A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations

The present study concerns the development of a new iterative method applied to a numerical continuation procedure for parameterized scalar nonlinear equations. Combining both a modified Newton’s technique and a stationary-type numerical procedure, the proposed method is able to provide suitable approximate solutions associated with scalar nonlinear equations. A numerical analysis of predictive capabilities of this new iterative algorithm is addressed, assessed, and discussed on some specific examples.


Introduction
The resolution of scalar nonlinear equations is an issue frequently encountered in many branches of physical sciences such as mechanics [1][2][3][4][5].Although, in the literature, the most used numerical methods are either the classical Newton's technique [3,4,6] or modified Newton-type procedures [7][8][9][10], they suffer from the main disadvantage of being held in check in the presence of critical points [11].In order to overcome this deficiency, we propose to develop a new iterative algorithm applied to a numerical continuation procedure [5] for providing the approximate solutions associated with parameterized scalar nonlinear equations.The presented algorithm is based on a modified Newton-type method coupled with a stationary numerical technique.This study is organized in the following manner: (i) in Section 2, the standard numerical continuation procedure is briefly recalled including some classical algorithms; (ii) in Section 3, the new proposed iterative numerical method is presented in detail; (iii) in Section 4, the predictive abilities associated with this new iterative algorithm are tested and evaluated on some examples.

Problem Statement.
We consider the parameterized scalar nonlinear equation E : (, ) ∈ R 2 → R in the following form: where  denotes the real-valued "solution" variable associated with the nonlinear problem under consideration and  is the real-valued scalar "parameter" variable.It is important to emphasize the following: (i) the parametrized scalar nonlinear equation E (see (1)) may include critical points ( cr ,  cr ) (see Figure 1); (ii) the couple (, ) can depend on another parameter  ∈ R such as (1) reading E((), ()) = 0; (iii) in the framework of solid mechanics, (1) represents the mechanical equilibrium equation and the "solution" variable  and the scalar "parameter" variable  denote the displacement and the mechanical load, respectively (i.e.,  ≡  and  ≡ ).Within this context, the natural parameter  is the physical time ; that is, (1) can be written as follows: E ( () ,  ()) = 0. (
For solving numerically (2), we consider the discrete-time interval [  ,  +1 ] and we perform a Taylor series expansion of the function E (representing here the mechanical equilibrium of solid) in the first order at point ( +1 +1 ,  +1 ) (with  +1 being fixed and constant): where (‖( +1 +1 −   +1 )‖) denotes the higher-order terms with the Landau notation (⋅) associated with the asymptotic behaviour of the function E (considering only the variable quantity ), ‖ +1 which is th iterative displacement of the incremental time  +1 ) for fixed and constant mechanical load  (at point  +1 which is the value of parameter of the incremental time  +1 ).It should be underscored that the variable ◼  (resp., ◼ +1 ) without the exponent  or ( + 1) represents a converged (resp., known) quantity at the incremental time   (resp.,  +1 ).

Standard Arclength and Pseudo-Arclength Procedures.
Similar to previous approach (see Section 2.2.1), we perform a Taylor series expansion of the function E (representing the mechanical equilibrium of solid) in the first order at point where (‖( +1 +1 −   +1 ), ( +1 +1 −   +1 )‖) denotes the higherorder terms with the Landau notation (⋅) associated with the asymptotic behaviour of the function E (considering the variable quantities (, )) and ) is the first-order partial derivative operator associated with the function E with respect to  (resp., ) at point   +1 (resp.,   +1 ) for fixed and constant displacement  (resp., mechanical load ) at point   +1 (resp.,   +1 ) which is th iterative variable of incremental time  +1 .
It may be stressed that there exist many other methods used for numerical continuation procedures; one of them, which is not present here, is called "normal flow algorithm" or "Davidenko's flow algorithm" (see [21,22] for more details); the mechanical equilibrium equation of solid E associated with the Davidenko's flow reads E(, ) = , where  denotes the perturbation parameter.

