The IntervalNewton procedure

This procedure allows one to take into account the specific structure of the system to solve when using the interval Newton method. In this method the classical algorithm, for example used by default in GradientSolve or HessianSolve, computes numerically for a box $X$ the interval evaluation of $M=J^{-1}(X_m)J(X)$ where $X_m$ is the mid-point of $X$. This numerical calculation does not allow to to take into account the dependency between the elements of $J$ when computing $M$. This procedure compute symbolically the matrix $M$ and re-arrange its elements (with MinimalCout) in order to reduce the dependency problem.

The syntax of this procedure is

IntervalNewton(func,nfunc,funcproc,njfunc,Jfuncproc,typemid,grad,grad3B,incl,vars,procname)

where

• func: the list of equations,
• nfunc: the number of equations that will be evaluated by funcproc. It may not be the same than the number of equations in func
• funcproc: the name of the procedure that is used to evaluate the equations. It may be the name of a procedure designed by the end-user with MakeF or the name of the procedure that will be used by the solving algorithm (usually "F")
• njfunc: the number of equations that will be used when computing the derivatives. It may not be the same than the number of equations in func
• Jfuncproc: the name of the procedure that is used to evaluate the derivative of the equations. It may be the name of a procedure designed by the end-user with MakeJ or the name of the procedure that will be used by the solving algorithm (usually "J")
• typemid: the interval Newton method uses a conditioning matrix. This flag will indicate which type of conditioning matrix is used. If 0 we use $J^{-1}(X_m)$, if 1 $Mid(J^{-1}(X))$, if 2 both above conditioning matrices will be used
• grad, grad3B: it is possible to use the derivatives of $M$ to improve the interval evaluation. If grad is set to 1 this will be used for all boxes in the algorithm. If grad3B is set to 1 it will also be used for the sub-boxes that are considered when using the 3B method
• incl: a list. If the end-user uses a specific procedure to compute the interval evaluation of func, then incl[1] should be set to 1. Similarly if it uses its own procedure to compute the derivatives of func, then incl[2] should be set to 1.
• vars: the list of variables. All $n$ unknowns in the $n$ elements of func must appear first in this list
• ProcName: the name of the simplification procedure that will be created in the file ProcName.C