**Preface to "Mathematical and Molecular Topology"**

Topology is one of the fundamental tools in relating entities. Topology naturally finds application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business and even the arts. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. Circa 1750, Euler stated the polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron), which may be regarded as the first theorem, signaling the birth of topology. Subjects included in topology are algebraic topology and graph theory. A related branch to graph theory is molecular topology (with concerns of either chemical or biological structure).

The Special Issue *Mathematical and Molecular Topology* received 10 submitted manuscripts from which 5 were accepted and published (50% success rate). Two manuscripts are related to mathematical topology, while the other three are related to molecular topology.

#### *Mathematical Topology Related Works*

A preclosure operator or Cech closure operator is a map between subsets of a set, similar ˇ to a topological closure operator, except that it is not required to be idempotent. π-normal, weakly π-normal and κ-normal generalizations of normality in Cech closure space are defined and ˇ characterized using canonically closed sets in [1]. An important result connects those spaces: the class of κ-normal spaces contains both the classes of weakly π-normal and almost normal Cech closure spaces.

A Banach space B is a complete normed vector space which, in terms of generality, lies in between a metric space (that has a metric, but no norm) and a Hilbert space (that has an inner-product, and hence a norm, that in turn induces a metric). In [2], the local convergence analysis of a fifth order method and its multi-step version in Banach spaces is studied. Starting from hypotheses based on the first Frechet-derivative only, the proposed approach provides a computable radius of convergence, ´ error bounds on the distances involved, and estimates on the uniqueness of the provided solution. Taylor expansions of higher-order derivatives may not exist or may be very expensive or impossible to compute, so approaches that use them do not produce such estimates. The authors provide numerical examples to validate the theoretical results. Basins of attraction are used to represent convergence domains of the methods and the boundaries of the basins reveal symmetric fractal-like shapes.

#### *Molecular Topology Related Works*

Graph algorithms, or algorithms operating on graphs is conceptually a branch of combinatorial algorithms having uses in many problems, from graph coloring, to fining a perfect matching and computing the lowest common ancestor, and to graph-based searching, routing and network theory. Complete subgraphs (or cliques) are subsets of vertices which are all adjacent one to the other, while maximal cliques are the largest such substructures in a graph. Maximal cliques of protein graphs serve to determine their similarity and function of the protein. In [3], improvements based on machine learning are added to a Maximum Clique Dynamic algorithm for finding the maximum clique in large graphs such as are protein graphs. The work is based on an algorithm published in 2007 [4] and has been widely used in bioinformatics since then, which uses an empirically determined parameter, Tlimit, that determines the algorithm's flow. In [3], the authors extended the MCQD algorithm with an initial phase of a machine learning-based prediction of the Tlimit parameter that is best suited for each input graph. The authors note that a such adaptability to graph types based on state-of-the-art machine learning is a novel approach that has not been used in most graph-theoretic algorithms. It is shown empirically that the resulting new algorithm MCQD-ML improves search speed on certain types of graphs, in particular molecular docking graphs used in drug design where they determine energetically favorable conformations of small molecules in a protein binding site. In such cases, the speed-up is twofold.

Entropy is a fundamental concept associated with measuring the state of disorder, randomness, or uncertainty. Clausius, Boltzmann, or Gibbs (statistical) entropy and Shannon's (information) entropy are practically one and the same. The values of entropy are key parameters driving the direction of spontaneous change for many commonplace events. In [5] the authors use various computational and mathematical techniques to calculate atom–bond connectivity entropy, atom–bond sum connectivity entropy, the newly defined Albertson entropy using the Albertson index, and the IRM entropy using the IRM index. An example of the calculation is given on H3BO3 by using the subdivision and line graph of the layer structure.

Complementing the molecular topology of a molecule, molecular geometry provides surface and structural representation, and is the key element differentiating among various molecules sharing the same topology. Various methods (from molecular mechanics and semi-empirical to ab initio and density functional theory) are involved in geometry optimization (energy minimization) of the molecules. Having as template a series of 20 amino acids with near-optimal geometry were used to reach the optimum geometries by using 39 methods (HF, MP2, B3LYP included) in [6]. Next, a pool of molecular descriptors was used to characterize each optimized geometric conformation and cluster analysis and principal component analysis were performed to get the similarities between the different optimization methods. As authors noted, the results after the analysis are classified into three main groups and can provide alternate selection accordingly to solve different types of problems.

Several topology problems on topics such cohomology, compactness, connectedness, homeomorphisms, homology, homotopy, symmetry and similarity are still to be explored to provide further insight on theoretical aspects of mathematical and molecular topology and their applications.

#### **References**


**Ria Gupta and Ananga Kumar Das \***

School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182320, India; 14dmt001@smvdu.ac.in or riyag4289@gmail.com

**\*** Correspondence: akdasdu@yahoo.co.in or ak.das@smvdu.ac.in

**Abstract:** New generalizations of normality in Cech closure space such as ˇ *π*-normal, weakly *π*normal and *κ*-normal are introduced and studied using canonically closed sets. It is observed that the class of *κ*-normal spaces contains both the classes of weakly *π*-normal and almost normal Cech ˇ closure spaces.

**Keywords:** closure space; canonically closed; weakly normal; almost normal; *π*-normal; weakly *π*-normal; *κ*-normal

**MSC:** 04A05; 54D15

#### **1. Introduction and Preliminaries**

It is evident from the literature that topological structures which are more general than the classical topology are more suitable for the study of digital topology, image processing, network theory, pattern recognition and related areas. Various generalized structures such as closure spaces, generalized closure spaces, Cech closure spaces, generalized topolo- ˇ gies (GT), weak structures (WS), Generalized neighborhood systems (GNS) etc. were introduced and studied in the past (see [1–5]). However, recently Cech closure spaces ˇ attracted the attention of researchers due to its possibility of application in other applied fields discussed above. Usefulness of this Cech closure setting in variety of allied fields ˇ such as digital topology, computer graphics, image processing and pattern recognition are available in the literature [6–9]. Cech closure space was defined by ˇ Cech [ ˇ 1], are obtained from Kuratowski [10] closure operator by omitting the idempotent condition. In this setting Galton [11] studied the motion of an object in terms of a function giving its position at each time and systematically investigated what a continuous motion looks like. J. Šlapal [6] observed that this structure is more suitable than others for application in digital topology because Cech closure spaces are well-behaved with respect to connect- ˇ edness. Allam et al. [12,13] introduced a new method for generating closure spaces via a binary relation which was subsequently used by G. Liu [14] to establish a one-to-one correspondence between quasi discrete closures and reflexive relation. Furthermore, J. Šlapal and John L. Pfaltz [15] studied network structures via associated closure operators. Higher separation axioms in Cech closure space was introduced by Barbel M. R. Stadler ˇ and F. Peter Stadler [16] in 2003 and discussed the concept of Urysohn functions, normal, regular, completely normal etc. in the form of neighborhood. In 2018 Gupta and Das [17] introduced higher separation axioms via relation. Since normality is an important topological property, many weak variants of normality introduced and studied in the past to properly study normality in general topology (See [18–22]). In the present paper, we introduced some variants of normality in Cech closure space as ˇ *π*-normal, weakly *π*-normal and *κ*-normal using canonically closed sets. It is observed that some characterizations of normality and almost normality which holds in topological spaces may not hold in

**Citation:** Gupta, R.; Das, A.K. Some Variants of Normal Cech Closure ˇ Spaces via Canonically Closed Sets. *Mathematics* **2021**, *9*, 1225. https:// doi.org/10.3390/math9111225

Academic Editors: Lorentz Jäntschi and Mihaela Tomescu

Received: 12 April 2021 Accepted: 24 May 2021 Published: 27 May 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Cech closure spaces. Further relation between newly defined notions and already defined ˇ notions was also investigated.

A closure space is a pair (*X*, *cl*), where *X* is any set and closure *cl* : *P*(*X*) → *P*(*X*) is a function associating with each subset *A* ⊆ *X* to a subset *cl*(*A*) ⊆ *X*, called the closure of *A*, such that *cl*(∅) = ∅, *A* ⊆ *cl*(*A*), *cl*(*A* ∪ *B*) = *cl*(*A*) ∪ *cl*(*B*). With any closure *cl* for a set *X* there is associated the interior operation *intcl*, usually denoted by *int*, which is a single-valued relation on *P*(*X*) ranging in *P*(*X*) such that for each *A* ⊆ *X*, *intcl*(*A*) = *X* − *cl*(*X* − *A*). The set *intcl*(*A*) is called the interior of *A* in (*X*, *cl*). In a closure space (*X*, *cl*), a set *A* is closed if *cl*(*A*) = *A* and open if its complement is closed i.e., if *cl*(*X* − *A*)=(*X* − *A*). In other words, a set is open if and only if *int*(*A*) = *A*. Additionally, from closure axioms we have *cl*(*A* ∩ *B*) ⊆ *cl*(*A*) ∩ *cl*(*B*) and *int*(*A*) ∪ *int*(*B*) ⊆ *int*(*A* ∪ *B*). In a Cech closure space a canonically closed (regularly closed) set is a closed set ˇ *A* of *X* such that *cl*(*int*(*A*)) = *A* and a canonically open (regularly open) set is an open set *U* of *X* such that *int*(*cl*(*U*)) = *U*.

**Definition 1.** *[23] A Cech closure space ˇ* (*X*, *cl*) *is said to be*


**Remark 1.** *The notion of normality defined above in the Definition 1 is different from the notion of normality defined in [1]. A closure space is said to be normal [1] if every pair of sets with disjoint closures are separated by disjoint neighborhoods. The disjoint sets considered by Cech for ˇ separation in the definition of normality are not necessarily closed sets and neighborhoods need not be open. Throughout the present paper, we have taken the notion of normality only in the sense of Definition 1.*

**Lemma 1.** *[1] If U and V are subsets of a closure space* (*X*, *cl*) *such that U* ⊆ *V then cl*(*U*) ⊆ *cl*(*V*)*.*

**Theorem 1.** *[23] Suppose* (*X*, *cl*) *is a Cech closure space such that ˇ int*(*cl*(*U*)) *is canonically open for every open set U. Then* (*X*, *cl*) *is weakly normal and almost normal implies* (*X*, *cl*) *is normal.*

## **2. Variants of Normal Cech Closure Space ˇ**

**Definition 2.** *Let* (*X*, *cl*) *be a Cech closure space then ˇ A is said to be π-closed if it is equal to the intersection of two canonically closed set.*

**Example 1.** *Let X* = {*a*, *b*, *c*, *d*} *be the set and define cl* : *P*(*X*) → *P*(*X*) *as cl*({*a*}) = {*a*}, *cl*({*b*}) = *cl*({*a*, *b*}) = {*a*, *b*}, *cl*({*c*}) = *cl*({*a*, *c*}) = *cl*({*c*, *d*}) = *cl*({*a*, *c*, *d*}) = {*a*, *c*, *d*}, *cl*({*d*}) = {*d*}, *cl*({*a*, *d*}) = {*a*, *d*}, *cl*({*b*, *c*}) = *cl*({*a*, *b*, *c*}) = *cl*({*b*, *c*, *d*}) = *cl*(*X*) = *X*, *cl*({*b*, *d*}) = *cl*({*a*, *b*, *d*}) = {*a*, *b*, *d*}, *cl*(∅) = ∅*. Here, the set A* = {*a*} *is π-closed as it is the intersection of two canonically closed set i.e.,* {*a*, *c*, *d*} *and* {*a*, *b*} *but* {*a*} *is not canonically closed. In this Cech closure space, <sup>ˇ</sup> cl*(*A*) = {*d*} <sup>=</sup> *<sup>A</sup> is closed but not <sup>π</sup>-closed as it is not equal to the intersection of two canonically closed set.*

The implications in Figure 1 are obvious from the definitions. However, none of these implications is reversible as shown in the above example.

canonically closed *π*-closed closed

**Figure 1.** Interrelation of types of closed sets.

**Definition 3.** *A Cech closure space ˇ* (*X*, *cl*) *is π-normal if for every two disjoint closed sets one of which is π-closed there exist two disjoint open sets U and V containing the closed set and the π-closed set respectively.*

It is obvious that in a Cech closure space ˇ (*X*, *cl*), every normal space is *π*-normal. However, the converse need not be true as shown below.

**Example 2.** *A Cech closure space which is ˇ π-normal but not normal. Let X* = *Y* ∪ {*p*, *q*} *be an infinite set, then any set A* ∈ *P*(*X*) *is one of the following four types of sets:*

*Type-I: A is finite in X. Type-II: A is infinite in Y such that p* ∈/ *A and q* ∈/ *A. Type-III:* (*Y* − *A*) *is finite and A contains either p or q. Type-IV:* (*Y* − *A*) *is finite and A contains both p and q.*

*Define cl* : *P*(*X*) → *P*(*X*) *by*

$$cl(A) = \begin{cases} \begin{array}{l} A\_{\prime} \\\\ A \cup \{p, q\}, & \text{if } A \text{ is of type-II;} \\\\ A \cup \{p, q\}, & \text{if } A \text{ is of type-III;} \\\\ A\_{\prime} \end{array} \end{cases}$$

*In this Cech closure space, type-I and type-IV sets are closed sets. A set ˇ U is open if U is an infinite set containing p and*/*or q whose complement is finite. Additionally, a finite set U in Y whose complement is infinite is an open set in X. In this space only two types of sets are canonically closed. i.e., (1) Every finite set in Y is canonically closed (2) a set containing both p and q whose complement is finite in Y is canonically closed. This space is π-normal but not normal because for two disjoint closed sets A* = *C* ∪ {*p*} *and B* = *D* ∪ {*q*}*, where C and D are finite in Y, there does not exist disjoint open sets satisfying the condition of normal Cech closure space. ˇ*

**Example 3.** *A space which is not π-normal. Let X be the set of integers defined by*

> *cl*({*x*}) = ⎧ ⎨ ⎩ *x*, *if x is even ;* {*x* − 1, *x*, *x* + 1}, *if x is odd .*

*and cl*(*A*) = *x*∈*A cl*(*x*)*.*

*This Cech closure space is not <sup>ˇ</sup> <sup>π</sup>-normal because for the <sup>π</sup>-closed set cl*(*A*) = {4} <sup>=</sup> *<sup>A</sup> and a closed set cl*(*B*) = {0, 1, 2} = *B there does not exist disjoint open sets containing cl*(*A*) *and cl*(*B*) *respectively.*

Following examples establish that the notion of weak normality defined earlier, and the notion of *π*-normality are independent notions.