A New Iterative Numerical Continuation Method
3.1.Proposed Algorithm.In this section, we present a new iterative numerical continuation procedure for approximating the solutions associated with any parameterized scalar nonlinear equations.The proposed iterative algorithm belongs to the family of predictor-corrector methods, and it uses both a modified Newton's method and a stationarytype numerical technique.The stationary procedure allows reducing the considered scalar nonlinear equation (see ( 1)) to only one explicit equation such as where † is the considered variable and ⋆ is the fixed and constant parameter.It should be stressed that the derivative of the function E (see (1)) checks that E (, ) = E (; ) + E (; ) , where  denotes the first-order total derivative operator (with (E/ †)( †; ⋆) ≡ (E/ †)( †; ⋆).When considering both a discrete-time interval [  ,  +1 ] and an orthonormal basis {ê  , ê }, the direction vector ⃗   associated with the tangent straight line T(;   ) at point (  ,   ) can be written as follows (with the stationary procedure (7)): with where ê denotes the unit vector of the basis such as ‖ê  ‖ = 1 (∀ = , ) and    are the components associated with the vector ⃗   in the orthonormal basis {ê  , ê }.It should be noted that, in (11), the term E(  ;   ) = E(  ,   ) = 0 since that (  ,   ) is the mechanical equilibrium point at the time   .
For crossing more easily some critical points ( cr ,  cr ) associated with the nonlinear function E, we introduce a new director vector ⃗   associated with the straight line H(;   ) at point (  ,   ) at the time   such as with where    (with  = , ) are the components associated with the vector ⃗   in the orthonormal basis {ê  , ê } and  and  are two parameters (,  ∈ R).
Using ( 12) and ( 13), we define the equation of the straight line H(;   +1 ) passing through the point (,   +1 ) and with the director vector ⃗   that must satisfy the following relation (see Figure 3): with International Journal of Engineering Mathematics  For the first iterative step (i.e.,  = 1), the iterative point  1 +1 must satisfy the following relation (with  0 +1 =   , ∀ = , ): where  is a parameter ( ∈ R * + ).
In line with (16), the iterative point  1 +1 checks (see Figure 3) with Using ( 6), ( 7) and ( 8), we have (with International Journal of Engineering Mathematics  In line with (19) and considering (7) combined with the updated iterative point  1 +1 obtained with ( 17) and ( 18), we can approximate the new iterative point  1 +1 as For the other iterative steps (i.e.,  > 1), we introduce the straight line W(;   ) passing thought the point ( 1 +1 , H( 1 +1 ;   )) and with the director vector ⃗   must satisfy the following relations (see Figure 3): with where    (with  = , ) are the components associated with the vector ⃗   in the orthonormal basis {ê  , ê }.The ( + 1)th iterative solution is obtained when

Some Numerical Examples
4.1.Preliminary Remarks.In the current section, we propose to test and evaluate the accuracy, efficiency, and robustness associated with the developed iterative method associated with the numerical continuation procedure in Section 3 on some scalar nonlinear equations.Moreover, all the numerical results of this section have been obtained with MATLAB software (see [7]).  with (b) For the other iterations ( ≥ 2),

New Iterative Numerical Continuation Algorithm (See
(ii) For the iterative solution   +1 ( ≥ 1), It is important to emphasize the following: (i) On the one hand, we consider only the case where Θ = Θ 1 for the new continuation algorithm (see ( 26)-( 30)) used in this section.
(ii) On the other hand, we introduce four types of Convergence Criterion (CC  ) (with  = 1, . . ., 4) in order to stop the iterative process associated with the new proposed algorithm: where  max represents the maximum number of iterations,  re and  1 (resp.,  2 ) are the tolerance parameters associated with the residue error of the function E and approximation error criterion of displacement  (resp., mechanical load ).In what follows, we consider the following values for each CC:  max = 30,  re = 10 −10 , and  1 =  2 = 10 −10 .

Sensibility Analysis.
In this section, we propose a sensibility analysis for evaluating the influence of different values assigned with the parameters (, , and ) used by the new iterative numerical continuation algorithm (see Section 3).All the numerical results associated with the scalar nonlinear functions ( 1 ), ( 2 ), and ( 3 ) (see Section 4.2) with different values of the parameters (, , and ) are presented in  In the light of all numerical results obtained in Sections 4.3.1 and 4.3.2, the new iterative numerical continuation algorithm is a relatively accurate, efficient, and robust method that allows passing specific critical points and providing suitable approximate solutions (  +1 ,   +1 ) associated with parameterized scalar nonlinear equations.

Conclusion
The present paper is devoted to a new iterative numerical continuation procedure for approximating the solutions associated with parameterized scalar nonlinear equations.Coupled with a modified Newton-type method and a stationary numerical technique, the presented algorithm is capable of providing satisfactory numerical solutions for scalar nonlinear equations using one control parameter.Through some illustrative examples, the predictive abilities of this new algorithm are tested, assessed, and discussed.
• ‖ is Euclidean norm associated with the quantity • (here, Euclidean distance reduces to the absolute value | • | since that there is only one-variable , i.e., ‖ variable  at the incremental time  +1 (with  ∈ N), and (E/)(  +1 ;  +1 ) ≡ (E/)(  +1 , )| = +1 is the firstorder partial derivative operator associated with the function E with respect to  (at point