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# Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

1 Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung, 202301, Taiwan

2 Department of Mathematics, College of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Alkharj, 11942, Saudi Arabia

3 Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom, 32511, Egypt

4 Department of Marine Engineering, National Taiwan Ocean University, Keelung, 202301, Taiwan

* Corresponding Author: Yung-Wei Chen. Email:

*Computer Modeling in Engineering & Sciences* **2023**, *135*(2), 1111-1130. https://doi.org/10.32604/cmes.2022.021655

**Received** 26 January 2022; **Accepted** 31 May 2022; **Issue published** 27 October 2022

## Abstract

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations (NAEs). This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms. We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system. Through the maximal orthogonal projection concept, to minimize a merit function within a selected interval of splitting parameters, the optimal parameters can be quickly determined. In each step, a linear system is solved by the Gaussian elimination method, and the whole iteration procedure is convergent very fast. Several numerical tests show the high performance of the optimal split-linearization iterative method (OSLIM).## Graphical Abstract

## Keywords

Nonlinear partial differential equations (PDEs) frequently appear in many critical natural sciences and engineering technology problems. Mathematical models in physics, mechanics, and other disciplines fields can be applied to describe physical phenomena. Although some phenomena may be modeled by the linear type PDEs, the assumption of more complex conditions on the material property and geometric domain of the problem requires the use of nonlinear type PDEs. In most situations, solving nonlinear PDEs analytically and exactly is impossible. Because of the crucial role of the nonlinear problems, the researchers provided different numerical methods to obtain the solutions of nonlinear PDEs, which are properly transformed into the nonlinear algebraic equations (NAEs) by using numerically discretized methods. As mentioned in [1–4], nonlinear optimization problem, nonlinear programming, nonlinear sloshing problem, and nonlinear complementarity problem (NCP) can be recast to the issues to solve NAEs, with the help of NCP-functions, for example, the Fischer-Burmeister NCP [5]. Indeed, the numerical solution of NAEs is one of the main problems in computational mathematics. Among the many numerical methods, Newton’s method and its improvements have now been widely developed; however, if the initial guess of the solution is incorrect, these algorithms may fail. In general, it is difficult for most NAEs to choose a good initial condition. Therefore, it is necessary to develop a more effective algorithm that is less sensitive to the initial guess of the solution and converges quickly.

For the vector form of the NAEs:

the Newton method is given by

where

The Newton method possesses a significant advantage in that it is quadratically convergent. However, the main drawbacks are sensitive to the initial point

where ν is a nonzero constant and q(t) may generally be a monotonically increasing function of t. The term

where A and

Let

the Jacobi iterative scheme for solving Eq. (4) is given by [12,13]

where

where U and L are a strict upper- and lower triangular matrix of A, respectively.

According to Eqs. (4) and (7), the SOR iterative method can be derived as follows [14,15]:

where w is a nonzero relaxation factor. When

Nowadays, there have different splitting techniques to decompose A as

where M is a non-singular matrix. Inserting Eq. (9) into Eq. (4), moving

For the LAE, there are many discussions about the split method to determine the best relaxation factor w in SOR [16–20]. In contrast, few results discuss the relationship between the split method and NAEs. In this paper, the split method based on the nonlinear term is applied to reduce the influence of nonlinear terms and further avoid selecting the parameters by a trail and error method. Hence, this article will propose a novel splitting-linearizing technique to iteratively solve the NAE and develop a new algorithm for determining the optimal splitting parameters.

2 A Split Linearization Method

In this paper, a new splitting idea is implemented and applied to decompose Eq. (1), and we develop an iterative method to solve it. In the general case, Eq. (1) can be expressed as follows:

where A and B are both

Then, we can decompose Eq. (11) to

The idea of the splitting technique in [21,22] has been proposed to solve a single nonlinear equation. Future, the nonlinear term

We give an initial guess

For

where

The above process is a novel splitting-linearizing method for solving the NAEs.

In this study, the Gaussian elimination method solves Eq. (14) to determine

For a linear system

to minimize the merit function has been derived and proved in [23–25] as follows:

to obtain a fast descent direction c for a given residual vector b used in the iterative solution of Eq. (16).

