Regularization of Computation of Solutions in a Neighborhood of the Branch Point

Abstract

In practice we can solve many important problems of the branching theory with sufficiently large precision only on the basis of the application of iterated methods using a PC. At present in the bibliography (see Vainberg and Trenogin [1], Krasnoselsky, Vainikko, Zabreiko and Rutizkii [1], Berger and Westreich [1], Chen and Christiances [1], Demoulin and Chen [1], Schroder [1]), anumber of schemes has been described for the construction of solutions of the equation

$$F\left( {x,\lambda } \right) = 0$$

(0.1)

in a neighborhood of the branch point λ ₀. Convergence of the corresponding methods having been proved on the assumption that the equation is precisely given and all computations are precisely developed. However, every real computation method is accompanied by various errors: 1) errors appearing at a result of errors in the definition of parameters of equation; 2) round-off errors; 3) errors in the method. The first group of errors can results in the perturbed (approximate) equation not even having real solutions in a neighborhood of the branch point, even though the exact equation has such solutions. If, nevertheless, we have the exact equation the computations have appeared unstable at the values of parameter near the branch point because of round-off errors: the linearized operator F _x (x ₀,λ0) has no bounded inverse. Finally, by decreasing the value of parameter (λ → λ ₀) the standard iterated methods are worsened, so that in practice the precision needed becomes unattainable. Thus the computation problem of solutions in a neighborhood of the branch point λ ₀ is found to be ill posed, and the methods in the works of Vainberg and Trenogin [1], Krasnoselsky, Vainikko, Zabreyko, Rutizkii [1], Berger and Westreich [1], Do-moulin and Chen [1] are unstable in the computational sense: small errors may give vise to a large distortion in the result.