Modified iterative double step length model for solving nonlinear systems with application to motion control

Abstract

The double-step-length procedure presents an inventive approach by incorporating two corrections during each iteration, which improves the robustness of the iterative process. The idea that if one correction fails, the other might still help guide the system in the right direction seems likely to reduce the chance of getting divergence, a common challenge in nonlinear solvers. This article presents a more effective approach to solving systems of nonlinear equations by combining the double-step length iterative method with the Picard-Mann hybrid technique. This combination improves performance. We streamline the process by employing a positive scalar to approximate the Jacobian matrix in Newton’s method, thereby eliminating the need to calculate derivatives. An inexact line search technique was used to determine the values of the two-step lengths. The proposed method converges globally under mild conditions. The numerical tests highlight its efficiency, proving that it is more effective than existing double-direction and length methods. Furthermore, we have effectively implemented this technique in motion control applications, showcasing its practicality and efficacy.

Introduction

Nonlinear systems of equations (NSE) play a vital role in a broad range of domains, including physics, engineering, economics, and applied mathematics, especially in solving systems of nonlinear differential equations^1,2,3 . These systems arise in scenarios where the relationship between variables cannot be accurately described using linear equations, making them more complex to solve^4,5,6. Researchers are actively working to develop efficient and reliable iterative methods for finding approximate solutions to these equations . A typical nonlinear system of equations can be expressed in the following form:

$$\begin{aligned} \mathfrak {F}(s) = 0, \end{aligned}$$

(1)

where $\mathfrak {F}: \mathbb {R}^n \rightarrow \mathbb {R}^n$ is a nonlinear mapping.

These problems can be solved using a variety of iterative strategies. These include methods that do not depend on derivatives^{7,8,9,10,11,12,13}, as well as Newton’s method (NM)¹⁴ and quasi-Newton methods (QNM)^15,16. Notably, Newton’s method is one of the most effective approaches because it is straightforward to implement and converges rapidly. However, it does require the calculation and storage of the Jacobian matrix (JM) at each iteration. These techniques employ a recursive formula to generate a sequence of points.

$$\begin{aligned} s_{k+1} = s_k +\alpha _k d_k, \quad k = 0, 1, \ldots \end{aligned}$$

(2)

where $\alpha _k$ is the step size, $d_k$ is the search direction, and the current and the previous iterations are $s_{k+1}$ and $s_{k}$, respectively^17,18. The direction $d_k$ of the NM is determined by solving:

$$\begin{aligned} \mathfrak {F}(s_k) + \mathfrak {F}^{'}(s_k) d_k = 0. \end{aligned}$$

Here, $\mathfrak {F}^{\prime }(s_k)$ is the Jacobian matrix of $\mathfrak {F}(s_k)$ at $s_k$. However, one limitation of Newton’s scheme is that it requires the derivative of JM to be computed at each iteration, which may not always be available or may not be precisely determined. In such cases, NM cannot be applied directly.

To solve this problem, researchers created QNM. These methods improve traditional optimisation techniques by replacing the JM or its inverse with a new approximation that updates during each step. This change makes the optimisation process more efficient. In QNM, the search direction is determined by:

$$\begin{aligned} d_k = -B_k^{-1} \mathfrak {F}(s_k). \end{aligned}$$

The problem (1) may be derived from an unconstrained optimisation problem¹⁹. Let $\mathfrak {h}$ be defined as a norm function represented by the equation:

$$\begin{aligned} \mathfrak {h}(s) = \frac{1}{2}\Vert \mathfrak {F}(s)\Vert ^{2}. \end{aligned}$$

(3)

Consequently, the nonlinear equations problem (1) can be equivalently expressed as the following global optimisation problem:

$$\begin{aligned} \min \mathfrak {h}(s), \quad s \in \mathbb {R}^n, \quad \mathfrak {h}:\mathbb {R}^n \rightarrow \mathbb {R}. \end{aligned}$$

(4)

The NM and QNM are among the classical and efficient algorithms for solving optimization problems. However, their major limitation lies in the requirement for significant matrix storage due to the calculation or estimation of the Jacobian at each iteration step. This drawback makes Newton-type algorithms less suitable for large-scale problems. To address this issue, numerous studies have developed matrix-free methods. One effective matrix-free algorithm is the Double Direction Method (DDM)²⁰, which generates sequences of iteration points via:

$$\begin{aligned} s_{k+1} = s_{k} + \alpha _{k} a_{k} + \alpha _{k}^{2} b_{k}, \end{aligned}$$

(5)

where $a_{k}$ and $b_{k}$ define the search directions. The main motivation behind the DDM algorithm is that it incorporates two corrections in the formula described by (5). correction If one of the correction schemes fails during the iterative process, the other serves as a backup mechanism to ensure convergence and stability of the system.

Recently, Petrovi$\acute{\textrm{c}}$ and Stanimirovi$\acute{\textrm{c}}$²⁰ introduced a new concept to the unconstrained optimization known as the accelerated double direction approach. This pioneering approach incorporates an acceleration parameter $\delta _{k} > 0$ to formulate the Hessian matrix approximation, which is expressed as follows:

$$\begin{aligned} \bigtriangledown ^{2} \mathfrak {h}(s_{k}) \approx \delta _{k} I, \end{aligned}$$

with $I$ denoting the identity matrix and $\bigtriangledown ^{2} \mathfrak {h}(s_{k})$ representing the Hessian of $\mathfrak {h}(s_{k})$. To improve the robustness of this algorithm, Petrovi$\acute{c}$ further refine the formula in²¹ by introducing a double-step length method (DSM) as follows:

$$\begin{aligned} s_{k+1} = s_{k} + \alpha _{k} a_{k} + \beta _k b_{k}. \end{aligned}$$

(6)

where $\alpha _{k}$ and $\beta _{k}$ denotes the two independently chosen step lengths, allowing for a more adaptive and flexible approach to convergence. Preliminary results in²¹ show that this modified algorithm demonstrates superior performance compared to the initial DDM, attaining a reduction in the iteration count required in addition to decrease in processor time.

