This package uses a fixed precision arithmetics with an accuracy of roughly . Different types of data structure are implemented in this package. For fixed value number:
BOOL, BOOL_VECTOR, BOOL_MATRIX, INT, INTEGER_VECTOR, INTEGER_MATRIX, REAL, VECTOR, MATRIX, COMPLEXfor intervals:
INTERVAL, INTERVAL_VECTOR, INTERVAL_MATRIXAll basic arithmetic operations can be used on interval-valued data using the same notation than for fixed numbers. Not that for vector and matrices the index start at 1: for an interval matrix represents the interval at the first row and first column of the interval matrix . The type INTERVAL_VECTOR will be used to implement the box concept, while the type INTERVAL_MATRIX will be used to implement the concept of list of boxes.
For the evaluation of more complex interval-valued function there are also equivalent function in the BIAS/Profil, whose name is usually obtained from their equivalent in the C language by substituting their first letter by the equivalent upper-case letter: for example the evaluation of where is an interval will be obtained by calling the function . We have also introduced in ALIAS some other mathematical operators whose names are derived from their Maple implementation: ceil, floor, round.
Table 2.1 indicates the substitution for the most used functions.
A special operator is defined in the procedure ALIAS_Signum:
formally it defines the signum operator of Maple defined as
The derivative of ALIAS_Signum is defined in the procedure ALIAS_Diff_Signum. Formally this derivative is 0 for any not equal to 0. In our implementation ALIAS_Diff_Signum() will return 0 except if is lower than ALIAS_Value_Sign_Signum in which case the procedure returns [-1e11,1e11].
The derivative of the absolute value is defined in the procedure ALIAS_Diff_Abs. If the interval X includes 0 the procedure returns [-1e11,1e11] otherwise it returns ALIAS_Signum(X).
Using the above procedures
when an user has to write an interval-valued function he has to
convert its C source code using the defined substitution. For example
if a function is written in C as:
We will denote by box a set of intervals which define the possible values of the unknowns. By extension and according to the context boxes may also be used to denote a set of such set. A function intervals will denote the interval values of a set of functions for a given box, while a solution intervals will denote the box which are considered to be solution of a system of functions.