Principle

The 3B method is an usually efficient method to reduce the width of the variable intervals. This method can be used for any algorithm of ALIAS and is embedded in the code of the solving procedures of the C++ library.

The principle of the method is quite simple. Let assume that we deal with a problem with $n$ variables $x_1,\ldots,x_n$ and that the range for the variable $x_i$ is [$a_1,b_1$] while $\epsilon$ is a small number. The procedure will examine if the problem may have a solution if the range for $x_i$ is reduced to [ $a_1,a_1+\epsilon$]. If it is not the case the range for $x_i$ may be reduced to [$a_2,b_2$]=[ $a_1+\epsilon,b_1$]. The procedure will then be repeated by using $2\epsilon$ as a new value for $\epsilon$ until the algorithm either determine that there is no solution to the problem or fails to show that that is no solution for the range [ $a_n,a_n+n\epsilon$]. The algorithm will then try to reduce the range for $x_i$ on the "right" side i.e. by examining if the problem has no solution for the range [ $b_1-\epsilon, b_1$]. This process is repeated in turn for each variable.

Within Maple the value of $\epsilon$ is given by the variable `ALIAS/Delta3B` which set an identical value of $\epsilon$ for all variables. Alternatively you may specify a specific value of $\epsilon$ for each variable by assigning a list to the variable `ALIAS/Delta3B_ARRAY` with as many elements as unknowns: each element of this list is the value of $\epsilon$ for each variable.

The 3B method will not be used for variables whose range have a width greater than a given threshold. This threshold is defined by the variable `ALIAS/Max3B` (default value: 5). You may assign a new value for this threshold by:

 
`ALIAS/Max3B`:=10:

Alternatively you may specify a specific threshold for each variable by using the array `ALIAS/Max3B_ARRAY`. For example:

 
`ALIAS/Max3B_ARRAY:=array([0.1,0.1,0.05]):

indicates that the 3B method will be used only if the width or the input intervals is lower than 0.1 for the 2 first unknowns and 0.005 for the third one.

As soon as the Max3B, Delta3B values have been set you activate the 3B method by setting the `ALIAS/3B` variable to 1 or 2. At iteration $n$ of the 3B method the algorithm will try to reduce the range for the variable by $n\epsilon$. Assume that $n$ is larger than 1 and the algorithm fails:

• if `ALIAS/3B` is set to 1 the algorithm will move to the next variable (or try to reduce the range on the "right" side if we dealing with the "left" side)
• if `ALIAS/3B` is set to 2: the algorithm was considering for the variable the range [ $a_k,a_k+n\epsilon$] when it has failed. Instead of moving to the next step the algorithm will consider the range [ $a_k,a_k+\epsilon$] and repeat the process until the method fails for a gain exactly equal to $\epsilon$