MAPLE library for the Interval Solver

All these procedures are available in The lib_IS.m library contains various procedures that may be useful for producing files for the parser. The following MAPLE programs are available:

• cout(eq): estimates the cost of the computation of eq in terms of number of arithmetic operations. This a rather rough approximation: it just compute the total number of addition, multiplication and function calls without any weight between these operators.
• minimal_cout(eq): returns an expression equivalent to eq but with a minimal cost of computation. Basically it consider the expanded form of the equation and compute its computational cost using the cout procedure. Then it consider the Horner forms of the equation using all possible ordering of the unknowns (if you have up to 6 unknowns). The cost of these forms are computed and as soon as a form as a lower cost then it is stored as the potential return for the function. If two forms have equal cost, then the form with the lower number of operations (obtained through the nops procedure) is retained.
• write_equation: this procedure takes as argument an equation or a list of equations and a name. The equation will be written in a minimal form in the formula file defined by the name
• write_gradient_equation: this procedures takes as as argument an equation or a list of equations and a name. The gradient of the equations in minimal form will be written in the formula gradient file defined by the name. If no third argument is provided the ordering of the gradient matrix with respect to the unknowns will be the one provided by the MAPLE procedure indets. A third argument may be provided and should be a list of unknown names which will be used for the ordering of the gradient matrix (i.e. if the list is [x1,x2,x3] for the equation eq the first equation will be deq/dx1, the second deq/dx2..). A fourth argument may be the list of unknowns and is used for parametric system for which the number of undefined variables may be larger than the number of unknowns (otherwise the procedure will return an error message indicating that the equation has more indeterminate than the number of unknowns given in the third argument).
• write_hessian_equation: a similar procedure than for the gradient except it computes the hessian matrix of the equations

An example of the use of this library follows:

 
readlib(lib_IS):
eq1:=y**2*z+2*x*y*t-2*x-z:
eq2:=-x**3*z+4*x*y**2*z+4*x**2*y*t+2*y**3*t+4*x**2-10*y**2+4*x*z-10*y*t+2:
eq3:=2*y*z*t+x*t**2-x-2*z:
eq4:=-x*z**3+4*y*z**2*t+4*x*z*t**2+2*y*t**3+4*x*z+4*z**2-10*y*t-10*t**2+2:
write_equation([eq1,eq2,eq3,eq4],"fcap"):
write_gradient_equation([eq1,eq2,eq3,eq4],"gcap",[x,y,z,t]):
write_hessian_equation([eq1,eq2,eq3,eq4],"hcap",[x,y,z,t]):

The formula file for the 4 equations will be written in the file fcap while the jacobian and hessian will be written in the file gcap, hcap. Note that the range file should define the ranges for the unknowns in the order x,y,z,t.