Tuning the Multithreaded Interval Method for Solving Underdetermined Systems of Nonlinear Equations

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

“…-various kinds of the interval Newton operator and switching between the componentwise Newton operator (for larger boxes) and Gauss-Seidel with inverse-midpoint preconditioner, for smaller ones, -a sophisticated heuristic to choose the bisected component [27], -an initial exclusion phase of the algorithm (deleting some regions, not containing solutions) -based on Sobol sequences [28], -an additional test based on quadratic approximation of a single equation and the Hansen's method [18] to solve quadratic equations with interval coefficients; see [29].…”

“…In Section 6 we consider two possible values of ε bc3 : 3 n and 6 n ; results for 1.5 n are not presented as they occurred to be less promising. As for ε Ncmp , the value 1 n was used -twice larger than in [27] to emphasize the importance of the componentwise operator, partially replaced by the BC3 procedure.…”

“…6: end if 7: if (right narrow (x, f, i, ε) results in "no solutions") then 8: return "no solutions" 9: end if 10: if (dist(x i , x old i ) >= ε equal ) then 11: modified = true 12: end if 13: until (not modified) 14: return x Operators Ncmp and GS are described, e.g., in [23,25,27]. For the sake of brevity, formulae are as follows:…”

“…In [27] the problem of choosing the proper variable for bisection has been discussed. We emphasized the insufficiency of earlier approaches (see, e.g., [9]) and proposed the heuristic, described by Algorithm 9.…”

“…Choosing the variable for bisection of x -heuristic from [27] Require: x {We assume the procedure gets sufficient info about the results of the Newton operator evaluation, also -see below} 1: FindMaxDiam(x, j max , w max return j max 6: else if (w max unn > 8 · w min ) then 7: return j max unn 8: end if 9: FindSmallestMaxMag(x, j , w) {Find the component with the smallest maximal magnitude of the Jacobi matrix in all rows} 10: if (w > 0.1) then 11: return j 12: else 13: return j max unn 14: end if…”