To further improve the convergence and robustness of the algorithm, Petrovi$\acute{\textrm{c}}$ et al.²² consider hybridizing the DDM with the Picard-Mann Hybrid Iterative Process (PMHP), presented by Khan²³. This hybridization utilizes the strengths of both schemes, which results in improved performance when addressing different benchmark problems. Incorporating the PMHP improves the complexity of the scheme, leading to improvements in both numerical efficiency and convergence of the scheme.

Definition 1.1

The PMHP is characterized by three relations:

$$\begin{aligned}&s_0=s\in \Omega , \end{aligned}$$

(7)

$$\begin{aligned}&s_{k+1}=T(y_k), \end{aligned}$$

(8)

$$\begin{aligned}&y_k=(1-\eta _k)s_k+\eta _kT(s_k), \quad k\ge 0, \end{aligned}$$

(9)

with $T: \Omega \longrightarrow \Omega$ denoting the mapping on a nonempty convex subset $\Omega$ of a normed space ${\bf E}$. The sequences $y_k$ and $s_k$ are follow from the iterative procedure defined in (9) and (8), and $\{\eta _k\}_{k\ge 0}$ is a sequence of positive values within the interval $(0,1)$.

Building on the hybridization rule introduced in^23,24 proposed a double-step length scheme equipped with a convergence procedure. This innovative approach was designed to refine the DSM described in²¹, thereby overcoming its earlier limitations.

The current literature regarding derivative-free DDMs for solving NSE is notably scarce. This identified gap in research prompted Halilu and Waziri⁹ to develop a derivative-free methodology that employs a DDM to effectively address the equation presented in (1), as articulated in equation (5). Subsequently, in²⁵, they developed a modified version of the algorithm in⁹. Their study establishes that the proposed method exhibits global convergence. The new scheme proposed by²⁵ lays foundation for reliable solutions to nonlinear equations without requiring derivatives. Building upon the existing framework, Abdullahi et al.¹¹ developed another modification DDM by revising the original formula in⁹ and incorporating a conjugate gradient scheme tailored for symmetric nonlinear equations. Their scheme not only retain the derivative-free property of the formula but also facilitated global convergence, which was achieved using the modified line search approach proposed by Li and Fukushima¹⁹. In another attempt, Halilu and Waziri²⁶ further modified the DDM discussed in (5) and established global convergence of the new scheme under some mild assumptions. Extensive computational experiments demonstrated the efficiency and robustness of the method. In their insightful studies presented in¹² and²⁷, the authors built upon the foundational concepts in equation (6) by introducing the DSM. This innovative method enhances the efficacy of the DDM described in⁹. Their results are particularly encouraging, indicating that the DSM achieves significantly faster convergence than the previous method. These findings not only highlight the potential of their approach to optimize numerical algorithms but also pave the way for further advancements in this area of research.

Motivated by the concepts presented by Petrovi$\acute{\textrm{c}}$ in²⁴, this study aims to develop an advanced hybrid DSM, as outlined in¹², for addressing the NSE using the PMHP proposed in²³. The objective is to enhance both the efficiency and accuracy of identifying solutions to Problem (1).

The contributions from this study are as follows:

• This paper introduces a new iterative algorithm for systems of nonlinear equations.

• To develop a derivative-free approach, the Jacobian matrix is approximated using the acceleration parameter.

• We propose a new algorithm that ensures global convergence.

• The correction parameter is derived using Picard-Mann iterative schemes to enhance the numerical performance of the proposed method.

• The proposed method is successfully applied to address issues related to motion control.

The organisation of this paper is outlined as follows: The subsequent section will detail the derivation of the proposed approach. A comprehensive convergence analysis of the suggested algorithm will be presented in “Convergence analysis”. “Numerical experiments” will illustrate various numerical tests and motion control applications of the proposed method. Finally, the article will be concluded in “Conclusion”, while the list of abbreviations is provided in “Abbreviations” section. In this paper, we refer to $\mathbb {R}^n$ as the $n$-dimensional real space, with the notation $\Vert \cdot \Vert$ representing the Euclidean norm.