**Example 4.** *A space which is weakly normal but not π-normal. Let X be the set of positive integers. Define cl* : *P*(*X*) → *P*(*X*) *as defined in Example 3. Here, the Cech closure space ˇ* (*X*, *cl*) *is weakly normal but not π-normal as shown in Example 3.*

**Example 5.** *A space which is π-normal but not weakly normal. Let X* = {*a*, *b*, *c*, *d*} *be the set and define cl* : *P*(*X*) → *P*(*X*) *as cl*({*a*}) = {*a*}, *cl*({*b*}) = {*b*}, *cl*({*c*}) = {*a*, *c*, *d*}, *cl*({*d*}) = {*d*}, *cl*({*a*, *b*}) = {*a*, *b*}, *cl*({*a*, *c*}) = {*a*, *c*, *d*}, *cl*({*a*, *d*}) = {*a*, *d*}, *cl*({*b*, *c*}) = *X*, *cl*({*b*, *d*}) = {*b*, *d*}, *cl*({*c*, *d*}) = {*a*, *c*, *d*}, *cl*({*a*, *b*, *c*}) = *X*, *cl*({*a*, *b*, *d*}) = {*a*, *b*, *d*}, *cl*({*a*, *c*, *d*}) = {*a*, *c*, *d*}, *cl*({*b*, *c*, *d*}) = *X*, *cl*(*X*) = *X*, *cl*(∅) = ∅*. Here,* (*X*, *cl*) *is a π-normal Cech closure space which fails to be weakly normal because for two ˇ disjoint closed sets A* = {*a*} = *cl*(*A*) *and B* = {*d*} = *cl*(*B*) *there does not exists an open set U such that cl*(*A*) ⊆ *U and int*(*cl*(*U*)) ∩ *B* = ∅*.*

**Theorem 2.** *If* (*X*, *cl*) *is a π-normal Cech closure space then for every ˇ π-closed set cl*(*A*) = *A and for every open set U containing cl*(*A*) *there exists an open set V such that cl*(*A*) ⊆ *V* ⊆ *cl*(*V*) ⊆ *U.*

**Proof.** Let *cl*(*A*) = *A* be a *π*-closed set and *U* be an open set containing *cl*(*A*). Since, (*X*, *cl*) is *π*-normal, there exist disjoint open sets *V* and *W* such that *cl*(*A*) ⊆ *V* and (*X* − *U*) ⊆ *W* implies *V* ⊆ (*X* − *W*). Thus, by Lemma 1, *cl*(*V*) ⊆ *cl*(*X* − *W*) implies *W* ⊆ *X* − *cl*(*V*). Therefore, (*X* − *U*) ⊆ *W* ⊆ *X* − *cl*(*V*) and hence *cl*(*A*) ⊆ *V* ⊆ *cl*(*V*) ⊆ *U*.

**Theorem 3.** *If* (*X*, *cl*) *is a π-normal Cech closure space then for every closed set ˇ cl*(*A*) = *A and for every π-open set U containing cl*(*A*) *there exists an open set V such that cl*(*A*) ⊆ *V* ⊆ *cl*(*V*) ⊆ *U.*

**Proof.** Let *cl*(*A*) = *A* be a closed set and *U* be a *π*-open set containing *cl*(*A*) implies (*X* − *U*) is a *π*-closed set which is disjoint from the closed set *A*. Since, (*X*, *cl*) is *π*normal, there exist disjoint open sets *V* and *W* such that *cl*(*A*) ⊆ *V* and (*X* − *U*) ⊆ *W*. Thus, *V* ⊆ (*X* − *W*) implies *cl*(*V*) ⊆ *cl*(*X* − *W*)=(*X* − *W*), and so, *W* ⊆ (*X* − *cl*(*V*)). Therefore, (*X* − *U*) ⊆ *W* ⊆ (*X* − *cl*(*V*)) and hence *cl*(*A*) ⊆ *V* ⊆ *cl*(*V*) ⊆ *U*.

**Definition 4.** *[24] A Cech closure space ˇ* (*X*, *cl*) *is said to be regular if for a closed set cl*(*A*) = *A and a point x* ∈/ *cl*(*A*) *there exist disjoint open sets U and V such that x* ∈ *U and cl*(*A*) ⊆ *V.*

**Definition 5.** *[1] A Cech closure space is said to be ˇ*


**Remark 2.** *In a Cech closure space, every normal ˇ T*<sup>1</sup> *space is regular and T*2*. but if we replace normal by π-normal then the result need not be true. Consider the space defined in Example 2 which is π-normal and T*<sup>1</sup> *but neither T*<sup>2</sup> *nor regular. The space is not T*<sup>2</sup> *because disjoint points* '*p' and* '*q' cannot be separated and is not regular because for closed set A* = *C* ∪ {*p*} *where C is finite in Y and a point* '*q' there does not exist disjoint open sets satisfying the required condition.*

**Definition 6.** *[24] A Cech closure space is said to be almost regular if for canonically closed set ˇ cl*(*int*(*A*)) = *A and a point x* ∈/ *cl*(*int*(*A*)) *there exist disjoint open sets U and V such that x* ∈ *U and cl*(*int*(*A*)) ⊆ *V.*

**Theorem 4.** *In a Cech closure space, every ˇ π-normal T*<sup>1</sup> *space is almost regular.*

**Proof.** let *cl*(*int*(*A*)) = *A* be a canonically closed set and *x* ∈/ *cl*(*int*(*A*)) be a point. Since the space is a *<sup>T</sup>*<sup>1</sup> Cech closure space, the singleton set <sup>ˇ</sup> {*x*} is closed. As every canonically closed set is *π*-closed, by *π*-normality there exist disjoint open sets *U* and *V* such that *cl*(*int*(*A*) <sup>⊆</sup> *<sup>U</sup>* and {*x*} ⊆ *<sup>V</sup>*. Hence (*X*, *cl*) is an almost regular Cech closure space. <sup>ˇ</sup>

**Definition 7.** *A Cech closure space is said to be weakly ˇ π-normal if for two disjoint π-closed sets there exist disjoint open sets separating them.*

**Definition 8.** *A Cech closure space is said to be ˇ κ-normal if for two disjoint canonically closed sets A and B there exist disjoint open sets U and V containing A and B respectively.*

From the definitions it is observed that every *π*-normal space is weakly *π*-normal, every weakly *π*-normal space as well as every almost normal space is *κ*-normal. Thus, the implications in Figure 2 are obvious but none of them is reversible which is exhibited below by Examples.

**Figure 2.** Interrelation of variants of normality.

**Example 6.** *A Cech closure space which is weakly ˇ π-normal but not π-normal. Let X* = {*a*, *b*, *c*, *d*} *be a set and define cl* : *P*(*X*) → *P*(*X*) *as cl*({*a*}) = *cl*({*b*}) = *cl*({*a*, *b*}) = {*a*, *b*}, *cl*({*c*}) = {*c*}, *cl*({*d*}) = *cl*({*c*, *d*}) = {*b*, *c*, *d*}, *cl*({*a*, *c*}) = *cl*({*b*, *c*}) = *cl*({*a*, *b*, *c*}) = {*a*, *b*, *c*}, *cl*({*a*, *d*}) = *cl*({*b*, *d*}) = *cl*({*a*, *b*, *d*}) = *cl*({*a*, *c*, *d*}) = *cl*({*b*, *c*, *d*}) = *cl*(*X*) = *X*, *cl*(∅) = ∅*. This space is vacuously weakly π-normal but not π-normal because for the π-closed set* {*a*, *b*} *and a closed set* {*c*}*, there does not exist disjoint open sets containing* {*a*, *b*} *and* {*c*}*.*

**Example 7.** *A Cech closure space which is weakly ˇ π-normal but not almost normal. Let X* = {*a*, *b*, *c*, *d*} *be the set and define cl* : *P*(*X*) → *P*(*X*) *as cl*({*a*}) = {*a*}, *cl*({*b*}) = *cl*({*a*, *b*}) = {*a*, *b*}, *cl*({*c*}) = *cl*({*a*, *c*}) = *cl*({*c*, *d*}) = *cl*({*a*, *c*, *d*}) = {*a*, *c*, *d*}, *cl*({*d*}) = {*d*}, *cl*({*a*, *d*}) = {*a*, *d*}, *cl*({*b*, *d*}) = *cl*({*a*, *b*, *d*}) = {*a*, *b*, *d*}, *cl*({*b*, *c*}) = *cl*({*a*, *b*, *c*}) = *cl*({*b*, *<sup>c</sup>*, *<sup>d</sup>*}) = *cl*(*X*) = *<sup>X</sup>*, *cl*(∅) = <sup>∅</sup>*. Clearly,* (*X*, *cl*) *is a Cech closure space which is vacuously <sup>ˇ</sup> weakly π-normal but not almost normal because for the canonically closed set cl*(*int*(*A*)) = {*a*, *b*} = *A and the closed set cl*(*B*) = {*d*} = *B there does not exist disjoint open sets containing A and B respectively.*

**Example 8.** *A Cech closure space which is ˇ κ-normal but not almost normal. The Cech closure space defined in Example ˇ 7 is vacuously κ-normal but not almost normal as shown in Example 7.*

**Example 9.** *A Cech closure space which is ˇ κ-normal.*

*Let X* = *Y* ∪ {*p*, *q*} *be an infinite set. Define cl* : *P*(*X*) → *P*(*X*) *as in Example 2. Here, the closure space* (*X*, *cl*) *is κ-normal as for two disjoint canonically closed sets there exist disjoint open sets containing them.*

**Example 10.** *A Cech closure space which is not ˇ κ-normal.*

*Let <sup>X</sup> be the set of integers and define cl* : *<sup>P</sup>*(*X*) <sup>→</sup> *<sup>P</sup>*(*X*) *as shown in Example 3. This Cech closure <sup>ˇ</sup> space* (*X*, *cl*) *is not κ-normal because for two disjoint canonically closed sets A* = {0, 1, 2} = *cl*(*int*(*A*)) *and B* = {4, 5, 6} = *cl*(*int*(*B*)) *there does not exist disjoint open sets containing them.*

**Theorem 5.** *If* (*X*, *cl*) *is a weakly π-normal Cech closure space then for every ˇ π-closed set A and for every π-open set U containing A there exists an open set V such that A* ⊆ *V* ⊆ *cl*(*V*) ⊆ *U.*

**Proof.** Let *cl*(*A*) = *A* be a *π*-closed set and *U* be a *π*-open set containing *cl*(*A*). Since, (*X*, *cl*) is weakly *π* normal, there exist disjoint open sets *V* and *W* such that *cl*(*A*) ⊆ *V* and (*X* − *U*) ⊆ *W*. Thus, *V* ⊆ *X* − *W* implies *cl*(*V*) ⊆ *cl*(*X* − *W*)=(*X* − *W*). Therefore, *A* ⊆ *V* ⊆ *cl*(*V*) ⊆ *U*.

**Theorem 6.** *If* (*X*, *cl*) *is a κ-normal Cech closure space then for every canonically closed set ˇ cl*(*int*(*A*)) = *A and for every canonically open set int*(*cl*(*U*)) = *U containing cl*(*int*(*A*)) *there exists an open set V such that cl*(*int*(*A*)) ⊆ *V* ⊆ *cl*(*V*) ⊆ *int*(*cl*(*U*))*.*

**Proof.** Proof of this theorem is similar to the proof of Theorem 5.

**Theorem 7.** *Suppose* (*X*, *cl*) *is a Cech closure space such that ˇ int*(*cl*(*U*)) *is canonically open for every open set U. Then* (*X*, *cl*) *is weakly normal and κ-normal implies* (*X*, *cl*) *is almost normal.*

**Proof.** let *cl*(*int*(*A*)) = *A* be a canonically closed set and *cl*(*B*) = *B* be a closed set disjoint from canonically closed set *cl*(*int*(*A*)) = *A*. Since, (*X*, *cl*) is a weakly normal Cech closure space, there exists an open set <sup>ˇ</sup> *<sup>U</sup>* such that *<sup>A</sup>* <sup>⊆</sup> *<sup>U</sup>* and *int*(*cl*(*U*)) <sup>∩</sup> *<sup>B</sup>* <sup>=</sup> <sup>∅</sup>. Since *int*(*cl*(*U*)) is canonically open, *X* − *int*(*cl*(*U*)) is canonically closed containing *cl*(*B*). Thus, by *κ*-normality there exist disjoint open sets *P* and *Q* such that *cl*(*int*(*A*)) ⊆ *P* and *cl*(*B*) <sup>⊆</sup> *<sup>X</sup>* <sup>−</sup> (*int*(*cl*(*U*))) <sup>⊆</sup> *<sup>Q</sup>*. Hence (*X*, *cl*) is an almost normal Cech closure space. <sup>ˇ</sup>

**Theorem 8.** *Suppose* (*X*, *cl*) *is a T*<sup>1</sup> *Cech closure space such that ˇ int*(*cl*(*U*)) *is canonically open for every open set U. Then* (*X*, *cl*) *is weakly π-normal and weakly normal implies* (*X*, *cl*) *is almost regular.*

**Proof.** let *cl*(*int*(*A*)) = *A* be a canonically closed set and *x* ∈/ *cl*(*int*(*A*)) be a point. Since (*X*, *cl*) is *T*1, the singleton set {*x*} is closed. By weak normality, there exists an open set *U* such that *A* ⊆ *U* and *int*(*cl*(*U*)) ∩ {*x*} = ∅. Since *int*(*cl*(*U*)) is canonically open, *X* − *int*(*cl*(*U*)) is canonically closed containing {*x*}. Thus, by weak *π*-normality, there exist disjoint open sets *P* and *Q* such that *cl*(*intA*) ⊆ *P* and {*x*} ⊆ *X* − (*int*(*cl*(*U*))) ⊆ *Q*. Hence (*X*, *cl*) is an almost regular Cech closure space. ˇ

It is clear from Example 11 that the *T*<sup>1</sup> axiom cannot be relaxed from the Theorem 8 as the space is weakly *π*-normal and weakly normal but not almost regular.

