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##

Integral with multiple variable

`ALIAS` offers 4 procedures to compute definite integral with more
than one variable.
int IntegrateMultiRectangle(
INTERVAL_VECTOR (* Function)
(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR (* Second_Derivative)
(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR & TheDomain,
int Iteration,
double Accuracy,
INTERVAL & Result)
int IntegrateMultiRectangle(
INTERVAL_VECTOR (* Function)
(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR (* Second_Derivative)
(int,int,INTERVAL_VECTOR &),
INTERVAL_MATRIX (* Gradient)(int, int,INTERVAL_VECTOR &),
INTERVAL_VECTOR & TheDomain,
int Iteration,
double Accuracy,
INTERVAL & Result)

In the second form `Gradient` is a procedure in `MakeJ` format
that allows to compute the derivatives of the second derivatives of
the function.
`ALIAS` offer also procedure based on Taylor expansion of the
function.
int IntegrateMultiTaylor(
INTERVAL_VECTOR (* CoeffInt)
(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR (* RestTaylor)
(int,int,INTERVAL_VECTOR &),
INTEGER_MATRIX &APOWERINT,
INTEGER_MATRIX &APOWERREM,
int nbrem,
int Order,
INTERVAL_VECTOR & TheDomain,
int Iteration,
double Accuracy,
INTERVAL & Result)
int IntegrateMultiTaylor(
INTERVAL_VECTOR (* CoeffInt)
(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR (* RestTaylor)
(int,int,INTERVAL_VECTOR &),
INTEGER_MATRIX &APOWERINT,
int Order,
INTERVAL_VECTOR & TheDomain,
int Iteration,
double Accuracy,
INTERVAL & Result)

The Taylor expansion of the function may be written as:

`CoeffInt`: a procedure in `MakeF` format that compute
the coefficients
`APOWERINT`: a table with the exponent
for
each term
`RestTaylor, APOWERREM`: in the first form they play the same
role than `CoeffInt`, `APOWERINT` for the remainder. There is
`nbrem` terms in the remainder
`RestTaylor`: in the second form a procedure in `MakeF`
format that compute the remainder
`Order`: the degree of the remainder

Jean-Pierre Merlet
2012-12-20