Preliminaries and algorithms

In this section, we present the motivation behind and the algorithm for the proposed method. First, we consider the accelerated DSM for the unconstrained optimisation problem defined in (4). Petrovi$\acute{\textrm{c}}$²¹ finds the minimiser of (4) using the DSM. By approximating the Hessian matrix through the acceleration parameter and employing a second-order Taylor expansion, the author derives the following acceleration parameter:

$$\begin{aligned} \delta _{k+1}^{ADSS}= 2\frac{\mathfrak {h}(s_{k+1})-\mathfrak {h}(s_k)+(\alpha _k\delta _k^{-1}+\beta _k)\Vert \nabla \mathfrak {h}(s_{k})\Vert ^2}{(\alpha _k\delta _k^{-1}+\beta _k)^2\Vert \nabla \mathfrak {h}(s_{k})\Vert ^2}. \end{aligned}$$

To enhance the convergence properties and the numerical results of the method in²¹, Petrovi$\acute{c}$ in²⁴ hybridised it with the PMHP by Khan²³, resulting in the following acceleration parameter.

$$\begin{aligned} \delta _{k+1}^{HADSS}= 2\frac{\mathfrak {h}(s_{k+1})-\mathfrak {h}(s_k)+\eta _k(\alpha _k\delta _k^{-1}+\beta _k)\Vert \nabla \mathfrak {h}(s_{k})\Vert ^2}{\eta _k^2(\alpha _k\delta _k^{-1}+\beta _k)^2\Vert \nabla \mathfrak {h}(s_{k})\Vert ^2}, ~~~~\eta _k\in (0,1). \end{aligned}$$

The enhancement not only improves the algorithm’s efficiency but also ensures greater accuracy in reaching solutions, thereby advancing the overall effectiveness of the method in practical applications. This improvement is made possible by the addition of the correction parameter $\eta _k$ in the scheme.

Now, let’s refer to DSM in¹². The approach created a derivative-free method for solving (1) via

$$\begin{aligned} \mathfrak {F}^{\prime }(s_{k})\approx \delta _{k}I, \end{aligned}$$

(10)

where $\delta _{k}>0$. The procedure in¹² generate the iterates of $\{s_k\}$ such that $s_{k+1}=s_{k}+x_k$, where $x_k=(\alpha _{k}+\beta _{k})a_{k}$ and the direction $a_{k}$ is given as

$$\begin{aligned} a_{k}=-\delta _{k}^{-1}\mathfrak {F}(s_k). \end{aligned}$$

(11)

The acceleration parameter is derived using a first-order Taylor expansion as follows:

$$\begin{aligned} \delta _{k+1}=\frac{v_k^Tv_k}{(\alpha _k+\beta _{k})v_k^Ta_k}, \end{aligned}$$

(12)

where $v_{k}=\mathfrak {F}(s_{k+1})-\mathfrak {F}(s_{k})$. The method presented in¹² converged globally. However, its numerical performance is unsatisfactory when $\delta _{k}$ approaches or equals 1. This constraint leads us to suggest a hybrid technique that produces better numerical results. Consequently, inspired by the approach in²⁴, we are interested in hybridising it with the PMHP. Therefore, we define the mapping $T$ in Definition 1.1 as

$$\begin{aligned} T(y_k) = y_k - (\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k). \end{aligned}$$

(13)

Therefore, from (8), (9), and (13), we have the following three relations:

$$\begin{aligned}&s_0=s\in \Omega ,\nonumber \\&s_{k+1}=y_k - (\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k), \end{aligned}$$

(14)

$$\begin{aligned}&y_k= (1-\eta _k)s_k+\eta _k(s_k - (\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k)), \quad k\ge 0. \end{aligned}$$

(15)

where $\eta _k\in (0,1)$ is a correction parameter. Consequently, we have the following lemma.

Lemma 2.1

Let the accelerated double-direction scheme be generated by the three term iterative process in (14) and (15), then the proposed scheme is given by

$$\begin{aligned} s_{k+1}=s_{k}-\rho _k(\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k), ~~\rho _k~\in (1,2). \end{aligned}$$

(16)

Proof

By substituting (15) into (14), we have

$$\begin{aligned} s_{k+1}&=(1-\eta _k)s_k+\eta _k(s_k - (\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k) - (\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k) \\&=s_k-\eta _ks_k+\eta _ks_k-\eta _k(\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k)-(\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k)\\&=s_k-(\eta _k+1)(\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k). \end{aligned}$$

Taking $(\eta _k+1)=\rho _k\in (1,2)$, we have (16). $\square$

Now, we are finding the updated value of the acceleration parameter by employing the first-order Taylor expansion of the proposed iteration (16).

$$\begin{aligned} \mathfrak {F}(s_{k+1})\approx \mathfrak {F}(s_k)+\mathfrak {F}^{\prime }(\xi )(s_{k+1}-s_{k}), \end{aligned}$$

where $\xi \in [s_{k}, s_{k+1}]$ can be expressed as $\xi =s_{k}+\varrho (s_{k+1}-s_{k}), \quad 0\le \varrho \le 1$. By selecting $\varrho =1$, we have

$$\begin{aligned} \mathfrak {F}(s_{k+1})\approx \mathfrak {F}(s_k)-\mathfrak {F}^{\prime }(s_{k+1})\rho _k(\alpha _k + \beta _{k}) \delta _k^{-1} \mathfrak {F}(s_k). \end{aligned}$$

(17)

From (10) and (17), it not difficult to verify

$$\begin{aligned} v_k\approx \delta _{k+1}x_k, \end{aligned}$$

(18)

$x_k=s_{k+1}-s_{k}$.

By multiplying both sides of Eq. (18) by the transpose of $v_k^T$, the proposed search direction and acceleration parameter are respectively defined as:

$$\begin{aligned}&d_{k}=-\rho _k\delta _{k}^{-1}\mathfrak {F}(s_k), \end{aligned}$$

(19)

$$\begin{aligned}&\delta _{k+1}=-\frac{\delta _{k}\rho _k^{-1}\Vert v_k\Vert ^2}{(\alpha _k+\beta _{k})\rho_kv_k^T\mathfrak {F}(s_k)}. \end{aligned}$$

(20)

Remark 2.2

In this paper, we adopt a consistent parameterisation by setting $\rho _k = \rho$ for all $k \ge 0$. The parameter $\rho$ is set to be in the interval $(1, 2)$. This selection is based on its influence on the algorithm’s convergence properties.