**Example 11.** *Let X* = {*a*, *b*, *c*} *be the set and define cl* : *P*(*X*) → *P*(*X*) *as cl*({*a*}) = {*a*}, *cl*({*b*}) = {*a*, *b*}, *cl*({*c*}) = {*a*, *c*}, *cl*({*a*, *b*}) = {*a*, *b*}, *cl*({*a*, *c*}) = {*a*, *c*}, *cl*({*b*, *c*}) = *cl*(*X*) = *X*, *cl*(∅) = ∅*. Clearly,* (*X*, *cl*) *is a Cech closure space which is weakly ˇ π-normal and weakly normal but not almost regular.*

**Definition 9.** *[24] A Cech closure space ˇ* (*X*, *cl*) *is said to be β-normal if for two disjoint closed sets cl*(*A*) = *A and cl*(*B*) = *B there exist disjoint open sets U and V whose closures are disjoint such that cl*(*A* ∩ *U*) = *cl*(*A*) *and cl*(*B* ∩ *V*) = *cl*(*B*)*.*

**Definition 10.** *[24] A Cech closure space is extremally disconnected (E. D) if for every open set ˇ U, cl*(*U*) *is open.*

**Example 12.** *A Space which is extremally disconnected.*

*Let X* = {*a*, *b*, *c*, *d*} *be the set. Define cl* : *P*(*X*) → *cl as cl*({*a*}) = *cl*({*a*, *c*}) = {*a*, *c*}, *cl*({*b*}) = {*b*}, *cl*({*c*}) = {*c*}, *cl*({*d*}) = *cl*({*b*, *d*}) = {*b*, *d*}, *cl*({*a*, *b*}) = *cl*({*a*, *b*, *c*}) = {*a*, *b*, *c*}, *cl*({*b*, *c*}) = {*b*, *c*}, *cl*({*c*, *d*}) = *cl*({*b*, *c*, *d*}) = {*b*, *c*, *d*}, *cl*({*a*, *d*}) = *cl*({*a*, *b*, *d*}) = *cl*({*a*, *c*, *d*}) = *cl*(*X*) = *X*, *cl*(∅) = ∅*. In this space, closure of every open set is open. Thus, the space is extremally disconnected.*

**Theorem 9.** *In an extremally disconnected Cech closure space ˇ* (*X*, *cl*)*, every β-normal space is κ-normal.*

**Proof.** Let *cl*(*int*(*A*)) = *A* and *cl*(*int*(*B*)) = *B* be two disjoint canonically closed sets. Thus, *cl*(*int*(*A*)) and *cl*(*int*(*B*)) are two disjoint closed sets. We must show (*X*, *cl*) is *κ*-normal. Since (*X*, *cl*) is *β*-normal, there exist disjoint open sets *U* and *V* such that *cl*(*cl*(*A*) ∩ *U*) =

*cl*(*A*), *cl*(*cl*(*B*) ∩ *V*) = *cl*(*B*) and *cl*(*U*) ∩ *cl*(*V*) = ∅. Thus, *cl*(*A*) = *cl*(*cl*(*A*) ∩ *U*) ⊆ *cl*(*U*) and *cl*(*B*) = *cl*(*cl*(*B*) ∩ *V*) ⊆ *cl*(*V*). By extremally disconnectedness of (*X*, *cl*), *cl*(*U*) and *cl*(*V*) are two disjoint open sets containing *cl*(*A*) and *cl*(*B*) respectively. Hence (*X*, *cl*) is *κ*-normal.

**Example 13.** *A Cech closure space which is ˇ κ-normal but not β-normal.*

*Let X* = *Y* ∪ {*p*, *q*} *be an infinite set. Define cl* : *P*(*X*) → *P*(*X*) *as in Example 2. Here, the closure space* (*X*, *cl*) *is κ-normal but not β-normal because for two disjoint closed sets cl*(*A*) = *C* ∪ {*p*} *and cl*(*B*) = *D* ∪ {*q*}*, where C and D are finite in Y, there does not exist disjoint open sets satisfying the condition of β-normal Cech closure space. ˇ*

**Example 14.** *Let X be an infinite set. Define cl* : *P*(*X*) → *P*(*X*) *as defined in [1] by*

$$cl(A) = \begin{cases} \begin{array}{c} A\_{\prime} & \text{if } A \text{ is finite;} \\\\ X\_{\prime} & \text{otherwise.} \end{array} \end{cases}$$

*Here,* (*X*, *cl*) *is a Cech closure space which is ˇ T*<sup>1</sup> *almost normal but not regular because for closed set cl*(*A*) = *A and a point disjoint from the closed set A there does not exist disjoint open sets separating them.*

The following theorem directly follows from the Theorem 1.

**Theorem 10.** *Suppose* (*X*, *cl*) *is a weakly normal Cech closure space such that ˇ int*(*cl*(*U*)) *is canonically open for every open set U. Then following are equivalent:*


#### **3. Discussion and Conclusions**

Closure space was first appeared in 1966 in the book "Topological Spaces" is popularly known as Cech closure space in the name of the author of the book E. ˇ Cech. After many ˇ decades of its introduction, it is now slowly becoming objects of increasing interest and importance. The purpose of this discussion is to discuss some important developments in this area in the last two decades. In 2003, some higher separation axioms including completely regular and completely normal spaces are studied in closure setting by Stadler et al. [16]. In 2008, Dimitrije Andrijevi´c and others [25] considered families of subset of a closure space equipped with different Vietoris-like topologies and studied properties such as connectedness and compactness of the space and its hyperspaces. Subsequently in 2010, they generalized the notions of the compact-open and graph topology to the set of functions between two Cech closure spaces [ ˇ 26]. Additionally, they investigated how the separation properties (*T*0, *T*<sup>1</sup> and regular) of the initial spaces are related to those of function spaces.

Recently, in 2021, Antonio Rieser [27] studied homotopy theory on the category of Cech closure spaces, whose objects are sets endowed with a ˇ Cech closure operator and ˇ whose morphisms are the continuous maps between them. They introduced some new classes of Cech closure structures on metric spaces, graphs, and simplicial complexes. ˇ

Another approach of generating closure spaces via a binary relation was also adopted by many researchers to address various issues in mathematics and other allied fields (see [12–15]). In [17], we have introduced and studied some new separation axioms on closure spaces generated through binary relations.

Apart from this, Junsheng Qiao [28] shown that the category of Cech closure spaces ˇ can be embedded in the category of stratified L-Cech closure spaces as a coreflective ˇ full subcategory. Perfilieva et al. [29] investigated the relationship between L-Fuzzy Cech closure spaces and L-Fuzzy co-topological spaces from the categorical viewpoint. ˇ Relational variants of categories related to L-Fuzzy closure spaces was studied in [30].

In this paper, we have defined and investigated few variants of normality in Cech clo- ˇ sure spaces using canonically closed sets. Normality is an important topological property, and its importance is due to its behaviour as it behaves differently from other separation axioms for subspaces and products. Additionally, the class of normal spaces are more general than the important class of compact Hausdorff spaces. Normality involves separation of closed sets by open sets. On the other hand, in digital image processing a picture needs to be segmented into subsets where relationship of these subset from other neighboring subsets and adjoining points plays a prominent role for the processing of images. Such types of relationships between sets/points are either geometrical or topological. Geometrical relation involves position of points whereas topological relation involves concepts such as adjacency, neighborhood, separation, connectedness and compactness. So, the possibility of application of the notions defined in this paper in digital topology and digital image processing cannot be ruled out.

**Author Contributions:** Investigation: R.G. and A.K.D.; Supervision: A.K.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** The first author received fellowship from Department of Science and Technology (DST), Government of India (IF140967).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are thankful to the referee for many valuable remarks that essentially improved the paper, and the first author is thankful to Department of Science and Technology (DST), Government of India for awarding INSPIRE fellowship (IF140967).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Deepak Kumar 1,\*, Sunil Kumar 2, Janak Raj Sharma <sup>1</sup> and Lorentz Jantschi 3,4,\***


**Abstract:** We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric.

**Keywords:** local convergence; nonlinear equations; Banach space; Fréchet-derivative

## **1. Introduction**

Let *X*, *Y* be Banach spaces and *D* ⊆ *X* be a closed and convex set. In this study, we locate a solution *x*∗ of the nonlinear equation

$$G(\mathfrak{x}) = 0,\tag{1}$$

where *G* : *D* ⊆ *X* → *Y* is a Fréchet-differentiable operator. In computational sciences, many problems can be written in the form of (1). See, for example, [1–3]. The solutions of such equations are rarely attainable in closed form. This is why most methods for solving these equations are usually iterative. The most well-known method for approximating a simple solution *x*∗ of Equation (1) is Newton's method, which is given by

$$\mathbf{x}\_{m+1} = \mathbf{x}\_m - G'(\mathbf{x}\_m)^{-1} G(\mathbf{x}\_m), \quad \text{for each } m = 0, 1, 2, \dots \tag{2}$$

and has a quadratic order of convergence. In order to attain the higher order of convergence, a number of modified Newton's or Newton-like methods have been proposed in the literature (see [2–20]) and references cited therein. In particular, Sharma and Kumar [18] recently proposed a fifth order method for approximating the solution of *G*(*x*) = 0 using the Newton–Chebyshev composition defined for each *n* = 0, 1, 2, . . . by

$$\begin{array}{rcl} y\_m &=& \pi\_m - \Gamma\_m G(\boldsymbol{x}\_m), \\\\ z\_m &=& y\_m - \Gamma\_m G(\boldsymbol{y}\_m), \end{array} \tag{3}$$

$$\boldsymbol{x}\_{m+1} \quad = \quad z\_m - \left(2\,I - \Gamma\_n[z\_{m\prime} \boldsymbol{y}\_m; \mathcal{G}]\right) \Gamma\_m \mathcal{G}(z\_m),$$

**Citation:** Kumar, D.; Kumar, S.; Sharma, J.R.; Jantschi, L. Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces. *Mathematics* **2021**, *9*, 2510. https://doi.org/10.3390/math9192510

Academic Editors: Juan Benigno Seoane-Sepúlveda and Alicia Cordero Barbero

Received: 14 August 2021 Accepted: 26 September 2021 Published: 7 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

where Γ*<sup>m</sup>* = *G* (*xm*)−1, and [*zm*, *ym* ; *G*] is the first order divided difference of *G*. The method has been shown to be computationally more efficient than existing methods of a similar nature.

The important part in the development of an iterative method is to study its convergence analysis. This is usually divided into two categories, namely the semilocal and local convergence. The semilocal convergence is based on the information around an initial point and gives criteria that ensure the convergence of iteration procedures. The local convergence is based on the information of a convergence domain around a solution and provides estimates of the radii of the convergence balls. Local results are important since they provide the degree of difficulty in choosing initial points. There exist many studies which deal with the local and semilocal convergence analysis of iterative methods such as [3–5,7–11,13,16,19,21–23]. The semilocal convergence of the method (3) in Banach spaces has been established in [18]. In the present work, we study the local convergence of this method and its multi-step version, including the computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution.

We summarize the contents of the paper. In Section 2, the local convergence (including radius of convergence, error bounds, and uniqueness results of method (3)) is studied. The generalized multi-step version is presented in Section 3. Numerical examples are performed to verify the theoretical results in Section 4. In Section 5, the basins of attractors are studied to visually check the convergence domain of the methods. Finally, some conclusions are reported in Section 6.

