nature of the functions: -1 for inequality , 0 for
equation, 1 for inequality , -2 for a function to be minimized, 2
for a function to be maximized and 10 for a function to be both
minimized and maximized (note that for an optimization problem the
function that has to be minimized must be the last function in the
list of function).
- the integer
`Exact` should be put to 1 as for a value
of 0 the procedure stop the evaluation of each box as soon
as the lower bound of the interval is negative and the upper bound
positive.
`AG` is an integer vector of size `m x n` which
indicates if the sign of some derivatives are already known (the
elements should then have the values -1, 0 or 1) or not (the value
must then be 2)
`AR` is a return matrix with the sign of the derivatives for
`Input`

The parameters `Type_Eq`, `AG, AR` may be omitted.
This procedure uses the derivatives for improving the interval
evaluation of the functions in two different ways:
- by taking into account of the monotonicity of the functions
- by using an interval evaluation of the functions based on their
Taylor expansion: it is therefore necessary to evaluate rightly the
derivatives of the functions

The best evaluation of the l-th equation may be computed with

INTERVAL Compute_Interval_Function_Gradient_Line(int l,int Dim_Var,
int Dimension_Eq,
INTERVAL_VECTOR (* IntervalFunction)(int,int,INTERVAL_VECTOR &),
INTERVAL_MATRIX (* Gradient)(int, int, INTERVAL_VECTOR &),
INTERVAL_VECTOR &Input,int Exact,INTEGER_VECTOR &AG)

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** Previous:** Jacobian matrix
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Jean-Pierre Merlet
2012-12-20