The proposed method algorithm is specified as follows:

Algorithm 1

Modified double step size approach (MIDSL).

Convergence analysis

We present how the proposed Algorithm 1 (MIDSL) converges globally in this section. To begin, let’s define the level set.

$$\begin{aligned} \Omega =\{s|\Vert \mathfrak {F}(s)\Vert \le \Vert \mathfrak {F}(s_{0})\Vert \}. \end{aligned}$$

(23)

Assumption 3.1

The following assumptions have been stated:

• 1. There exists $s^*\in \mathbb {R}^n$ such that $\mathfrak {F}(s^*)=0$.

• 2. The function $\mathfrak {F}(s)$ is smooth in some neighborhood say Q of $s^*$ containing $\Omega$.

• 3. The Jacobian of $\mathfrak {F}(s)$ is bounded symmetric and positive definite on Q, i.e., there exist positive constants $H_1>H_2>0$ such that

  $$\begin{aligned} \Vert \mathfrak {F}^\prime (s)\Vert \le H_1, \quad \forall s\in Q, \end{aligned}$$

  (24)

  and

  $$\begin{aligned} H_2\Vert d\Vert ^{2}\le d^{T}\mathfrak {F}^\prime (s)d, \quad \forall s\in Q, d\in \mathbb {R}^n. \end{aligned}$$

  (25)

Remark 3.2

We make the following remark:

Assumption 3.1 implies that there exist constants $H_1>H_2>0$ such that

$$\begin{aligned} & H_2\Vert d\Vert \le \Vert \mathfrak {F}^\prime (s)d\Vert \le H_1\Vert d\Vert , \quad \forall s\in Q, d\in \mathbb {R}^n. \end{aligned}$$

(26)

$$\begin{aligned} & H_2\Vert s-v\Vert \le \Vert \mathfrak {F}(s)-\mathfrak {F}(v)\Vert \le H_1\Vert s-v\Vert \quad \forall s,v\in Q. \end{aligned}$$

(27)

Since $\rho ^{-1}\delta _{k}I$ approximates $\mathfrak {F}^\prime (s_{k})$ along $x_{k}$, the following assumption can be made.

Assumption 3.3

$\rho ^{-1}\delta _{k}I$ is a good approximation to $\mathfrak {F}^\prime (s_{k})$, i.e.,

$$\begin{aligned} \Vert (\mathfrak {F}^\prime (s_{k})-\rho ^{-1}\delta _{k}I)d_{k}\Vert \le \epsilon \Vert \mathfrak {F}(s_{k})\Vert , \end{aligned}$$

(28)

where, $\varepsilon \in (0,1)$ is a small quantity.

Lemma 3.4

Let $\{s_{k}\}$ be produced by the MIDSL algorithm, supposing Assumption 3.3is true. Then $d_{k}$ is a sufficient descent direction for $\mathfrak {h}(s_{k})$ at ${x_{k}}$ i.e.,

$$\begin{aligned} \triangledown \mathfrak {h}(s_{k})^Td_{k}<c\Vert \mathfrak {F}(s_{k})\Vert ^{2}, ~~c>0. \end{aligned}$$

(29)

Proof

From (3), (19), and (28), we have

$$\begin{aligned} \begin{aligned} \triangledown \mathfrak {h}(s_{k})^Td_{k}&=\mathfrak {F}_{k}^T \mathfrak {F}^\prime (s_{k})d_{k} \\&=\mathfrak {F}(s_{k})^T[(\mathfrak {F}^\prime (s_{k})-\rho ^{-1}\delta _{k}I)d_{k}-\mathfrak {F}(s_{k})]\\&=\mathfrak {F}(s_{k})^T(\mathfrak {F}^\prime (s_{k})-\rho ^{-1}\delta _{k}I)d_{k}-\Vert \mathfrak {F}(s_{k})\Vert ^{2}. \end{aligned} \end{aligned}$$

(30)

Applying the Cauchy–Schwarz inequality, we have,

$$\begin{aligned} \begin{aligned} \triangledown \mathfrak {h}(s_{k})^Td_{k}&\le \Vert \mathfrak {F}(s_{k})\Vert \Vert (\mathfrak {F}^\prime (s_{k})-\rho ^{-1}\delta _{k}I)d_{k}\Vert -\Vert \mathfrak {F}(s_{k})\Vert ^{2} \\ &\le -(1-\epsilon )\Vert \mathfrak {F}(s_{k})\Vert ^{2}. \end{aligned} \end{aligned}$$

(31)

Since $\epsilon \in (0,1)$, taking $c=1-\epsilon$, Lemma 3.4 is true.

We can conclude from Lemma 3.4 that the norm function $\mathfrak {h}(s_{k})$ is a descent along $d_{k}$, which means that $\Vert \mathfrak {F}(s_{k+1})\Vert \le \Vert \mathfrak {F}(s_{k})\Vert$ is true. $\square$

Lemma 3.5

Let $\{s_{k}\}$ be produced by the MIDSL algorithm, supposing Assumption 3.3is true. Then $\{s_{k}\}\subset \Omega$.