#### **2. Local Convergence**

The local convergence analysis of method (3) is presented in this section. Let *L*<sup>0</sup> > 0, *L* > 0, *L*<sup>1</sup> > 0, and *M* ≥ 0 be given parameters. It is convenient to generate some functions and parameters for the local convergence study that follows. Define function *g*1(*t*) on interval [0, <sup>1</sup> *L*0 ) by

$$g\_1(t) = \frac{Lt}{2(1 - L\_0 t)}$$

$$g\_2 = \dots$$

and parameter

$$r\_1 = \frac{2}{2L\_0 + L} < \frac{1}{L\_0}.\tag{4}$$

Then, we have that *g*1(*r*1) = 1 and 0 ≤ *g*1(*t*) ≤ 1 for each *t* ∈ [0,*r*1). Moreover, define the function *g*2(*t*) and *h*2(*t*) on interval [0, <sup>1</sup> *L*0 ) by

$$\lg\_2(t) = \left(1 + \frac{M}{1 - L\_0 t}\right) \lg\_1(t)$$

and

$$h\_2(t) = \wp\_2(t) - 1.$$

We have that *<sup>h</sup>*2(0) = <sup>−</sup><sup>1</sup> <sup>&</sup>lt; 0 and *<sup>h</sup>*2(*r*1) = *<sup>M</sup>* <sup>1</sup>−*L*0*r*<sup>1</sup> <sup>&</sup>gt; 0. According to the intermediate value theorem, function *h*2(*t*) has zeros in the interval (0,*r*1). Denote such zeros by *r*2. Finally, define functions *K*(*t*), *g*3(*t*), and *h*3(*t*) on the interval [0, <sup>1</sup> *L*0 ) by

$$\begin{aligned} K(t) &= 1 + \frac{1}{1 - L\_0 t} \left( L\_0 + L\_1 t (\mathcal{g}\_2(t) + \mathcal{g}\_1(t)) \right) t, \\\\ \mathcal{g}\_3(t) &= \left( 1 + \frac{MK(t)}{1 - L\_0 t} \right) \mathcal{g}\_2(t) \\\\ h\_3(t) &= \mathcal{g}\_3(t) - 1. \end{aligned}$$

and

We have that *<sup>h</sup>*3(0) = <sup>−</sup><sup>1</sup> <sup>&</sup>lt; 0 and *<sup>h</sup>*3(*r*2) = *MK*(*r*2) <sup>1</sup>−*L*0*r*<sup>2</sup> <sup>&</sup>gt; 0. According to the intermediate value theorem, function *h*3(*t*) has zeros in (0,*r*2). Denote such zeros by *r*<sup>3</sup> of function *h*3(*t*) in interval [0,*r*2). Set

$$r = \min\{r\_i\}, \quad i = 1, 2, 3. \tag{5}$$

Then, we obtain that

Then, for each *t* ∈ [0,*r*)

$$0 < r \le r\_1. \tag{6}$$

$$0 \le g\_1(t) \le 1,\tag{7}$$

$$0 \le \gcd\_2(t) \le 1\tag{8}$$

and

$$0 \le \gcd\_{\mathbb{S}}(t) \le 1. \tag{9}$$

Let *U*(*v*, *ρ*) and *U*¯ (*v*, *ρ*) symbolise the open and closed balls in *X*, with a radius *ρ* > 0 and a centre *v* ∈ *X*.

Using the above notations, we then describe the local convergence analysis of method (3).

**Theorem 1.** *Suppose G* : *D* ⊆ *X* → *Y is a Fréchet-differentiable function. Let* [., .; *G*] : *X* × *X* → *L*(*Y*) *be the divided difference operator. Consider that there exist x*<sup>∗</sup> ∈ *D, L*<sup>0</sup> > 0*, L* > 0*, L*<sup>1</sup> > 0*, and M* ≥ 1*, such that for each x*, *y* ∈ *D*

$$G(\mathfrak{x}^\*) = 0,\ G(\mathfrak{x}^\*)^{-1} \in L(\mathcal{Y}, X),\tag{10}$$

$$\left\|\left\|G'(\mathbf{x}^\*)^{-1}\left(G'(\mathbf{x}) - G'(\mathbf{x}^\*)\right)\right\| \le L\_0 \|\mathbf{x} - \mathbf{x}^\*\|,\tag{11}$$

$$\|\|G'(\mathbf{x}^\*)^{-1}(G'(\mathbf{x}) - G'(y))\|\| \le L \|\mathbf{x} - y\|\|,\tag{12}$$

$$\|\|G'(\mathfrak{x}^\*)^{-1}G'(\mathfrak{x})\|\|\leq M,\tag{13}$$

$$\|\|G'(\mathbf{x}^\*)^{-1}([\mathbf{x}, y; G] - G'(\mathbf{x}^\*))\|\| \le L\_1(\|\|\mathbf{x} - \mathbf{x}^\*\| + \|y - \mathbf{x}^\*\|),\tag{14}$$

*and*

$$
\bar{\mathcal{U}}(\mathfrak{x}^\*, r) \subset D,\tag{15}
$$

*where r is defined by (5). Then, for each m* = 0, 1, ...*, the sequence* {*xm*} *generated by method (3) for x*<sup>0</sup> ∈ *U*(*x*∗,*r*) − {*x*∗} *is well defined, stays in U*(*x*∗,*r*)*, and converges to x*∗*. Furthermore, the following estimates hold:*

$$\|y\_m - \mathbf{x}^\*\| \le g\_1(\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\mathbf{x}\_m - \mathbf{x}^\*\| < \|\mathbf{x}\_m - \mathbf{x}^\*\| < r,\tag{16}$$

$$\|\|z\_{m} - \mathbf{x}^\*\|\| \le \mathbf{g}\_2(\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\|\mathbf{x}\_m - \mathbf{x}^\*\|\| < \|\|\mathbf{x}\_m - \mathbf{x}^\*\|\| < r \tag{17}$$

*and*

$$\|\|\mathbf{x}\_{m+1} - \mathbf{x}^\*\|\| \le \mathcal{g}\_3(\|\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\|\mathbf{x}\_m - \mathbf{x}^\*\|\|,\tag{18}$$

*where the* "*g*" *functions are defined previously. Furthermore, if there exists <sup>T</sup>* <sup>∈</sup> [*r*, <sup>2</sup> *L*0 ) *such that <sup>U</sup>*¯ (*x*∗, *<sup>T</sup>*) <sup>⊂</sup> *D, then x*<sup>∗</sup> *is the only solution of G*(*x*) = <sup>0</sup> *in <sup>U</sup>*¯ (*x*∗, *<sup>T</sup>*)*.*

**Proof.** We shall show the estimates (16)–(18) using mathematical induction. Using (4), (11), and the hypotheses *x*<sup>0</sup> ∈ *U*(*x*∗,*r*) − {*x*∗}, we obtain that

$$\left\|\left\|G'(\mathbf{x}^\*)^{-1}(G(\mathbf{x}\_0) - G(\mathbf{x}^\*))\right\|\right\| \le L\_0 \left\|\mathbf{x}\_0 - \mathbf{x}^\*\right\| < L\_0 r < 1. \tag{19}$$

It follows from (19) and the Banach Lemma [3] that *G* (*x*0)−<sup>1</sup> ∈ *<sup>L</sup>*(*Y*, *<sup>X</sup>*) and

$$\|\|G'(\mathbf{x}\_0)^{-1}G'(\mathbf{x}^\*)\|\| \le \frac{1}{1 - L\_0 \|\|\mathbf{x}\_0 - \mathbf{x}^\*\|} < \frac{1}{1 - L\_0 r}.\tag{20}$$

Hence, *y*<sup>0</sup> is well defined for *m* = 0. Then, by using (4), (7), (12), and (20), we have

$$\begin{split} \|y\_0 - \mathbf{x}^\*\| &\le \|\mathbf{x}\_0 - \mathbf{x}^\* - G'(\mathbf{x}\_0)^{-1} G(\mathbf{x}\_0)\| \\ &\le \|G'(\mathbf{x}\_0)^{-1} G'(\mathbf{x}^\*)\| \left\| \int\_0^1 G'(\mathbf{x}^\*)^{-1} [G'(\mathbf{x}^\* + \theta(\mathbf{x}\_0 - \mathbf{x}^\*)) - G'(\mathbf{x}\_0)] \right\| \|d\theta \\ &\quad \times \|\mathbf{x}\_0 - \mathbf{x}^\*\| \\ &\le \frac{L\|\mathbf{x}\_0 - \mathbf{x}^\*\|^2}{2(1 - L\_0 \|\|\mathbf{x}\_0 - \mathbf{x}^\*\|)} \\ &= g\_1(\|\|\mathbf{x}\_0 - \mathbf{x}^\*\|) \|\mathbf{x}\_0 - \mathbf{x}^\*\| < \|\mathbf{x}\_0 - \mathbf{x}^\*\| < r, \end{split} \tag{21}$$

which shows (16) for *m* = 0 and *y*<sup>0</sup> ∈ *U*(*x*∗,*r*).

Notice that for each *θ* ∈ [0, 1] and *x*<sup>∗</sup> + *θ*(*x*<sup>0</sup> − *x*∗) − *x*∗ = *θx*<sup>0</sup> − *x*∗ < *r*. That is, *x*<sup>∗</sup> + *θ*(*x*<sup>0</sup> − *x*∗) ∈ *U*(*x*∗,*r*). We can write

$$G(\mathbf{x}\_0) = G(\mathbf{x}\_0) - G(\mathbf{x}^\*) = \int\_0^1 G'(\mathbf{x}^\* + \theta(\mathbf{x}\_0 - \mathbf{x}^\*))(\mathbf{x}\_0 - \mathbf{x}^\*)d\theta. \tag{22}$$

Then, using (13) and (21), we have

$$\begin{split} \|\|G'(\mathbf{x}^\*)^{-1}G(\mathbf{x}\_0)\|\| &= \left\| \int\_0^1 G'(\mathbf{x}^\*)^{-1}G'(\mathbf{x}^\* + \theta(\mathbf{x}\_0 - \mathbf{x}^\*))(\mathbf{x}\_0 - \mathbf{x}^\*)d\theta \right\|\\ &\leq M\|\|\mathbf{x}\_0 - \mathbf{x}^\*\|. \end{split} \tag{23}$$

Similarly, we obtain

$$\|\|G'(\mathbf{x}^\*)^{-1}G(y\_0)\|\| \le M\|\|y\_0 - \mathbf{x}^\*\|\|,\tag{24}$$

$$\|\|G'(\mathfrak{x}^\*)^{-1}G(z\_0)\|\| \le M\|z\_0 - \mathfrak{x}^\*\|.\tag{25}$$

Using the second substep of method (3), (8), (20), (21), (27), and (24), we obtain that

$$\begin{split} \|z\_{0} - \mathbf{x}^{\*}\| &\leq \quad \|y\_{0} - \mathbf{x}^{\*}\| + \|G'(\mathbf{x}\_{0})^{-1}G(y\_{0})\| \\ &= \quad \|y\_{0} - \mathbf{x}^{\*}\| + \|G'(\mathbf{x}\_{0})^{-1}G'(\mathbf{x}^{\*})\| \|\|G'(\mathbf{x}^{\*})^{-1}G(y\_{0})\| \\ &\leq \quad \|y\_{0} - \mathbf{x}^{\*}\| + \frac{M\|y\_{0} - \mathbf{x}^{\*}\|}{1 - L\_{0}\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|} \\ &\leq \quad \left(1 + \frac{M}{1 - L\_{0}\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|}\right) \|y\_{0} - \mathbf{x}^{\*}\| \\ &\leq \quad \left(1 + \frac{M}{1 - L\_{0}\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|}\right) \mathcal{g}\_{1}(\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|) \|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| \\ &\leq \quad \mathcal{g}\_{2}(\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|) \|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| < \,\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| < r. \end{split} \tag{26}$$

Which shows (17) for *m* = 0 and *z*<sup>0</sup> ∈ *U*(*x*∗,*r*).

Next, we have the linear operator *A*<sup>0</sup> = 2*I* − *G* (*x*0)−1[*y*0, *x*0; *G*]; by using (11), (14), and (20), we obtain

*A*<sup>0</sup> = 2*I* − *G* (*x*0)−1[*z*0, *<sup>y</sup>*0; *<sup>G</sup>*] ≤ 1 + *G* (*x*0)−<sup>1</sup> *G* (*x*0) − [*z*0, *y*0; *G*] ≤ 1 + *G* (*x*0)−1*G* (*x*∗)*G* (*x*∗)−1(*G* (*x*0) − [*z*0, *y*0; *G*]) ≤ 1 + *G* (*x*0)−1*G* (*x*∗)*G* (*x*∗)−<sup>1</sup> *G* (*x*0) − *G* (*x*∗) + *G* (*x*∗) − [*z*0, *y*0; *G*] ≤ 1 + *G* (*x*0)−1*G* (*x*∗) *G* (*x*∗)−1(*G* (*x*0) − *G* (*x*∗)) + *G* (*x*∗)−1(*G* (*x*∗) − [*z*0, *y*0; *G*]) <sup>≤</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup> 1−*L*<sup>0</sup>*x*0−*x*∗ *L*<sup>0</sup>*x*<sup>0</sup> − *x*∗ + *L*<sup>1</sup> *z*<sup>0</sup> − *x*∗ + *y*<sup>0</sup> − *x*∗ <sup>≤</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup> 1−*L*<sup>0</sup>*x*0−*x*∗ *L*<sup>0</sup>*x*<sup>0</sup> − *x*∗ + *L*<sup>1</sup> *g*2(*x*<sup>0</sup> − *x*∗) + *g*1(*x*<sup>0</sup> − *x*∗) *x*<sup>0</sup> − *x*∗ <sup>≤</sup> <sup>1</sup> <sup>+</sup> <sup>2</sup> 1−*L*<sup>0</sup>*x*0−*x*∗ *L*<sup>0</sup> + *L*<sup>1</sup> *g*2(*x*<sup>0</sup> − *x*∗) + *g*1(*x*<sup>0</sup> − *x*∗) *x*<sup>0</sup> − *x*∗ = *K*(*x*<sup>0</sup> − *x*∗). (27)

Then, using Equations (4), (9), (25), and (26), we obtain that

$$\begin{split} \|\mathbf{x}\_{1} - \mathbf{x}^{\*}\| &\leq \quad \|z\_{0} - \mathbf{x}^{\*}\| + \|A\_{0}\|\|G'(\mathbf{x}\_{0})^{-1}G(z\_{0})\| \\ &= \quad \|z\_{0} - \mathbf{x}^{\*}\| + \|A\_{0}\|\|G'(\mathbf{x}\_{0})^{-1}G'(\mathbf{x}^{\*})\| \|\|G'(\mathbf{x}^{\*})^{-1}G(z\_{0})\| \\ &\leq \quad \|z\_{0} - \mathbf{x}^{\*}\| + \frac{MK(\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|)\|z\_{0} - \mathbf{x}^{\*}\|}{1 - L\_{0}\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|} \\ &\leq \quad \left(1 + \frac{MK(\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|)}{1 - L\_{0}\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|}\right) \|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| \\ &\leq \quad \left(1 + \frac{MK(\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|)}{1 - L\_{0}\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\|}\right) \|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| \\ &\leq \quad \lesssim\_{\mathbf{C}} \quad \|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| \|\|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| < \|\mathbf{x}\_{0} - \mathbf{x}^{\*}\| < r\_{\prime} \end{split} \tag{28}$$

which proves the (18) for *m* = 0 and *x*<sup>1</sup> ∈ *U*(*x*∗,*r*). By simply replacing *x*0, *y*0, *z*0, and *x*<sup>1</sup> by *xm*, *ym*, *zm*, and *xm*+<sup>1</sup> in the preceding estimates, we arrive at (16)–(18). Then, from the estimates *xm*+<sup>1</sup> − *x*∗ < *xm* − *x*∗ < *r*, we deduce that lim*m*→∞*xm* = *x*<sup>∗</sup> and *xm*+<sup>1</sup> ∈ *U*(*x*∗,*r*).

