The BiCenteredForm procedure

This procedure is based on the interval evaluation of equations using centered form with as centers two possible centers which may be optimal (see the ALIAS C++ manual for the calculation of these centers). The centered form uses the fact the interval evaluation of a function $f$ in the $n$ unknown $X=\{x_1,\ldots,x_n\}$ is included in $f(c)+J(X)(X-c)$ where $c$ is point called the center. As mentioned earlier two possible centers are calculated. Furthermore the calculation of the product $J(X)(X-c)$ may involve several occurrences of the same variables, that are not detected when computing the product numerically. This procedure allows, as an option, to compute symbolically the product and to re-arrange it in order to try to reduce multiple occurrences.

The syntax of this procedure is:

 
BiCenteredForm(func,funcproc,Jfuncproc,vars,custom,procname)

where

• func: a list of equations
• funcproc: the name of a procedure in MakeF format that computes the interval evaluation of func
• Jfuncproc :the name of a procedure in MakeJ format that computes the interval evaluation of the derivatives of func
• vars: a list of variable names
• custom: a string that indicate how funcproc and Jfuncproc are obtained.
  • "none": the procedure will assume that funcproc and Jfuncproc are the procedures F and J that are generated by GradientSolve and HessianSolve
  • "F": the procedure will assume that funcproc is a customized procedure that, for example, has been generated by MakeF
  • "J": the procedure will assume that Jfuncproc is a customized procedure that, for example, has been generated by MakeJ
  • "FJ": the procedure will assume that funcproc and Jfuncproc are customized procedures
• procname: the name of the simplification procedure that will be created in the file procname.C

In addition the procedure admits a 7th optional argument "prod". In that case the procedure will compute symbolically the product $J(X)(X-c)$ and re-arrange its terms. Both the numerical form and the one resulting from the symbolic calculation will be used in the simplification procedure.