Proof

From Lemma 3.4, we have $\Vert \mathfrak {F}(s_{k+1})\Vert \le \Vert \mathfrak {F}(s_{k})\Vert$. Furthermore, for all k,

$$\begin{aligned} \Vert \mathfrak {F}(s_{k+1})\Vert \le \Vert \mathfrak {F}(s_{k})\Vert \le \Vert \mathfrak {F}(s_{k-1})\Vert \le \ldots \le \Vert \mathfrak {F}(s_{0})\Vert . \end{aligned}$$

This means that $\{s_{k}\}\subset \Omega$. $\square$

Lemma 3.6

Let $\{s_{k}\}$ be produced by the MIDSL algorithm, supposing Assumption 3.3is true. Then there exists a constant $H_2>0$ such that for all k,

$$\begin{aligned} v_{k}^Tx_{k}\ge H_2\Vert x_{k}\Vert ^{2}, \end{aligned}$$

(32)

where $x_k=(\alpha _k+\beta _k)d_k$.

Proof

From mean-value theorem and (25),

$v_{k}^Tx_{k}=x_{k}^T(\mathfrak {F}(s_{k+1})-\mathfrak {F}(s_{k}))=x_{k}^{T}F^\prime (\xi )x_{k}\ge H_2\Vert x_{k}\Vert ^{2}$.

Where $\xi =s_{k}+\kappa (s_{k+1}-s_{k})$ , $\kappa \in (0,1)$.

As shown in the related Lemma 3.6 and (27), the following inequality is true.

$$\begin{aligned} \frac{v_{k}^Tx_{k}}{\Vert x_{k}\Vert ^2}\ge H_2, \quad \quad \frac{\Vert v_{k}\Vert ^2}{v_{k}^Tx_{k}}\le \frac{H_1^{2}}{H_2}. \end{aligned}$$

(33)

$\square$

Lemma 3.7

Let $\{s_{k}\}$ be produced by the MIDSL algorithm, supposing Assumption 3.3is true. Then we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert (\alpha _k+\beta _k)d_{k}\Vert =0, \end{aligned}$$

(34)

and

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert (\alpha _k+\beta _k)\mathfrak {F}(s_{k})\Vert =0. \end{aligned}$$

(35)

Proof

From (21) for all $k>0$

$$\begin{aligned} \begin{aligned} \omega _{2}\Vert (\alpha _k+\beta _k)d_{k}\Vert ^{2}&\le \omega _{1}\Vert (\alpha _k+\beta _k)\mathfrak {F}(s_{k})\Vert ^{2}+\omega _{2}\Vert (\alpha _k+\beta _k)d_{k}\Vert ^{2}\\&\le \Vert \mathfrak {F}(s_{k})\Vert ^{2}-\Vert \mathfrak {F}(s_{k+1})\Vert ^{2}+\chi _{k}\Vert \mathfrak {F}(s_{k})\Vert ^{2}. \end{aligned} \end{aligned}$$

(36)

By summing the above inequality, we have

$$\begin{aligned} \begin{aligned} \omega _{2}\sum _{i=0}^{k}\Vert (\alpha _i+\beta _i)d_{i}\Vert ^{2}&\le \sum _{i=0}^{k}\left( \Vert \mathfrak {F}(s_{i})\Vert ^{2}-\Vert \mathfrak {F}(s_{i+1})\Vert ^{2}\right) + \sum _{i=0}^{k}\chi _{i}\Vert \mathfrak {F}(s_{i})\Vert ^{2}, \\ &=\Vert \mathfrak {F}(s_{0)}\Vert ^{2}-\Vert \mathfrak {F}(s_{k+1})\Vert ^{2}+\sum _{i=0}^{k}\chi _{i}\Vert \mathfrak {F}(s_{i})\Vert ^{2}, \\ &\le \Vert \mathfrak {F}(s_{0})\Vert ^{2}+\Vert \mathfrak {F}(s_{0})\Vert ^{2}\sum _{i=0}^{k}\chi _{i}, \\ &\le \Vert \mathfrak {F}(s_{0})\Vert ^{2}+\Vert \mathfrak {F}(s_{0})\Vert ^{2}\sum _{i=0}^{\infty }\chi _{i}. \end{aligned} \end{aligned}$$

(37)

Based on the level set and the fact that the sequence $\{\chi _{k}\}$ meets the criterion in (22), it follows that the series $\displaystyle \sum _{i=0}^{\infty }\Vert (\alpha _i+\beta _i)d_{i}\Vert ^{2}$ converges. This leads to the conclusion in (34). By applying the same reasoning as above, but now considering $\omega _{1}\Vert (\alpha _k+\beta _k)\mathfrak {F}_{k}\Vert ^{2}$ on the left side, we arrive at (35). $\square$

Lemma 3.8

Let $\{s_{k}\}$ be produced by the MIDSL algorithm, supposing Assumption 3.3is true. Then there exists a constant $M_1>0$ such that for all$k>0$,

$$\begin{aligned} \Vert d_{k}\Vert \le M_1. \end{aligned}$$

(38)

