Iterative method for solving nonlinear singular problems

The aim of our work is to present a method (p-factor method) for solving nonlinear equations of the form $$\begin{aligned} F(x)=0, \quad F:\mathbb {R}^n\rightarrow \mathbb {R}^m, \end{aligned}$$F(x)=0,F:Rn→Rm,in singular (irregular) case, i.e. when the matrix $$F'(x)$$F′(x) is singular at an initial point $$x_0$$x0 of an iterative sequence $$\{x_k\}$${xk}, $$k=1,2,\ldots $$k=1,2,… We investigate conditions that have to be fulfilled at the initial point $$x_0$$x0 to obtain the existence of solution of nonlinear equation $$F(x)=0$$F(x)=0, prove convergence of the presented p-factor method and give estimation of convergence rate for this method.


Introduction
We consider the problem of a construction of a numerical method (or an iterative sequence {x k }, k = 1, 2, . . .) for solving a system of nonlinear equations where F : R n → R m , m ≤ n and F is singular at the initial point x 0 . The problem (1) is called regular at the point x 0 if rank F (x 0 ) = m. Otherwise, it is called nonregular (singular). Usually, for proving convergence of the numerical method and obtaining an estimation of a convergence rate, a local convergence is supposed, i.e. the initial point x 0 of the iterative sequence is assumed to belong to some neighborhood U ε (x * ) = {x : x − x * < ε}, ε > 0 of a solution x * to the system (1) and moreover this solution has to exist. For this reason, an important problem in construction numerical method one can faced is to prove the existence of solution in some neighborhood of the initial point and moreover to fix assumptions on a neighborhood of the starting point so that the constructed numerical method is convergent.
For the classical Newton (or Gauss-Newton) method in regular case, where F (x 0 ) −1 R = F (x 0 ) F (x 0 )F (x 0 ) T −1 , the existence of a solution x * in neighborhood of the initial point x 0 is guaranteed by Theorem 1 from the Sect. 2. And if the assumptions of Theorem 1 are valid, for the Newton method we have geometrical estimation of convergence rate: However, one of the main requirements for the method convergence, existence of solutions and estimation of the convergence rate is nonsingularity of the Jacobian matrix F (x 0 ). In a singular case, when F (x 0 ) −1 R does not exist, the situation is much more complicated.
In the paper we present a numerical method ( p-factor method) for solving equations in the case when F (x 0 ) is singular. We prove a theorem on existence of solutions and justify convergence and estimation of the rate of convergence to the described method.
One could wonder why we propose a new method instead of choosing another initial point and then using the classical method. It is because of the fact that even if one choose a starting point which is not singular it could happened that the next iteration might give rejection effect, that is x 1 − x * x 0 − x * . The method described in this paper allows to avoid such effects and gives quadratic rate of convergence.
The construction of p-regularity introduced in [7] gives new possibilities for description and investigation of the set of solutions in the singular case. We use some of the methods to prove the conditions of the existence of local solutions to nonlinear singular equations in [4] and then some other results in the case with non-zero p-kernel (see [5]) and also give a description of a tangent cone in singular case [6].
Throughout this paper we assume that the mapping F : R n → R m is continuously p-times differentiable on R n and its pth order derivative at x ∈ R n will be denoted as F ( p) (x) (a symmetric multilinear map of p copies of R n to R m ) and the associated p-form is Let X and Y are sets. We denote by 2 Y the set of all subsets of the set Y . Any mapping Φ : X →2 Y is said to be a multimapping (or a multivalued mapping) from X into Y . For a linear operator Λ : Furthermore, we shall use the "norm" of such right inverse operator and so In our further considerations, under notion Λ −1 we shall mean just right inverse operator (multivalued) with the norm defined by (3).
We call h ∈ R n a tangent vector to a set M ⊆ R n at x * ∈ M if there exist ε > 0 and a function r The collection of all tangent vectors at x * is called the tangent cone to M at x * and it is denoted by We recall two auxiliary lemmas. The first of these lemmas is a "multivalued" generalization of the contraction mapping principle. By dist(A 1 , A 2 ) we mean the Hausdorff distance between sets A 1 and A 2 .
Lemma 1 (Contraction multimapping principle [2]) Assume that we are given a multimapping where the sets Φ(z) are non-empty and closed for any z ∈ U ε (z 0 ). Further, assume that there exists a number θ, 0 < θ < 1 such that Then, for every number ε 1 which satisfies the inequality (4)

