Mathematical background

Let consider a system of $n$ equations in $m$ unknowns $F_i(x_1,\ldots,x_m)=0$ and assume that $x_1$ may vary in the range $[\underline{x_1},\overline{x_1}]$ and that it is considered as the parameter of this system. Using the solving algorithms of ALIAS (or any other method) we are able to determine the real roots of this system, if any, for a given value of the parameter, for example for $x_1=\underline{x_1}$. Let assume that we have found $k$ real roots $X_1,\ldots, X_k$. Assume now that we change the value of $x_1$ to $\underline{x_1}+\epsilon$. Using Kantorovitch theorem (see section 3.1.2) we are able to determine if Newton method with as initial estimate $X_i$ will converge toward a solution of the new system and the radius of convergence $\alpha_i$ such that the solution is unique in the ball $B_i =[X_i-\alpha_i, X_i+\alpha_i]$ (alternatively we may also use Moore theorem, see section 2.10). If this is the case and if the intersection of the $B_i$ is empty, then we are able to compute in a certified way the new $k$ solutions of the new system. Otherwise we will repeat the procedure with a smaller value for $\epsilon$ until the method succeed. A failure case will be when the system become singular at a point (for example when 2 branches are collapsing). If this happen we will change the value of $x_1$ until we find a new set of solutions which satisfy Kantorovitch theorem and start again the process.

This method enable to follow all the branches for which initial points have been found when solving the initial system. We may then store these points in an array, for example at regular step for the value of the parameter.

By default we use Kantorovitch theorem to follow the branches but by setting the global variable
ALIAS_Use_Kraw_Continuation to 1 we will use Moore test.