Proof

From (19), (20), and (27), we have

$$\begin{aligned} \begin{aligned} \Vert d_{k}\Vert&=\left\| -\rho \frac{(\alpha _{k-1}+\beta _{k-1})v_{k-1}^Td_{k-1}\mathfrak {F}(s_{k})}{v_{k-1}^Tv_{k-1}}\right\| \\ &=\left\| -\rho \frac{v_{k-1}^Tx_{k-1}\mathfrak {F}(s_{k})}{\Vert v_{k-1}\Vert ^{2}}\right\| \\ &\le \frac{\rho \Vert \mathfrak {F}(s_{k})\Vert \Vert x_{k-1}\Vert \Vert v_{k-1}\Vert }{H_2^{2}\Vert x_{k-1}\Vert ^{2}} \\ &\le \frac{\rho \Vert \mathfrak {F}(s_{k})\Vert H_1\Vert x_{k-1}\Vert }{H_2^{2}\Vert x_{k-1}\Vert } \\ &\le \frac{\rho \Vert \mathfrak {F}(s_{k})\Vert H_1}{H_2^{2}} \\ &\le \frac{\rho \Vert \mathfrak {F}(s_{0})\Vert H_1}{H_2^{2}}. \end{aligned} \end{aligned}$$

(39)

Taking $M_1=\frac{\rho \Vert \mathfrak {F}(s_{0})\Vert H_1}{H_2^{2}}$, we have (38). $\square$

Theorem 3.9

Let $\{s_{k}\}$ be produced by the MIDSL algorithm, supposing Assumption 3.3is true. Assume further for all $k>0$,

$$\begin{aligned} (\alpha _{k}+\beta _{k})\ge \lambda \frac{|\mathfrak {F}(s_{k})^{T}d_{k}|}{\Vert d_{k}\Vert ^{2}}, \end{aligned}$$

(40)

where $\lambda$ is some positive constant. Then

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \mathfrak {F}(s_{k})\Vert =0. \end{aligned}$$

(41)

Proof

From Lemma 3.8 we have (38). Also, from (34) and the boundedness of $\{\Vert d_{k}\Vert \}$, we have

$$\begin{aligned} \lim _{k\rightarrow \infty }(\alpha _{k}+\beta _{k})\Vert d_{k}\Vert ^{2}=0, \end{aligned}$$

(42)

from (40) and (42) we have

$$\begin{aligned} \lim _{k\rightarrow \infty }|\mathfrak {F}(s_{k})^{T}d_{k}|=0. \end{aligned}$$

(43)

Also, from (19) we have,

$$\begin{aligned} & \mathfrak {F}(s_{k})^{T}d_{k}=-\rho \delta _{k}^{-1}\Vert \mathfrak {F}(s_{k})\Vert ^{2}, \end{aligned}$$

(44)

$$\begin{aligned} & \begin{aligned} \Vert \mathfrak {F}(s_{k})\Vert ^{2}&= |-\mathfrak {F}(s_{k})^{T}d_{k}\rho ^{-1}\delta _{k}| \\ &\le \rho ^{-1}|\delta _{k}||\mathfrak {F}_{k}^{T}d_{k}|. \end{aligned} \end{aligned}$$

(45)

Since

$$\begin{aligned} \delta _{k}^{-1}=\frac{(\alpha _{k}+\beta _{k})v_k^Td_k}{v_{k-1}^Tv_{k-1}} =\frac{v_{k-1}^{T}x_{k-1}}{\Vert v_{k-1}\Vert ^{2}}\ge \frac{h\Vert x_{k-1}\Vert ^{2}}{\Vert v_{k-1}\Vert ^{2}} \ge \frac{H_2\Vert x_{k-1}\Vert ^{2}}{H_1^{2}\Vert x_{k-1}\Vert ^{2}}=\frac{H_2}{H_1^{2}}. \end{aligned}$$

Then,

$$\begin{aligned} |\delta _{k}^{-1}|\ge \frac{H_2}{H_1^{2}}. \end{aligned}$$

Therefore from (45) we have,

$$\begin{aligned} \Vert \mathfrak {F}(s_{k})\Vert ^{2}\le |\mathfrak {F}(s_{k})^{T}d_{k}|\left( \frac{H_1^{2}}{\rho H_2}\right) . \end{aligned}$$

(46)

Consequently,

$$\begin{aligned} 0\le \Vert \mathfrak {F}(s_{k})\Vert ^{2}\le |\mathfrak {F}(s_{k})^{T}d_{k}|\left( \frac{H_1^{2}}{\rho H_2}\right) \longrightarrow 0. \end{aligned}$$

(47)

Hence,

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \mathfrak {F}(s_{k})\Vert =0. \end{aligned}$$

(48)

The proof is complete. $\square$

Numerical experiments

In this section, the study discusses the computational efficiency of the methods on a system of nonlinear equations with application to motion control problems, illustrating the efficiency and relevance of the proposed method real-world situations.

Experiments on system of nonlinear equations

The robustness and efficiency of the new (MIDSL) scheme was assessed by comparing with other existing algorithms including:

• 1. A double step length method (IDS) presented in¹².

• 2. A modified double-direction method (MDFDD) is presented in²⁵.

The computer codes used in this study were developed using Matlab version 9.4.0 (R2018a) and executed on a PC that features an 8 GB of RAM and 1.80 GHz CPU processor. All three methods run under similar line search method, as detailed in equation (21). Throughout the experiments, the parameters for the IDS and MDFDD algorithms were set according to the settings outlined in¹² and²⁵, respectively. For the MIDSL method, the following parameter values were used: $\omega _{1} = \omega _{2} = 10^{-4}$, $\rho = 1.2$, $q = 0.2$, $r = 0.3$ and $\chi _{k}=\displaystyle \frac{1}{(k+1)^{2}}$. The program execution terminates if the iteration count exceeds 1000 or if $\Vert \mathfrak {F}(s_{k})\Vert \le 10^{-5}$. The symbol ’–’ denotes a failure when the number of iterations exceeds 1000 without discovering a point of $s_k$ that meets the established stopping criterion. To demonstrate the comprehensive numerical experiments conducted with the three methods, we applied these techniques to six benchmark test problems, using various initial points and dimensions ranging from 1000 to 100,000.