Finally, we show the uniqueness part; let *Q* = <sup>1</sup> <sup>0</sup> *G* (*y*<sup>∗</sup> + *t*(*x*<sup>∗</sup> − *y*∗))*dt* for some *<sup>y</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>U</sup>*¯ (*x*∗,*r*) with *<sup>G</sup>*(*y*∗) = 0. Using (15), we obtain that

$$\|G'(\mathbf{x}^\*)^{-1}(Q - G'(\mathbf{x}^\*))\| \quad \le \int\_0^1 L\_0 \|y^\* + t(\mathbf{x}^\* - y^\*) - \mathbf{x}^\*\| dt$$

$$\le \int\_0^1 (1 - t) \|\mathbf{x}^\* - y^\*\| dt \tag{29}$$

$$\le \frac{L\_0}{2} T < 1.$$

It follows from (29) that *Q* is invertible. Then, from the identity 0 = *G*(*x*∗) − *G*(*y*∗) = *Q*(*x*<sup>∗</sup> − *y*∗), we deduce that *x*<sup>∗</sup> = *y*∗.

**Remark 1.** *By (11) and the estimate*

$$\begin{aligned} \|\|G'(\mathfrak{x}^\*)^{-1}G'(\mathfrak{x})\|\| &= \|\|G'(\mathfrak{x}^\*)^{-1}(G'(\mathfrak{x}) - G'(\mathfrak{x}^\*)) + I\|\| \\ &\le 1 + \|\|G'(\mathfrak{x}^\*)^{-1}(G'(\mathfrak{x}) - G'(\mathfrak{x}^\*))\|\| \\ &\le 1 + L\_0 \|\|\mathfrak{x} - \mathfrak{x}^\*\|\| \end{aligned}$$

*condition (13) can be dropped and be replaced by*

$$\mathcal{M}(t) = 1 + L\_0 t$$

*<sup>M</sup>*(*t*) = *<sup>M</sup>* <sup>=</sup> 2, *since t* <sup>∈</sup> [0, <sup>1</sup> *L*0 ).

#### **3. Generalized Method**

The multistep version of (3) consisting of *<sup>q</sup>* <sup>+</sup> 1, (*<sup>q</sup>* <sup>∈</sup> <sup>N</sup>), steps is expressed as

$$\begin{cases} z\_m^{(0)} = y\_m - \Gamma\_m G(y\_m), \\ z\_m^{(1)} = z\_m - \Psi(x\_m, y\_m, z\_m) G(z\_m), \\ z\_m^{(2)} = z\_m^{(1)} - \Psi(x\_m, y\_m, z\_m) G(z\_m^{(1)}), \\ \dots \\ z\_m^{(q-1)} = z\_m^{(q-2)} - \Psi(x\_m, y\_m, z\_m) G(z\_m^{(q-2)}), \\ z\_m^{(q)} = x\_{m+1} = z\_m^{(q-1)} - \Psi(x\_m, y\_m, z\_m) G(z\_m^{(q-1)}), \end{cases} \tag{30}$$

where *ym* = *xm* − Γ*mG*(*xm*), *z* (0) *<sup>m</sup>* = *zm*, *ψ*(*xm*, *ym*, *zm*)=(2*I* − Γ*m*[*zm*, *ym* ; *G*])Γ*m*, and Γ*<sup>m</sup>* = *G*(*xm*)−1.

Next, we show that the generalized scheme (30) possesses convergence order 2*q* + 3.

#### *3.1. Order of Convergence*

The definition of divided difference is required to derive (30) convergence order. Recalling the result of Taylor's expansion on vector functions (see [24]) for this:

**Lemma 1.** *<sup>G</sup>* : *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup> be r-times Fréchet differentiable in a convex set <sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup> then for any x*, *<sup>h</sup>* <sup>∈</sup> <sup>R</sup>*n*, *the following expression holds:*

$$G(\mathbf{x} + h) = G(\mathbf{x}) + G'(\mathbf{x})h + \frac{1}{2!}G''(\mathbf{x})h^2 + \frac{1}{3!}G'''(\mathbf{x})h^3 + ... + \frac{1}{(r-1)!}G^{(r-1)}(\mathbf{x})h^{r-1} + \mathbb{R}\_{\mathbf{r}\_f} \tag{31}$$

*where*

$$||\mathcal{R}\_r|| \le \frac{1}{r!} \sup\_{0 \le t \le 1} ||G^{(r)}(\mathbf{x} + th)|| \ ||h||^r \text{ and } h^r = (h, h, \dots, h).$$

The divided difference operator [·, · ; *<sup>G</sup>*] : *<sup>D</sup>* <sup>×</sup> *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* <sup>×</sup> <sup>R</sup>*<sup>n</sup>* −→ *<sup>L</sup>*(R*n*) is defined by (see [24])

$$\mathbb{E}\left[\mathbf{x} + h, \mathbf{x}; \mathbf{G}\right] = \int\_0^1 \mathbf{G}'(\mathbf{x} + th) \, dt, \, \forall \mathbf{x}, h \in \mathbb{R}^n. \tag{32}$$

When we expand *G* (*x* + *th*) in the Taylor series at point *x* and integrate, we obtain

$$\mathcal{G}\left[\mathbf{x} + h, \mathbf{x}; \mathbf{G}\right] = \int\_0^1 G'(\mathbf{x} + th) \, dt = G'(\mathbf{x}) + \frac{1}{2} G''(\mathbf{x})h + \frac{1}{6} G'''(\mathbf{x})h^2 + O(h^3). \tag{33}$$

where *<sup>h</sup><sup>i</sup>* = (*h*, *<sup>h</sup>*, *<sup>i</sup>* . . ., *<sup>h</sup>*), *<sup>h</sup>* <sup>∈</sup> <sup>R</sup>*n*.

Let *em* = *xm* − *x*∗. Expanding *G*(*xm*) in a neighbourhood of *x*<sup>∗</sup> and assuming Γ = *G* (*x*∗)−<sup>1</sup> exists, we obtain

$$\mathbf{G}(\mathbf{x}\_m) = \mathbf{G}'(\mathbf{x}^\*) (\boldsymbol{\varepsilon}\_m + A\_2(\boldsymbol{\varepsilon}\_m)^2 + A\_3(\boldsymbol{\varepsilon}\_m)^3 + A\_4(\boldsymbol{\varepsilon}\_m)^4 + A\_5(\boldsymbol{\varepsilon}\_m)^5 + O((\boldsymbol{\varepsilon}\_m)^5)),\tag{34}$$

where *Ai* = <sup>1</sup> *<sup>i</sup>*! <sup>Γ</sup>*G*(*i*)(*x*∗) <sup>∈</sup> *Li*(R*n*, <sup>R</sup>*n*) and (*em*)*<sup>i</sup>* = (*em*,*em*, *<sup>i</sup>* . . .,*em*), *em* <sup>∈</sup> <sup>R</sup>*n*, *<sup>i</sup>* <sup>=</sup> 2, 3, . . . Additionally,

$$G'(\mathbf{x}\_m) = G'(\mathbf{x}^\*) (I + 2A\_2 \mathbf{e}\_m + 3A\_3 (\mathbf{e}\_m)^2 + 4A\_4 (\mathbf{e}\_m)^3 + O((\mathbf{e}\_m)^4)),\tag{35}$$

$$G''(x\_m) = G'(x^\*) (2A\_2 + 6A\_3 \varepsilon\_m + 12A\_4 (\varepsilon\_m)^2 + O((\varepsilon\_m)^3)),\tag{36}$$

*or*

$$G^{\prime\prime\prime}(\mathbf{x}\_m) = G^{\prime}(\mathbf{x}^\*) (6A\_3 + 24A\_4e\_m + O((e\_m)^2)).\tag{37}$$

The inversion of *G* (*xm*) yields

$$\begin{split} G'(\mathbf{x}\_{\mathfrak{m}})^{-1} &= \begin{pmatrix} I & -2A\_2\mathbf{e}\_{\mathfrak{m}} + (4A\_2^2 - 3A\_3)(\mathbf{e}\_{\mathfrak{m}})^2 - (4A\_4 - 6A\_2A\_3 - 6A\_3A\_2 + 8A\_2^3)(\mathbf{e}\_{\mathfrak{m}})^3 \\\\ + O((\mathbf{e}\_{\mathfrak{m}})^4)) \Gamma. \end{pmatrix} \end{split} \tag{38}$$

We are in a position to investigate scheme (30)'s convergence behaviour. As a result, the following theorem is established:

**Theorem 2.** *Suppose that*

*(i) G* : *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup> is many times differentiable mapping. (ii) There exists a solution x*<sup>∗</sup> ∈ *D of equation G*(*x*) = 0 *such that G* (*x*∗) *is nonsingular. Then, sequence* {*xn*} *generated by method (30) for x*<sup>0</sup> ∈ *D converges to x*<sup>∗</sup> *with order* 2*q* + 3*, <sup>q</sup>* <sup>∈</sup> <sup>N</sup>*.*

**Proof.** Employing (34) and (38) in the Newton iteration *ym*, we obtain that

$$\begin{aligned} \vec{e}\_{m} &= \ y\_{m} - \mathbf{x}^\* = A\_2 \boldsymbol{e}\_{m}^2 + (2A\_2^2 - A\_3)\boldsymbol{e}\_{m}^3 + (4A\_2^3 - 4A\_2A\_3 - 3A\_3A\_2 + 3A\_4)\boldsymbol{e}\_{m}^4 \\ &- (8A\_2^4 + 6A\_3^2 + 6A\_2A\_4 + 4A\_4A\_2 - 8A\_2^2A\_3 - 6A\_2A\_3A\_2 - 6A\_3A\_2^2)\boldsymbol{e}\_{m}^5 + O(\epsilon\_m^6). \end{aligned} \tag{39}$$

The Taylor series of *G*(*ym*) about *x*<sup>∗</sup> yields

$$G(y\_m) = G'(\mathbf{x}^\*) (\tilde{\boldsymbol{\varepsilon}}\_m + A\_2 \tilde{\boldsymbol{\varepsilon}}\_m^2 + A\_3 \tilde{\boldsymbol{\varepsilon}}\_m^3 + A\_4 \tilde{\boldsymbol{\varepsilon}}\_m^4 + O(\tilde{\boldsymbol{\varepsilon}}\_m^5)),\tag{40}$$

Substituting (38)–(40) in first step of (30), we obtain

$$\bar{\epsilon}\_{\mathfrak{m}} = z\_{\mathfrak{m}} - \mathbf{x}^\* = 2A\_2^2 \epsilon\_{\mathfrak{m}}^3 + (4A\_2 A\_3 - 9A\_2^3 + 3A\_3 A\_2) \epsilon\_{\mathfrak{m}}^4 + O(\epsilon\_{\mathfrak{m}}^5). \tag{41}$$

Using Equations (35)–(37) in (33) for *x* + *h* = *zm*, *x* = *ym*, and *h* = *e*¯*<sup>m</sup>* − *e*˜*m*, it follows that

$$[z\_{m\prime}y\_m; G\ ] = G'(\mathfrak{x}^\*) \left(I + A\_2(\mathfrak{e}\_m + \mathfrak{e}\_m) + O((\mathfrak{e}\_m)^2, (\mathfrak{e}\_m)^2)\right),$$

and

$$
\Gamma\_m[z\_m, y\_m; G] = I - 2A\_2 e\_m + (4A\_2^2 - 3A\_3)(\varepsilon\_m)^2 + A\_2(\overline{\varepsilon}\_m + \overline{\varepsilon}\_m) + O((\varepsilon\_m)^3).
$$