Let

The minimization in Eq. (17) is equivalent to maximize

which is the orthogonal projection of

To optimize w in Eqs. (14) and (15),

where

Let

and insert them into Eq. (17), we can derive

We can search the minimal value of

in a given interval

In the OSLIM, there is an interesting feature that by using the Cauchy-Schwarz inequality in Eq. (20),

it readily leads to

In general,

The procedures of the OSLIM to solve the NAEs are summarized as follows:

(i) Give n and nonlinear equations,

(ii) Give an initial guess

(iii) For

If

To further assess the performance of OSLIM, we investigate it using the convergence order. For solving a scalar equation

where r is a solution of

For solving the NAEs, we can generate the sequence

where

We first consider [26]:

Two roots are (1, 1) and (1, −1) obviously; however, there exist other roots.

As noted by Yeih et al. [26], the Newton method fails because the Jacobi matrix

is singular at

Using the OSLIM to solve this problem, a translation is applied as follows:

and choose

Eq. (37) can be written as

of which

By using the OSLIM, we take

Considering two-variable nonlinear equations [28] as follows:

Hirsch et al. [28] used the continuous Newton algorithm to calculate this problem; however, the numerical results are inaccurate. Under the same situation, the FTIM [6], modified Newton method [9], and RNBA [27] have been applied and obtained three solutions from three different initial guessing values. In Liu et al. [29], this problem solved by the optimal vector drive algorithm (OVDA) can find five solutions. As stated in the literature [6], the attracting sets for several fixed points of problem are undefined. Therefore, various methods may give different solutions, even with the same initial value.

Eqs. (40) and (41) can be written as

or written as

Eqs. (42) and (43) are of the form in Eq. (14), but with different B. For the uniqueness of B, for example, we can put the product term

To compare with the result obtained by Atluri et al. [9], we take

Considering a system of three algebraic equations with three variables:

Obviously

By using the OSLIM, we take

Here, considering the following boundary value problem:

The exact solution is

By using the finite difference discretisation of u at the grid points, we can obtain

where

Setting n = 39,

Then, considering a test example given by Krzyworzcka [30] as follows:

To convince describe the coefficient matrices of B as Eq. (21),

Setting

The following example is given by Roose et al. [31]:

The OSLIM can adopt two different formulations to solve Eq. (53). The two different formulations lead to different coefficient matrices of B. First, we adopt

To convince desdescribee coefficient matrices of B as Eq. (21),

where the number with a superscript k denotes the kth step value, and it is inserted in the coefficient matrix when we derive the linear system. We modify the term

in the second line to

such that we have a second formulation:

To convince describe the coefficient matrices of B as Eq. (21),

Setting n = 10,

Setting n = 10,

Considering the following a two-dimensional nonlinear Bratu equation:

and the domain is an ellipse with semi-axes lengths a = 1 and b = 0.5. The Dirichlet boundary condition is imposed. Due to u > 0, we decompose it to

where

After collocating points inside the domain and on the boundary, simultaneously, to satisfy the Eq. (56) and the boundary condition equation, we can derive NAEs to determine the

By using the OSLIM, we take

When evaluating the performance of newly developed iterative solutions for solving nonlinear algebraic equations (NAE), there are two critical factors: accuracy and convergence speed. In this article, we first decomposed the nonlinear terms in the NAEs through a splitting parameter and then we linearized the NAEs around the values at the previous step and derived a system of linear algebraic equations (LAEs). Through the maximum orthogonal projection concept at each iterative step, we select the optimal value of the splitting parameter and use the Gaussian elimination method to solve the LAE. The features of the proposed method are summarized as follows: (a) The merit function is in terms of the coefficient matrix, right-side vector and the value of unknown vector at the previous step. (b) It is easy to minimize the search within a preferred interval through some operations. (c) The current OSLIM is insensitive to the initial guess and without needing the inversion of the Jacobian matrix at each iterative step, which can save much of the computational time. We have successfully applied the OSLIM to solve the NAEs resulting from the nonlinear boundary value problem in one- and two-dimensional spatial spaces. Through the numerical test, the convergence order of the OSLIM is between linear and quadratic convergence.

Funding Statement: The corresponding author appreciates the financial support provided by the Ministry of Science and Technology, Taiwan, ROC under Contract No. MOST 110-2221-E-019-044.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

Liu, C., El-Zahar, E. R., Chen, Y. (2023). Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method.*CMES-Computer Modeling in Engineering & Sciences, 135(2)*, 1111–1130.