The initial guesses presented in Table 1 were utilized in the experiments:

Table 1 Initial points.

and the subsequent test problems were selected based on the SNE to illustrate the performance of the proposed algorithm.

Problem 1⁷

$\mathfrak {F}_{1}(s)=s_{1}-e^{\cos \left( \frac{s_{1}+s_{2}}{n+1}\right) }$,

$\mathfrak {F}_{i}(s)=s_{i}-e^{\cos \left( \frac{s_{i-1}+s_{i}+s_{i+1}}{n+1}\right) }$,

$\mathfrak {F}_{n}(s)=s_{n}-e^{\cos \left( \frac{s_{n-1}+s_{n}}{n+1}\right) }$,    $i=2,3,\ldots ,n-1$.

Problem 2¹²

$\mathfrak {F}_{i}(s)=(1-s_{i}^{2})+s_{i}(1+s_{i}s_{n-2}s_{n-1}s_{n})-2$,    $i=1,2,\ldots ,n$.

Problem 3⁹

$\mathfrak {F}_{i}(s)=s_i-3s_i\left( \frac{\sin s_i}{3}-\frac{33}{50}\right) +2$,    $i=1,2,\ldots ,n$.

Problem 4²⁷

$\mathfrak {F}_{i}(s)=s_{i}- \left( 1- \frac{c}{2n} \sum\nolimits_{j=1}^{n}\frac{\mu _is_j}{\mu _i+\mu _j}\right) ^{-1}$,    $i=1,2,\ldots ,n, \quad j=1,2,\ldots ,n$ with $c\in [0, 1)$ and $\mu =\frac{i-0.5}{n}$. (In our experiment we take c =0.1).

Problem 5²⁵

$\mathfrak {F}_{i}(s)=2s_{i}-\sin |s_{i}|$,    $i=1,2,\ldots ,n.$

Problem 6²⁵

$\mathfrak {F}_{1}(s)=(s_1^2+s_2^2)s_1-1,$

$\mathfrak {F}_{i}(s)=(s_{i-1}^2+2s_{i}^2+s_{i+1}^2)s_{i}-1,$

$\mathfrak {F}_{n}(s)=(s_{n-1}^2+s_{n}^2)s_n$,    $i=2,3,\cdots ,n-1$.

Table 2 Numerical experiments for problems 1 & 2 with respect to MIDSL, IDS & MDFDD methods.

Table 3 Numerical experiments for problems 3 & 4 with respect to MIDSL, IDS & MDFDD methods.

Table 4 Numerical experiments for problems 5 & 6 with respect to MIDSL, IDS & MDFDD methods.

Fig. 1

Performance profile based on the number of iterations within a factor $\tau$ of the best time.

Fig. 2

Performance profile based on the function evaluation within a factor $\tau$ of the best time.

Fig. 3

Performance profile with respect to CPU time (in seconds) within a factor $\tau$ of the best time.

Detailed results of the numerical experiment are reported in Tables 2, 3 and 4. In evaluating these methods, we focused on three key metrics: the number of iterations (Iter), function evaluations (Fun), CPU time in seconds (Time(s)) required for each approach to attain a solution, and the residual norm $\Vert \mathfrak {F}(s_{k})\Vert$. Below, we now analyse these metrics and their significance.

The number of iterations represents the cycles the algorithm undergoes to find a solution. Typically, a lower number of iterations indicates a more efficient algorithm. The tables show that the MIDSL method requires fewer iterations than both the IDS and MDFDD methods, demonstrating its effectiveness in solving the problem. Function evaluations denote the frequency with which the algorithm computes the value of the objective function during its operation. While the function evaluations for the proposed methods are considerably higher than those for the IDS method, the proposed methods still achieve a solution more rapidly than IDS. CPU time refers to the actual duration required by the algorithm to compute a solution, influenced by the complexity of each iteration and the overall number of iterations. In practical terms, a shorter CPU time signifies a faster and more efficient method. Our findings indicate that the MIDSL consistently outperforms the other methods in terms of CPU time, often arriving at a solution more quickly than both IDS and MDFDD. Moreover, the accuracy of the solution is evaluated through the residual norm, which indicates how well the computed solution meets the equations of the original problem. The tables reveal that all three methods achieve a sufficiently low residual norm, ensuring their effectiveness in solving problem (1). Upon analysing the results, it becomes evident that all three methods seek to address the issue defined by equation (1). Nevertheless, the proposed method not only converges to the answer more rapidly but also requires significantly fewer iterations and less CPU processing time than those used for comparison. In addition, the MIDSL method demonstrates its superiority by successfully solving all the problems that the MDFDD method was unable to tackle. A notable example is found in Problem 4, where the MDFDD method failed to arrive at a solution, while the proposed method delivered effective results. This highlights the reliability and efficiency of the proposed method in addressing the nonlinear problems.