As a result, we arrive at the conclusion

$$\Psi(\mathbf{x}\_{\mathfrak{m}}, y\_{\mathfrak{m}}, z\_{\mathfrak{m}}) = \left(1 - 5A\_2^2(e\_{\mathfrak{m}})^2 + 2(10A\_2^3 - 4A\_2A\_3 - 3A\_3A\_2)(e\_{\mathfrak{m}})^3 + O((e\_{\mathfrak{m}})^4)\right) G'(\mathbf{x}^\*)^{-1}.\tag{42}$$

In addition, we have

$$G(z\_m) = G'(\mathfrak{x}^\*) (\mathbb{E}\_m + O(\left(\mathbb{E}\_m\right)^2)).\tag{43}$$

Using (42) and (43) in the second step of method (30), it follows that

$$
\varepsilon\_{\mathfrak{m}}^{(1)} - \mathfrak{x}^\* = 10A\_2^4 (\mathfrak{e}\_{\mathfrak{m}})^5 + O\left((\mathfrak{e}\_{\mathfrak{m}})^6\right). \tag{44}
$$

The expansion of *G*(*z* (*q*−1) *<sup>m</sup>* ) about *<sup>x</sup>*<sup>∗</sup> yields

$$G(z\_m^{(q-1)}) = G'(\mathbf{x}^\*) \left( (z\_m^{(q-1)} - \mathbf{x}^\*) + A\_2 (z\_m^{(q-1)} - \mathbf{x}^\*)^2 + \cdots \right). \tag{45}$$

Then, we have

$$\begin{split} \mathfrak{g}(\mathbf{x}\_{\mathfrak{m}}, y\_{\mathfrak{m}}, z\_{\mathfrak{m}}) G(z\_{\mathfrak{m}}^{(q-1)}) &= \quad \left( \mathbf{I} - 5A\_2^2 (\epsilon\_{\mathfrak{m}})^2 + 2(10A\_2^3 - 4A\_2 A\_3 - 3A\_3 A\_2)(\epsilon\_{\mathfrak{m}})^3 + \mathcal{O}((\epsilon\_{\mathfrak{m}})^4) \right) G'(x^\*)^{-1} \\ &\quad \times G'(x^\*) \left( (z\_{\mathfrak{m}}^{(q-1)} - x^\*) + A\_2 (z\_{\mathfrak{m}}^{(q-1)} - x^\*)^2 + \cdots \right) \\ &= \quad \left( z\_{\mathfrak{m}}^{(q-1)} - x^\* \right) - 5A\_2^2 (z\_{\mathfrak{m}}^{(q-1)} - x^\*)(\epsilon\_{\mathfrak{m}})^2 + A\_2 (z\_{\mathfrak{m}}^{(q-1)} - x^\*)^2 + \ldots \end{split} \tag{46}$$

Using (46) in (30), we obtain

$$z\_{\mathfrak{m}}^{(q)} - \mathbf{x}^\* = \mathfrak{5}A\_2^2 (z\_{\mathfrak{m}}^{(q-1)} - \mathbf{x}^\*)(e\_{\mathfrak{m}})^2 - A\_2 (z\_{\mathfrak{m}}^{(q-1)} - \mathbf{x}^\*)^2 + \cdots \ . \tag{47}$$

As we know from (44) that *z* (1) *<sup>m</sup>* − *<sup>x</sup>*<sup>∗</sup> = <sup>10</sup>*A*<sup>4</sup> <sup>2</sup>(*em*)<sup>5</sup> + *<sup>O</sup>* (*em*)<sup>6</sup> , from (47) for *q* = 2, 3, we therefore have

$$\begin{aligned} z\_m^{(2)} - \mathfrak{x}^\* &= 5A\_2^2 (\mathfrak{e}\_m)^2 (z\_m^{(1)} - \mathfrak{x}^\*) + \cdots \\ &= 50A\_2^6 (\mathfrak{e}\_m)^7 + O\left((\mathfrak{e}\_m)^8\right) \end{aligned}$$

and

$$\begin{split} z\_{m}^{(3)} - \mathfrak{x}^\* &= 5A\_2^2 (\mathfrak{e}\_m)^2 (z\_m^{(2)} - \mathfrak{x}^\*) + \cdots \\ &= 250 A\_2^8 (\mathfrak{e}\_m)^9 + O\left( (\mathfrak{e}\_m)^{10} \right). \end{split}$$

Proceeding by induction, it follows that

$$e\_{m+1} = z\_m^{(q)} - x^\* = 2 \cdot 5^q A\_2^{2q+2} (e\_m)^{2q+3} + O\left( (e\_m)^{2q+4} \right).$$

This completes the proof of Theorem 2.

**Remark 2.** *Note that method (3) utilizes three functions, one derivative, and one inverse operator per full iteration and converges to the solution with the fifth order of convergence. The generalized scheme (30) based on (3) (for q* = 1*) generates the methods with increasing convergence orders* 5, 7, 9, ... *corresponding to q* = 1, 2, 3, ... *at an additional cost of one function evaluation per each iteration. This fulfils the main aim of developing higher order methods, keeping computational cost under control.*

#### *3.2. Local Convergence*

Along the same lines as method (3), we offer the local convergence analysis of method (30). Define *g*¯2, *λ*, *μ*, and *h<sup>μ</sup>* on the interval [0,*r*2) by

$$
\lg\_2(t) = \frac{K(t)}{1 - w\_0(t)},
$$

$$
\lambda(t) = 1 + \lg\_2(t)M,
$$

$$
\mu(t) = \lambda^q(t)\lg\_2(t)t^{\lambda - 1}
$$

and

$$h\_{\mu}(t) = \mu(t) - 1.$$

We have that *hμ*(0) < 0. Suppose that

$$
\mu(t) \to +\infty \text{ or a positive number as } t \to r\_2^-. \tag{48}
$$

Denote by *r*(*q*) the smallest zero on the interval (0,*r*2) of function *h<sup>μ</sup>* . Define *r*<sup>∗</sup> by

$$r^\* = \min\{r\_1, r^{(q)}\}.\tag{49}$$

**Proposition 1.** *Suppose that the conditions of Theorem 2 hold. Then, sequence* {*xm*} *generated for x*<sup>0</sup> ∈ *U*(*x*∗,*r*∗) − {*x*∗} *by method (30) is well defined in U*(*x*∗,*r*∗)*, remains in U*(*x*∗,*r*∗)*, and converges to x*∗*. Moreover, the following estimates hold:*

$$\|\|y\_m - \mathbf{x}^\*\|\| \le \mathcal{g}\_1(\|\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\|\mathbf{x}\_m - \mathbf{x}^\*\| \le \|\|\mathbf{x}\_m - \mathbf{x}^\*\| < r^\*,$$

$$\|\|z\_m - \mathbf{x}^\*\|\| \le \mathcal{g}\_2(\|\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\|\mathbf{x}\_m - \mathbf{x}^\*\| \le \|\mathbf{x}\_m - \mathbf{x}^\*\|,$$

$$\|\|z\_m^{(i)} - \mathbf{x}^\*\|\| \le \lambda^i(\|\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\|z\_m - \mathbf{x}^\*\|\tag{50}$$

$$\le \lambda^i(\|\|\mathbf{x}\_m - \mathbf{x}^\*\|) \mathcal{g}\_2(\|\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\|\mathbf{x}\_m - \mathbf{x}^\*\|\|^\lambda$$

$$\le \|\|\mathbf{x}\_m - \mathbf{x}^\*\|\|, \quad i = 1, 2, \dots, q - 1,$$

*and*

$$\begin{split} \|\mathbf{x}\_{k+1} - \mathbf{x}^\*\| &= \|\mathbf{z}\_m^{(q)} - \mathbf{x}^\*\| \quad \le \lambda^q (\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\mathbf{z}\_m - \mathbf{x}^\*\| \\ &\le \mu (\|\mathbf{x}\_m - \mathbf{x}^\*\|) \|\mathbf{x}\_m - \mathbf{x}^\*\|. \end{split} \tag{51}$$

*Furthermore, x*<sup>∗</sup> *is the only solution of G*(*x*) = 0 *in D*<sup>1</sup> = *D* ∩ *U*(*x*∗,*r*∗)*.*

**Proof.** Only new estimations (50) and (51) will be shown. We show the first two estimations using the evidence of Theorem 1. Then, we will be able to obtain that

$$\begin{split} \|\Psi(\mathbf{x}\_{m}, y\_{m}, z\_{m})G'(\mathbf{x}^{\*})\| &\leq \quad \|\left(2I - G'(\mathbf{x}\_{m})^{-1}[z\_{m}, y\_{m}; G]\right)G'(\mathbf{x}\_{m})^{-1}G'(\mathbf{x}^{\*})\| \\ &\leq \quad \|\left(2I - G'(\mathbf{x}\_{m})^{-1}[z\_{m}, y\_{m}; G]\right)\| \|\|G'(\mathbf{x}\_{m})^{-1}G'(\mathbf{x}^{\*})\| \\ &\leq \quad \frac{K(\|\mathbf{x}\_{m} - \mathbf{x}^{\*}\|)}{1 - w\_{0}(\|\|\mathbf{x}\_{m} - \mathbf{x}^{\*}\|)} \\ &\leq \quad \mathcal{g}\_{2}(\|\mathbf{x}\_{m} - \mathbf{x}^{\*}\|). \end{split} \tag{52}$$

Moreover, we have

$$\begin{split} \|\boldsymbol{z}^{(1)} - \boldsymbol{\mathsf{x}}^{\*}\| &= \|\boldsymbol{z}\_{m} - \boldsymbol{\mathsf{x}}^{\*} - \boldsymbol{\psi}(\boldsymbol{x}\_{m}, \boldsymbol{y}\_{m}) \boldsymbol{G}(\boldsymbol{z}\_{m})\| \\ &\leq \|\boldsymbol{z}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\| + \|\boldsymbol{\psi}(\boldsymbol{x}\_{m}, \boldsymbol{y}\_{m}, \boldsymbol{z}\_{m}) \boldsymbol{G}'(\boldsymbol{\mathsf{x}}^{\*})\| \|\boldsymbol{M}^{\prime}(\boldsymbol{\mathsf{x}}^{\*})^{-1} \boldsymbol{G}(\boldsymbol{z}\_{m})\| \\ &\leq \|\boldsymbol{z}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\| + \bar{\boldsymbol{g}}\_{2} (\|\boldsymbol{x}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\|) \boldsymbol{M} \|\boldsymbol{z}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\| \\ &\leq \lambda (\|\boldsymbol{x}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\|) \|\boldsymbol{z}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\| \\ &\leq \mu (\|\boldsymbol{x}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\|) \|\boldsymbol{x}\_{m} - \boldsymbol{\mathsf{x}}^{\*}\|. \end{split}$$

Similarly, we obtain

$$\begin{split} \|z\_{m}^{(2)} - \mathbf{x}^\*\| &\leq \lambda (\|\mathbf{x}\_{\mathfrak{m}} - \mathbf{x}^\*\|) \|z\_{m}^{(1)} - \mathbf{x}^\*\| \\ &\leq \lambda^2 (\|\mathbf{x}\_{\mathfrak{m}} - \mathbf{x}^\*\|) \|z\_{\mathfrak{m}} - \mathbf{x}^\*\| \\ &\dots + \dots + \dots + \dots \\ \|z\_{\mathfrak{m}}^{(i)} - \mathbf{x}^\*\| &\leq \lambda^i (\|\mathbf{x}\_{\mathfrak{m}} - \mathbf{x}^\*\|) \|z\_{\mathfrak{m}} - \mathbf{x}^\*\| \\ \|\mathbf{x}\_{\mathfrak{m}+1} - \mathbf{x}^\*\| &\leq \|z\_{\mathfrak{m}}^{(q)} - \mathbf{x}^\*\| \leq \lambda^q (\|\mathbf{x}\_{\mathfrak{m}} - \mathbf{x}^\*\|) \|z\_{\mathfrak{m}} - \mathbf{x}^\*\| \\ &\leq \mu (\|\mathbf{x}\_{\mathfrak{m}} - \mathbf{x}^\*\|) \|\mathbf{x}\_{\mathfrak{m}} - \mathbf{x}^\*\|) .\end{split}$$

That is, we have *xm*, *ym*,*zm*, *z* (*i*) *<sup>m</sup>* ∈ *U*(*x*∗,*r*∗), *i* = 1, 2, . . . , *q*, and

$$\|\mathbf{x}\_{m+1} - \mathbf{x}^\*\| \le \bar{\mathcal{c}} \|\mathbf{x}\_m - \mathbf{x}^\*\|,\tag{53}$$

where *c*¯ = *μ*(*x*<sup>0</sup> − *x*∗) ∈ [0, 1), so lim*m*→<sup>∞</sup> *xm* = *x*<sup>∗</sup> and *xm*+<sup>1</sup> ∈ *U*(*x*∗,*r*∗). The uniqueness result is standard, as shown in Theorem 1.

