Solving triangular algebraic systems by means of simultaneous iterations (Q1805909)

Weierstrass' method for the simultaneous computation of all zeros of a polynomial has attracted much interest. It is connected also with the names Dochev, Durand, and Kerner. Also the reviewer has made a small contribution [with \textit{H. Späth} in Z. Angew. Math. Mech. 47, 409-410 (1967; Zbl 0155.46702)]. The simultaneous computation avoids a shortcoming you have in a sequential computation of the roots. A restart of a Newton iteration without a deflation may result in a recomputation of the same solution. A restart with explicit deflation produces an error of the solution which is not easily handled. In this paper that was communicated by C. Brezinski, a generalization is studied for which such a motivation is not present. Triangular algebraic systems \[ P_1 (x_1)=0,~P_2 (x_1, x_2)=0,~P_3 (x_1, x_2, x_3)=0,~\ldots \] are treated similarly. An iteration of Newton type for the simultaneous computation of all solutions is proposed. The possible decoupling of the system is only used for a simplification of the formulation. Therefore the disadvantages of not-decoupling the system dominate. Moreover the determinants are not reduced in the way that is possible for one complex variable.