Elements of p-regularity theory
Let us consider the case when the regularity condition fails but the mapping F is p-regular. We construct the p-factor operator under the assumption that R m is decomposed into an orthogonal sum where Y 1 = ImF (x 0 ), and the remaining spaces are defined as follows. Let Z 1 = R m , Z 2 = Y ⊥ 1 , and let P Z 2 : Y → Z 2 be the orthogonal projection onto Z 2 . Let Y 2 be a linear span of the image of the quadratic map P Z 2 F (x 0 )[·] 2 . More generally, define inductively, The order p is the minimum number for which (5) holds.
Define the following mappings and is called the p-factor operator along h at the point x 0 .
Sometimes it is convenient to use the following equivalent definition of p-factor We also introduce the corresponding multivalued right inverse operator, and nonlinear operator Let us introduce the following set We can write it also as follows where For our further considerations we need a generalization of the notion of regular mapping.

Definition 2
We say that the mapping F is p-regular at x 0 along h if rankΛ h = m.

Definition 3
We say that the mapping F is p-regular at x 0 if either it is p-regular along every h from the set H p (x 0 ) \ {0} or H p (x 0 ) = {0}.

Regular case
In regular case the tangent cone to a solution set of the Eq. (1) equals to the kernel of the first derivative of the mapping F, i.e. T 1 M(x 0 ) = Ker F (x 0 ). To investigate regular nonlinear problems one can apply classical results, such as Lyusternik theorem, implicit function theorem, Lagrange-Euler optimality conditions. Besides description of the solution set and formulation of optimality conditions, very important problem is to give a guarantee of existence of a solution in some neighborhood of an initial point.
We quote one of the modifications of the theorem about existence of solutions of Eq. (1) in regular case (see e.g. [1]).
are valid then the equation F(x) = 0 has a solution x * ∈ U ε (x 0 ), the method (2) converges to x * and the following estimation holds: For the proof see [4].

Singular case
We can generalize the above results and give the necessary conditions that have to be valid for existence of solutions in the singular case, i.e. when F (x 0 ) is singular. First, let us recall two lemmas that are essential in the proof of Theorems 2 and 3.
For the singular case we have the following generalization of the Theorem 1.
Theorem 2 Let F : R n → R m , F ∈ C p+1 (U ε (x 0 )), ε > 0 and there exists h, such that h ∈ Ψ −1 (−F(x 0 )) \ {0},ĥ = h h and F is p-regular at x 0 along h. Moreover, assume that for a given ε the following conditions are satisfied For the proof see [4]. Notice that if p = 1 then Theorem 2 reduces to the Theorem 1. If p ≥ 2 then F (x 0 ) is singular and we have singular case.
Remark 1 (Sequential p-regularity) The assumptions of Theorem 2 can be relax if we consider the sequence: Namely, instead of p-regularity of F at x 0 we can assume p-regularity F on {ξ k }, k = 1, 2, . . .. It means that instead of surjectivity of Λ h we can require that ImΛ h = Y 1 ⊕ · · · ⊕ Y p and f i (

Remark 2
We choose x ∈ R n as a representative of the set Consider the following method ( p-factor method) Theorem 3 Suppose that for F, x 0 , h, ε, δ the assumptions of Theorem 2 are valid. Then the sequence defined in (8) converges to the solution x * ∈ U ε (x 0 ) and the following estimation of convergent rate holds: Proof Consider the following equivalent form of the sequence (8).
We can decompose the space R 2 as follows R 2 = Y 1 ⊕ Y 2 , where Y 1 = t 0 : t ∈ R and Y 2 = 0 r : r ∈ R . Then we calculate norms of some elements to examine if the assumptions of Theorem 2 are satisfied. We obtain δ = √ 10 6 +1