#### **4. Numerical Examples**

Here, we shall demonstrate the theoretical results of local convergence which we have proved in Sections 2 and 3. To do so, the methods of the family (30) of order five, seven, and nine are chosen. Let us denote these methods by *M*5, *M*7, and *M*9, respectively. The divided difference in the examples is computed by [*x*, *y* ; *F*] = <sup>1</sup> <sup>0</sup> *F* (*y* + *θ*(*x* − *y*))*dθ*. We consider three numerical examples, which are presented as follows:

**Example 1.** *Let us consider <sup>B</sup>* <sup>=</sup> <sup>R</sup>*m*−<sup>1</sup> *for natural integer <sup>m</sup>* <sup>≥</sup> <sup>2</sup>*. <sup>B</sup> is equipped with the max-norm <sup>x</sup>* <sup>=</sup> *max*1≤*i*≤*m*−<sup>1</sup>*xi. The corresponding matrix norm is <sup>A</sup>* <sup>=</sup> *max*1≤*i*≤*m*−<sup>1</sup> <sup>∑</sup>*j*=*m*−<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> |*aij*| *for A* = (*aij*)1≤*i*,*j*≤*m*−1*. Consider the two-point boundary value problem on interval [0, 1]:*

$$\begin{cases} \ v'' + v^{3/2} = 0, \\ v(0) = v(1) = 0. \end{cases} \tag{54}$$

Let us denote Δ = 1/*m*, *ui* = Δ*i*, and *vi* = *V*(*ui*) for each *i* = 0, 1, ... , *m*. We can write the discretization of *v* at points *ui* in the following form:

$$v\_{i}^{\prime\prime} \simeq \frac{v\_{i-1} - 2v\_{i} + v\_{i+1}}{\Delta^{2}} \text{ for each } i = 2, 3, \dots, m - 1.$$

Using the initial conditions in (54), we obtain that *v*<sup>0</sup> = *vm* = 0, and (54) is equivalent to the system of the nonlinear equation *F*(*v*) = 0 with *v* = (*v*1, *v*2, ... , *vm*−1) in the following form:

$$\begin{cases} \Delta^2 v\_1^{3/2} - 2v\_1 + v\_2 = 0, \\\ v\_{i-1} + \Delta^2 v\_i^{3/2} - 2v\_i + v\_{i+1} = 0 \text{ for each } i = 2, 3, \dots, m - 1. \end{cases} \tag{55}$$

Using (55), the Fréchet-derivative of operator *F* is given by

$$F'(v) = \begin{pmatrix} \frac{3}{2} \Delta^2 v\_1^{1/2} - 2 & 1 & 0 & \dots & 0\\ 1 & \frac{3}{2} \Delta^2 v\_1^{1/2} - 2 & 1 & \ddots & 0\\ 0 & 1 & \ddots & \ddots & \vdots\\ \vdots & & \ddots & \ddots & \ddots & 1\\ 0 & \dots & 0 & 1 & \frac{3}{2} \Delta^2 v\_1^{1/2} - 2 \end{pmatrix}.$$

Choosing *m* = 11, the corresponding solution is *x*<sup>∗</sup> = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0)*T*, and we have *L*<sup>0</sup> = *L* = *L*<sup>1</sup> = 3.942631477 and *M* = 2. The parameters using method (30) are given in Table 1.

**Table 1.** Numerical results for example 1.


Thus, it follows that the above-considered methods of scheme (30) converge to *x*∗ and remain in *U*¯ (*x*∗,*r*∗).

**Example 2.** *Scholars have determined that the speed of blood in a course is an element of the distance of the blood from the conduit's focal pivot (Figure 1). As per Poiseuille's law, the speed (*cm/s*) of blood that is r cm from the focal hub of a supply route is given by the capacity*

$$S(r) = \mathbb{C}(\mathbb{R}^2 - r^2),\tag{56}$$

*where R is the range of the course, and C is a consistent that relies upon the thickness of the blood and the tension between the two closures of the vein. Assume that for a specific course,*

$$\text{C} = 1.76 \times 10^5 \text{ cm/s}$$

*and*

*<sup>R</sup>* <sup>=</sup> 1.2 <sup>×</sup> <sup>10</sup>−<sup>2</sup> cm.

**Figure 1.** Cut-away view of an artery.

Using the numerical values, the problem reduces to

$$f\_2(\mathbf{x}) = 25.344 - 176.000\mathbf{x}^2 = 0,$$

where *x* = *r*.

The graph of the function *f*2(*x*) is shown in Figure 2.

**Figure 2.** Graph of *f*2(*x*).

The zero of *f*2(*x*) = 0 is *x*<sup>∗</sup> = 0.012; then, we have *L*<sup>0</sup> = *L* = *L*<sup>1</sup> = 84.2803 and *M* = 5280. The parameters using method (30) are given in Table 2.

It follows that the above-considered methods of scheme (30) will converge to *x*∗ and remain in *U*¯ (*x*∗,*r*∗) if *r*<sup>∗</sup> is chosen as shown in Table 2.


**Table 2.** Numerical results for example 2.

**Example 3.** *Consider the quasi-one-dimensional isentropic flow of a perfect gas through a variablearea channel, shown in Figure 3.*

**Figure 3.** In quasi-one-dimension flows, the stream tube cross section area is allowed to vary in one direction *A* = *A*(*x*).

The relationship between the Mach number *M* and the flow area *A*, derived by Zucrow and Hoffman [25], is given by

$$\varepsilon = \frac{A}{A^\*} = \frac{1}{M} \left( \frac{2}{\gamma + 1} \left( 1 + \frac{\gamma - 1}{2} M^2 \right) \right)^{(\gamma + 1)/2(\gamma - 1)}\text{.}\tag{57}$$

where *A*∗ is the choking area (i.e., the area where *M* = 1), and *γ* is the specific heat ratio of the flowing gas shown in Figure 4.

**Figure 4.** The area–Mach-number relation.

For each value of *ε*, two values of *M* exist, one less than unity (i.e., subsonic flow) and one greater than unity (i.e., supersonic flow). For the values of *ε* = 5.00 and *γ* = 1.4, Equation (57) becomes

$$f\_{\mathbb{B}}(\mathbf{x}) = \mathbf{5} - \frac{0.578704(1 + 0.2\mathbf{x}^2)^3}{\mathbf{x}}.\tag{58}$$

where *x* = *M*. The graph of the function *f*3(*x*) is shown in Figure 5, and the zero is *x*∗ = 0.116689. Then, we have that

$$L = L\_0 = L\_1 = 8.137146, \quad \text{and} \quad M = 0.610065.$$

**Figure 5.** Graph of *f*3(*x*).

The parameters using method (30) are given in Table 3.

**Table 3.** Numerical results for example 3.


The computed values of *r*∗ show that the considered methods of the scheme (30) will converge to *x*<sup>∗</sup> and remain in *U*¯ (*x*∗,*r*∗).

#### **5. Study of Complex Dynamics of the Method**

To view the geometry of the methods of the family (30) of five, seven, and nine order methods, in the complex plane, we present the attraction of basins of the roots by performing the methods on some functions (see Table 4). The basins are displayed in Figures 6–8 concerning capacities. To draw basins, we use square shapes *<sup>R</sup>* <sup>∈</sup> <sup>C</sup> of size [−2, 2] × [−2, 2] and allot various shadings to the basins. The dark region is appointed to the focuses for which the strategy is disparate.

**Table 4.** Comparison of performance based on basins of attraction of methods.


**Figure 6.** Basins of attraction of *M*5, *M*7, and *M*<sup>9</sup> for polynomial *P*1(*z*).

**Figure 7.** Basins of attraction of *M*5, *M*7, and *M*<sup>9</sup> for polynomial *P*2(*z*).

**Figure 8.** Basins of attraction of *M*5, *M*7, and *M*<sup>9</sup> for polynomial *P*3(*z*).

#### **6. Conclusions**

In this work, we have extended the utilization of technique (3) by introducing its assembly investigation and complex elements. Rather than using different procedures depending on the higher subordinate request just as a Taylor series, we have utilized only a subsidiary of request one, since this actually shows up in the technique. One more benefit of our methodology is the calculation of uniqueness balls where the repeats lie just as appraisals on *xn* − *x*∗. These objectives are accomplished utilizing our Lipschitz-like conditions. The hypothetical outcomes so determined are confirmed on some useful issues. Finally, we have checked the security of the technique through utilizing a complex element apparatus, specifically a bowl of fascination.

**Author Contributions:** Conceptualization, S.K. and J.R.S.; Formal analysis, S.K. and L.J.; Investigation, S.K. and L.J.; Methodology, D.K.; Writing-original draft, D.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Technical University of Cluj-Napoca open access publication grant.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We would like to express our gratitude to the anonymous reviewers for their help with the publication of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Donatella Bálint <sup>1</sup> and Lorentz Jäntschi 1,2,\***


**Abstract:** Various methods (Hartree–Fock methods, semi-empirical methods, Density Functional Theory, Molecular Mechanics) used to optimize a molecule structure feature the same basic approach but differ in the mathematical approximations used. The geometry optimization procedure calculates the energy at an initial geometry of a molecule and then proceeds to search a new geometry with a lower energy. Using the 3D structures collected from the PubChem database, 20 amino acid geometry optimization calculations were performed with several methods. The purpose of the study was to analyze these methods (39) to find the relationship between them and to determine which to use under different circumstances. Cluster analysis and principal component analysis were performed to evaluate the similarities between the different methods. The results after the analysis can classified into three main groups and can be selected accordingly to solve different types of problems.

**Keywords:** Gaussian; optimization; geometry; molecular modeling; amino acids

## **1. Introduction**

A basis set is essentially a finite number of atomic-like functions, over which the molecular orbital is formed via linear combination of atomic orbitals (LCAO). There are multiple choices for the basis set, such as Slater type orbitals [1] (STOs) or Gaussian-type orbitals [2] (GTOs). The wave functions are also called "stationary states" or "energy eigenstates"; in chemistry they are called "atomic orbitals" or "molecular orbitals". Consequently, they are important in molecular modeling [3,4].

Stationary states be described by the time-independent Schrödinger equation:

$$H\psi = E\psi,\tag{1}$$

where *ψ* is the state vector of the quantum system, *E* is the energy, and *H* is the Hamiltonian operator.

In the time-independent Schrödinger equation, the operation may produce specific values for the energy called energy eigenvalues. In addition to its role in determining system energies, the Hamiltonian operator generates the time evolution of the wavefunction in the form:

$$H\psi = j\hbar \frac{\partial}{\partial t'} \tag{2}$$

where the *j* constant is the imaginary unit, is the reduced Planck constant, and *t* is time.

The Schrödinger equation provides a method for calculating the wave function of a system and its dynamic change over time. The equation is a wave equation in terms of the wave function which predicts analytically and precisely the probability of events or outcome. The spatial part needs to be solved for in time-independent problems, because the time-ependent phase factor is always the same.

**Citation:** Bálint, D.; Jäntschi, L. Comparison of Molecular Geometry Optimization Methods Based on Molecular Descriptors. *Mathematics* **2021**, *9*, 2855. https://doi.org/ 10.3390/math9222855

Academic Editor: Yang-Hui He

Received: 28 September 2021 Accepted: 7 November 2021 Published: 10 November 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The Schrödinger Equation (1) for molecular systems can only be solved approximately [5]. The energy operator can be replaced by the energy eigenvalue *E*, so the timeindependent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator. Approximation methods can be classified into ab initio or semi-empirical categories.

An unknown one-electron function, such as an orbital *ψ<sup>i</sup>* can be expanded in a set of known functions *χ<sup>k</sup>* (*k* = 1, 2, . . . , *M*), the basis set:

$$
\psi\_i = \sum\_{k=1}^{M} \mathfrak{c}\_{ki} \chi\_k. \tag{3}
$$

In Hartree–Fock (HF) and Kohn–Sham density function theory (DFT), the coefficients *cki* are determined by minimizing the total energy, which by traditional methods lead to a matrix eigenvalue problem that is solved iteratively to provide a self-consistent field (SCF) solution.

The foundations of the orbital theory were laid by Hartree, Fock, and Slater. If the 2*n* electrons in a molecule are assigned to a set of n molecular orbitals *ψ<sup>i</sup>* (*i* = 1, ... , *n*), the corresponding many electron wavefunction is:

$$\psi = (2n!)^{-1/2} \det[(\psi\_-1|a)(\psi\_-1|\beta)(\psi\_-2|a)\dots]. \tag{4}$$

The *ψ<sup>i</sup>* are orthonormal and α and β are spin functions.

Slater-Type Orbitals (STOs) and Gaussian-Type Orbitals (GTOs) are used to describe AOs (atomic orbitals). STOs describe the shape of AOs more accurately than GTOs, but GTOs feature an advantage: they are much easier to compute. In fact, calculating multiple GTOs and combining them to describe an orbital is faster than calculating an STO. This is why combinations of GTOs are usually used to describe STOs, which, in turn, describe AOs.

The simplest and standard basis set in the Gaussian Program is Slater-Type- Orbitals simulated by three Gaussian functions each (STO-3G). Generally, if n < 3 the calculations produce poor results, in consequence, STO-3G is called the minimal basis set. We use minimal basis sets for qualitative results, very large molecules, or quantitative results for very small molecules (atoms) [6]. STOs represent the exact solutions for hydrogenlike atoms and provide a better representation than Gaussian functions for multielectron systems on a function-to-function comparison.

The most commonly used bases set for geometry optimization is 3-21G [7–9]. This method uses three Gaussians for the core orbitals and a two/one split for the valence functions. Usually, d orbitals for all heavy (non-hydrogen) atoms are added to improve a basis set. The polarization basis sets are those that include the d orbitals; they are indicated by the symbol "\*". A further development is the 6-31G\*\* basis, in which a set of p orbitals is added to each hydrogen in the 6-31G\* basis set [10].

A number of methods are used to optimize the geometry of molecules: empirical force field methods (molecular mechanics, a cheaper method in terms of computational speed, able to provide exceptional structural parameters), semi-empirical methods (to solve the Schrödinger equation, with certain approximations and description of the electron properties of atoms and molecules), and ab initio methods (e.g., Hartree–Fock, Post-Hartree-Fock, and Density Functional Theory) [6].