Leveraging the performance profile established by Dolan and Mor$\acute{\textrm{e}}$²⁸, we have generated Figures 1, 2 and 3 to provide a detailed overview of the performance and efficiency of three distinct methods. Each figure conveys essential information about the fraction $P(\tau )$ of problems for which a particular method operates within a factor $\tau$ of the optimal execution time. From Figs. 1 and 3 , it is clear that the curves representing the MIDSL method outperform those of both the IDS method and the MDFDD method. In most cases, the MIDSL method delivers superior performance. Nevertheless, Fig. 2 reveals instances where the IDS method significantly surpasses the MIDSL method for a specific set of problems, highlighting that while the MIDSL method is generally more effective, there are particular scenarios in which it may not be the optimal choice. Ultimately, these observations suggest that the proposed MIDSL method excels in efficiency, evidenced by a reduction in both the number of iterations required and the CPU time consumed (measured in seconds). The graphical representations in the figures demonstrate that our method is adept at solving large-scale nonlinear systems of equations, establishing its value in practical applications.

Application of the proposed method in motion control models

In this section, the problems arising from motion control involving three-joint planar robotic manipulators are considered. Since (1) is equivalent to the global optimisation problem (4), we can apply the MIDSL algorithm to solve this problem. The kinematics model provides a detailed framework for understanding the orientation and position of each component within a robotic system. This model represents the dynamics of the various components and their interactions, facilitating a comprehensive analysis of the robot’s overall functionality and performance. The model can be described as follows.

$$\begin{aligned} \varphi (\theta )=\left[ \begin{array}{c} a_1+a_2+a_3 \\ b_1+b_2+b_3 \\ \end{array} \right] , \end{aligned}$$

(49)

where the kinematics are mapped using the function $\varphi (\cdot )$, $a_1=l_1\cos (\theta _1)$, $a_2=l_2\cos (\theta _1 + \theta _2)$, $a_3=l_3\cos (\theta _1 + \theta _2+\theta _3)$, $b_1=l_1\sin (\theta _1)$, $b_2=l_2\sin (\theta _1 + \theta _2)$, and $b_3=l_3\cos (\theta _1 + \theta _2+\theta _3)$. Additionally, the $j^{th}$ rod’s lengths are $l_1$, $l_2$ and $l_3$. The vectors $\varphi (\theta )\in \mathbb {R}^3$ and $\theta \in \mathbb {R}^3$ correspond to the end effector position vector and joint angle vector, respectively. Now, at any time interval $t_k\in [0, t_f]$, the minimisation issue may be addressed as

$$\begin{aligned} \min _{\theta \in \mathbb {R}^3}\frac{1}{2}\Vert \varphi (\theta )-u_{t_k}\Vert ^2, \end{aligned}$$

(50)

where $t_f$ is the end of task duration and $u_{t_k}\in \mathbb {R}^2$ denotes the vector of the desired path at time $t_k$. According to²⁹, the rod lengths are chosen to be equal to 1, and a Lissajous curve is tracked to control the end effector as

$$\begin{aligned} u_{t_k}=\left[ \begin{array}{c} \frac{3}{2}+\frac{2}{5}\sin \left( \frac{\pi t_k}{5}\right) \\ \frac{\sqrt{(}3)}{2}+\frac{2}{5}\sin \left( \frac{\pi t_k}{5}+\frac{\pi }{3}\right) \\ \end{array} \right] . \end{aligned}$$

(51)

We set the following parameters for this experiment: $t_f=10$ seconds, $\omega _{1} = \omega _{2} = 10^{-4}$, $\rho = 1.2$, $q = 0.2$, $r = 0.3$, and the starting point $\theta _0=[\theta _1,\theta _2,\theta _3]=[0,\frac{\pi }{3},\frac{\pi }{2}]^T.$ For fairness of comparison, the parameters of IDS and MDFDD are chosen precisely as specified in¹² and²⁵, respectively. Furthermore, the work duration [0, 10] is divided into 200 equal portions.

Figures 4, 5, 6 and  7 provide a comprehensive visualisation of the robot trajectories generated by the MIDSL algorithm. The patterns observed in these figures indicate that the MIDSL algorithm not only meets the given task objectives but does so with high efficiency. Specifically, Figures 6 and 7 highlight the precision of three algorithms by showing that the errors incurred in MIDSL and MDFDD are both approximately $10^{-5}$, whereas one in IDS algorithm is around $10^{-1}$. This minimal error margin emphasises the reliability and effectiveness of our proposed method in achieving accurate results in robotic trajectory planning. Overall, these findings support the practical applicability of the MIDSL algorithm in real-world problems.

Fig. 4

Manipulator trajectories.

Fig. 5

End effector trajectory and desired path.

Fig. 6

Tracking errors on the horizontal x-axis.

Fig. 7

Tracking errors on the vertical y-axis.

Conclusion

This study constructed a hybridized step size algorithm for solving systems of nonlinear equations, utilizing the Picard-Mann hybrid iterative scheme as articulated in²³. The proposed scheme was derived by incorporating a correction parameter that improve convergence. An important feature of the proposed algorithm is its derivative-free scheme, which enhance the efficiency of the algorithm for large-scale problems. To evaluate the efficiency of the new scheme, the study conducted extensive numerical experiment on a set of large-scale test problems. Preliminary findings show that the proposed algorithm outperformed other existing algorithms by consistently requiring fewer iteration count and less CPU time. The study further extended the new scheme to solve motion control model with three degrees of freedom, demonstrating its practical applicability. Findings from the application problem demonstrate the method’s effectiveness in real-world situations. Future studies aim to explore the construction of a double-step size algorithm that does not rely on the Lipschitz assumption. This approach if successful will further improve the efficiency and robustness of the method.