John A. People [11] pioneered the development of ab initio methods using Slater type bases sets or Gaussian orbitals to model the wave function. He defined models, selecting a combination of methods and bases sets, and compared the experimental results of the analysis. With his team, he established an extended basis of contracted Gaussian functions that considers the same properties but is still simple enough to be widely applied to organic molecules [10]. Gaussian-type atomic orbitals have been used broadly to calculate atomic and molecular wavefunctions. They were involved in the growth of one of the most common computational chemistry packages, the Gaussian programs.

For ab initio methods, the first step is a single-determinant SCF (self-consistent field) calculation. Ab initio quantum chemistry methods present the challenge of solving the electronic Schrödinger equation based on the positions of the nuclei and the number of electrons to provide valuable data.

Their quality depends on the basis set used. The Hartree and Hartree–Fock methods can be regarded as reference methods for many calculations in complex systems. The first solutions to be obtained are used in the next iteration. Hartree-Fock equations must be solved by an iterative procedure and offer the second set of solutions. This approach, SCF, continues as long as the energies of all the electrons remain unaffected. Almost all ab initio calculations use GTO basis sets.

Pure density functional theory (DFT) [12,13] methods are characterized by pairing an exchange functional with a correlation functional. Most current DFT studies use BP86, B3LYP, or BPW91 functionals.

The combination of the method and the basis set determines the chemistry model as Gaussian, specifying a level of theory. HF methods are considered the default if no other keywords are mentioned. Most methods also require a basis set; if no basis set keyword is specified, then STO-3G is used automatically. Some examples of basis sets are: STO-3G, 3-21G, 6-21G, and 6-31G. Single first-polarization functions can also be requested by using the usual \* or \*\* notation. 6-31G\* (or 6-31G(d)) is 6-31G with additional d polarization functions on non-hydrogen atoms; 6-31G\*\* (or 6-31G(d, p)) is 6-31G\* plus p polarization functions for hydrogen [5]. The + and ++ diffuse functions are accessible with some basis sets. 6-31+G is 6-31G plus diffuse s and p functions for non-hydrogen atoms; 6-31++G also features diffuse functions for hydrogen. Thom Dunning introduced optimized basis sets with correlated wavefunctions: cc (correlations-consistent basis) or pV (polarized valence basis) [14]. The prefix aug (augmented) can be used to add diffuse functions. Which a basis set is used, it is related to the purpose of the calculation and the molecules to be studied. Even a large basis set is not always a guarantee of agreement with the experimental data [13,15].

Different approaches [16–18] to the comparison of basis sets agree that, even if they are similar, basis sets cannot be generalized. Some recommendations we found in the articles studied and by consulting Gaussian tutorials are:

A large basis set is not always the best (ex: cc-pVQZ is overkill for Hartree-Fock).

The minimal basis set (STO-3G) allows the analysis of the largest molecules while having the lowest resolution/quality for quantum level. In general, cc-pVDZ is equivalent to or worse than 6-31G (d, p).

cc-pVTZ is better than 6-311G(d,p) or similar.

The convergence of ab initio methods is time-consuming.

The following bases sets are approximately equivalent:

6-31G → cc-pVDZ 6-311G → aug-cc-pVDZ 6-31+G(d) → cc-pVTZ 6-311+G(d) → aug-cc-pVTZ 6-31++G(d,p) → cc-pVQZ 6-311++G(d,p) → aug-cc-pVQZ

Due to the many basis sets and optimization methods, it is very difficult to find the optimal approach for scientific calculations. The choice of basis set for chemical calculations can have a major impact on the quality of the results, particularly for correlated ab initio methods [19]. The choice can be made based on the knowledge related to the design, development, and optimization of the latest developments in the field. For example, applications of basis sets are in the simulation and optimization of ultrasonic non-destructive tests, which are highly important in structural materials such as fiber composites, but also in columnar grained stainless steels [20]. Another approach could be functional cluster analysis (FCA) for multidimensional functional datasets, using orthonormalized Gaussian basis functions, which can be applied for example, to protein structures [21].

The purpose of this study was to analyze 39 optimization methods to find the relationship between them and determine which to use under different circumstances. Cluster

analysis, Statistical analysis (ANOVA), and principal component analysis (PCA) were performed to evaluate the similarities between the different methods.

#### **2. Materials and Methods**

The 20 amino acid structures (3D) shown in Table 1 were collected from the PubChem compound database [22].


**Table 1.** Amino acids used as input data to our algorithm.

These 20 amino acids feature different forms, isomers, enantiomers, and conformers. In biological systems, amino acids feature the same chirality; most are levorotatory (*L*) and not dextrorotatory (*D*). Using the *L* conformer of these compounds, geometry optimizations were performed on the structures (Table 2). The most frequent procedure to establish the basis functions describing the occupied atomic orbitals by HF/DFT optimization followed by addressing the issue of polarization functions subsequently. Whatever optimization method is used, it defines a local minimum, and it is possible that optimization starting from different initial exponents will lead to different outcomes.

**Table 2.** Geometry optimization methods used in the calculations.


The workflow is represented in the next figure (Figure 1). After the Gaussian program made the calculations based on the 39 methods selected, a family of molecular descriptors (FMPI- Fragmental Matrix Property Indices) [23] was also calculated to evaluate the degree of similarity between the methods. The results were submitted to cluster, PCA, and other statistical analyses.

**Figure 1.** The working algorithm.

We collected the structures (3D) of the 20 essential amino acids (*L* conformers) from PubChem databases (.sdf files) and analysed them with the Gaussian program, taking into consideration the following steps, also shown in Figure 1:


With a homemade \*php program we generated .hin files from the .sdf files and generated the molecular descriptors (FMPI). FMPI molecular descriptors are the improved version of SMPI (Szeged Matrix Property Indices) [24,25] descriptors. With SMPI, distance matrix are calculated, and then for each pair of (distinct) atoms the atoms closer to the first than to the second atom of the pair are collected into a matrix [15]. The improvement made to SMPI is the extension of the principle applied in Szeged fragments to the other two matrices collecting fragments from molecules for pairs of atoms.

Therefore, the gene sequence of FMPI was increased from SMPI with one gene and the number of descriptors was multiplied by three (arriving at 4536) [23]. After we obtained 4536 descriptors for every amino acid, we used the Statistica program to perform the clustering and PCA analysis.

#### **3. Results and Discussion**

After applying the algorithm described in the Methodology section, a principal component analysis (PCA) and a clustering analysis were performed. The next figure (Figure 2) shows the first and second component as the result of the PCA analysis.


**Figure 2.** The explained variation (R2X) and the predictive variation (Q2X) of the PCA components.

Each principal component is a linear combination of the variables from the whole data set. A total of 90,720 descriptors for each component and each method were analysed, or a total of 3.538.080 descriptors.

The result of the PCA analysis indicated that the principal components (Figure 2) explained our large amount of data at 99.8851%, which reflected the variance of the data.

The first component accounted for a maximum amount of total variance (71.25%) in the data analysed. The second component accounted for the maximum variance that was not explained by the first component (14.9%). The third component also accounted for the maximum variance (6.51%) after the first two.

The R2X describes the predictive accuracy and takes values between 0 and 1. The more significant a principal component, the larger its R2X. The explained variance (R2Xadj) is simply the explained variation (R2X) adjusted for the degrees of freedom.

The quality assessment, goodness-of-prediction (Q2) statistic is typically reported as a result of cross-validation and provides a qualitative measure of consistency between the predicted and original data. As we add more variables to the PCA analysis, the value of Q2 increases. Large values of Q2 indicates a relevant and significant analysis.

In the next figure (Figure 3), a score loading plot, the distribution of component 1 versus component 2 is represented. The plot indicates that the similar methods are indeed roughly grouped together. Furthermore, the loadings define the orientation of the principal components in space. The loading vectors are p1 and p2. In our case, the first three components explained most of the data. In the next figure, a score loading plot, the distribution of component 3 versus component 2 is represented (Figure 4).

The classification of the methods into four categories (Semi-Empirical, Density Functional Theory, Molecular Mechanics Møller–Plesset Perturbation Theory, Coupled-Cluster Theory and Hartree–Fock) in the Section 2 is not entirely valid if we take into consideration the similarity between them. The degree of similarity between the methods grouped the data into the main categories presented above, but also into different and mixed groups. The PCA and cluster analyses produced comparable results.

The cluster analysis dendrogram (Figure 5) shows the Euclidean distances between the 39 methods compared. The single linkage or nearest neighbour technique is one of the simplest hierarchical clustering methods. The Euclidian distance between the methods due to the large data set (3.538.080 variables) was very high. To compare these methods, the standardization of the linkage distance was chosen on the X-axis. (Dlink/Dmax) \*100 represents the linkage distances (Dlink) divided by maximum linkage distance (Dmax).

**Figure 3.** Score plot showing the distribution of the methods in the two principal components.

**Figure 4.** Score plot showing the distribution of the methods in the principal components p2 and p3.

The similarity between the optimization methods varies between the basis sets used. After obtaining the results of the PCA and clustering analyses, a classification can be made within the several different groups. The difference between the optimization methods was minimal; the tree clustering shows the relationship among them. For an extensive analysis, the data should be selected from different groups to obtain various results from multiple points of view.

**Figure 5.** Clustering results.

Several studies use hybrid methods in their analysis [26–28] in order to obtain considerably better results. Davidson and Feller [28] in 1986 described a few criteria upon which a selection of the basis sets could be made, although since then many other methods have been introduced in computational chemistry. Because different theoretical methods and molecular properties have different basis set demands, different computer architectures and algorithms have different efficiency requirements, and the desired accuracy varies with the application, it is not possible to design one 'optimum' basis set.

Cramer [29] discussed the evolution of basis sets from the most widely used splitvalence basis sets, such as 3-21G, 6-21G, 4-31G, 6-31G, and 6-311G [30], to modern examples of basis sets, such as cc-pCVDZ, cc-pCVTZ, etc. [31]. Comparing all the sets of comparisons, it is evident that the geometries for the molecules containing second-row elements are considerably more difficult to predict accurately than those for simpler organics. For example, it was found that AM1 is less successful when extended to these species than PM3 [32]. Furthermore, DFT methods feature limitations, such as different trends and high error accuracy [33].

The effort to determine the 'best' combinations of methods and basis sets that produce statistically good results for certain molecules and properties has become especially pronounced with the proliferation of the modern methods. Geometry optimization and energy minimization are fundamental tasks in molecular modelling and drug design. The failure to minimize energy and/or optimize geometry is directly converted to wrong molecular descriptors [34].

Because we used a very large data set, the results are more explicable if we divide them into different subgroups. After we performed the cluster and PCA analyses for every subgroup the following results were obtained.

For the semi-empirical methods, two principal components explain most of the data (Figure 6), and thecluster analysis showed the same tendency. The results can be divided into three main groups: am1; indo, cndo; and pm6, pm3mm, pm3, pddg. In conclusion, if we use one method from each group, this should be enough to describe our data.

For the Density Functional Theory methods, the statistical analysis looks a little different, because the dataset was larger this time. Most of the analyzed methods were part of this family.

**Figure 6.** Score plot showing the distribution of the methods in the principal components p1 and p2 and cluster analysis.

Figure 7 demonstrates that the DFT methods are similar to each other, but also some 'outlier' methods can be observed. The methods can be divided into four major groups, and three methods, which are positioned separately.

**Figure 7.** Score plot showing the distribution of the methods in the principal components p1 and p2 and cluster analysis.

In the Møller–Plesset Perturbation Theory methods, one principal component was identified (Figure 8). The methods are divided into two main groups, with one (mp2-3-21g) remaining a basis set.

**Figure 8.** Score plot showing the distribution of the methods in the principal components p1 and p2 and cluster analysis.

The most widely used optimization calculation is the Hartree–Fock method. Based on our analysis, we identified two principal components (Figure 9) and two main groups.

**Figure 9.** Score plot showing the distribution of the methods in the principal components p1 and p2 and cluster analysis.

The other methods, which are part of Coupled-Cluster Theory and Molecular Mechanics, could not be analysed separately because of the small dataset they represented. One method (CCSD) in Coupled-Cluster Theory and two methods (UFF, Dreiding) in Molecular Mechanics Theory did not reveal statistical significance if we analysed them alone. They are included in the first analysis, where all the methods are examined.

We performed another statistical analysis: the Single-Factor ANOVA test.

The reason for performing ANOVA was to see whether any difference existed between the groups for particular variables. The null hypothesis states that there was no significant difference between the methods analysed, based on the molecular descriptors calculated.

The *p*-value was 0.9995 > 0.05, so we accepted the null hypothesis, and concluded that there were no significant differences between the methods. In the Figure 10, the results of the ANOVA indicate that we cannot reject the null hypothesis.


**Figure 10.** ANOVA test results.

#### **4. Conclusions**

In conclusion, we can state that the size of the basis set does not reflect its applicability in different circumstances. It is not possible to find the best basis set, only a couple of basis sets that fit our dataset. If we use different basis sets we obtain different results. Therefore, care must be taken to select the correct basis set. What makes the difference in results are the different selections and the correct use of optimization methods.

To find the best geometry optimization method to use in different situations, we must know are related. Two similar methods excluded each other in the analysis because they provided almost the same results. The results of our analysis show the correlation and the degree of relationship between the methods studied. The reclassification of the 39 examined methods facilitates the selection of the best basis sets for different study areas.

**Author Contributions:** Conceptualization, D.B.; Methodology, L.J.; Supervision, L.J.; Writing original draft, D.B.; Writing—review & editing, L.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Technical University of Cluj-Napoca open access publication grant.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

