Solving fuzzy complex system of linear equations using eigenvalue method

Abstract

The occurrence of imprecision in the real world is inevitable due to some unexpected situations. The imprecision is often involved in any engineering design process. The imprecision and uncertainty are often interpreted as fuzziness. Fuzzy systems have an essential role in the uncertainty modeling, which can formulate the uncertainty in the actual environment. In this paper, a new method based on linear algebra for resolution of a fuzzy complex system of linear equations is presented. Moreover, a new algorithm is designed to find all the solutions of the system based on the eigenvalue method. Thanks to our approach, we can find all the solutions of the system at a time. Also, we are able to determine when the solution sets of the system are nonempty sets.To illustrate the easy application of the proposed method, numerical examples are given and the obtained results are discussed.

Keywords

Triangular fuzzy number fuzzy complex number fuzzy complex system of linear equations Gröbner bases eigenvalues

1 Introduction

Complex system of linear equations plays a vital role in real life problems such as optimization, current flow, economics and engineering. Hence, its resolution has an essential role in the fields. A general complex system of linear equations may be written as CZ = W, where C and W are standard complex matrices and Z is the unknown complex vector. For the sake of simplicity, variables and parameters of these systems are defined exactly in the modeling. But in actual practice, the parameters and variables may be uncertain or vague and those are found, in general, by some experiment or experience. Hence, to overcome the uncertainty, one may use the complex numbers as fuzzy. Accordingly, fuzzy complex system of linear equations is consideredas $[C] {\tilde{Z}} = {\tilde{W}},$ where the coefficient matrix [C] is an usual complex matrix, ${\tilde{W}}$ is a column vector of fuzzy complex number and ${\tilde{Z}}$ is the vector of fuzzy complex unknown.

Fuzzy complex number was first proposed by Buckley [9]. Qiu et al. [12, 13] discussed the sequence and series of fuzzy complex numbers and their convergence. Solution of fuzzy complex system of linear equations was described by Rahgooy et al. [20] and applied to circuit analysis problem. Jahantigh et al. [17] developed a numerical procedure for complex fuzzy linear systems. Behera and Chakraverty [3] proposed a new and simple center and width based method to solve fuzzy real and complex system of linear equations. Solution sets of complex linear interval systems were investigated by Hladik [16]. Householder method was used by Djanybekov [2] for the solution of interval complex linear systems. In interval complex linear systems, the coefficient matrix was also taken as interval by the authors [2]. Further, Candau et al. [21] analyzed the complex interval arithmetic using polar form. Fuzzy modeling and identification procedure was implemented by Cao et al. [19] for the analysis and design of complex control systems. Filev [7] applied fuzzy approach to the control of nonlinear systems. Further, Petkovic and Petkovic [18] investigated the complex interval arithmetic and applied it to various example problems. The authors [18], presented a circular form of the interval complex number. Complex interval arithmetic was also studied by Rokne and Lancaster [14]. Recently, Behera and Chakraverty [4] proposed a new method to solve a general fuzzy complex system of linear equations. In the original system, the elements of unknown variable vector and right hand side vector are considered as complex fuzzy number. Initially, the general system is solved by adding and subtracting the left and right bounds of the fuzzy complex unknown and right hand side fuzzy complex vector, respectively. Subsequently, the above obtained solutions are used to get the final solution of the general fuzzy complex system of linear equations.

We propose an approach to solve the fuzzy complex system of linear equations based on the eigenvalue method. In the eigenvalue method, the computation of solutions of a system is done independently with respect to each other. In the proposed method, the computation of solutions of a fuzzy complex system of linear equations leads to finding the eigenvalues of a matrix. Hence, we can use the useful tools in the linear algebra such as converting the matrix into a triangular matrix via elementary row operations and applying the determinant properties. An algorithm is also proposed to find the solutions of the system based on the eigenvalue method. In this approach, suppose that I is a zero-dimensional ideal in $ℝ [x_{1}, \dots, x_{n}]$ . This means that the quotient ring $\frac{ℝ [x_{1}, \dots, x_{n}]}{I}$ is a finite dimensional vector space on $ℝ$ . Given any polynomial f, the matrix representing the following linear mapping $\begin{matrix} θ_{f} : \frac{ℝ [x_{1}, \dots, x_{n}]}{I} \to \frac{ℝ [x_{1}, \dots, x_{n}]}{I}, \\ [g] \mapsto [f . g], \end{matrix}$ with respect to a basis of $\frac{ℝ [x_{1}, \dots, x_{n}]}{I}$ is denoted by m_f. Stickelberger’s Theorem states that eigenvalues of m_f are the values of f evaluated at the solutions of the polynomial system. In particular, evaluating the polynomials f = x_i, i = 1,. . . , n gives the coordinates of the solutions of the polynomial system. The main idea of the proposed approach is based on converting the fuzzy complex system of linear equations into a crisp system and getting a polynomial system of 8n equations and 6n unknowns such that the solutions of the new system may be obtained by a successful scheme of solving systems. In this approach, we can compute the Gröbner basis for the ideal generated by the new system with respect to reverse graded lexicographic order. Then, the eigenvalue method is applied to find the solutions of the system. Moreover, a sufficient and necessary condition is presented for existence of the solution of a fuzzy complex system of linear equations.

The rest of the paper is set out as follows. In Section 2, preliminaries on fuzzy arithmetic, fuzzy complex arithmetic, polynomials and Gröbner basis are given and the eigenvalue method for solving a system of polynomial equations is illustrated and reviewed. In Section 3, a fuzzy complex system of linear equations is introduced. The main algorithm to find solutions of the fuzzy complex system of linear equations is suggested in Section 3. The proposed method is illustrated by solving some examples in Section 4. Last section includes the conclusion.

2 Preliminaries

This section contains three subsections. Preliminaries on fuzzy arithmetic and fuzzy complex arithmetic are introduced in the first subsection. The basic concepts related to polynomials and Gröbner basis are also introduced in the second subsection. Finally, we introduce the eigenvalue method to solve a system of polynomial equations in the third subsection.

2.1 Fuzzy arithmetic and fuzzy complex arithmetic

In this subsection, fuzzy arithmetic and fuzzy complex arithmetic are reviewed. The details can be found in [8].

Definition 2.1. A fuzzy subset $\tilde{u}$ of $ℝ$ is defined by its membership function $μ_{\tilde{u}} : ℝ ⟶ [0, 1],$ which assigns a real number in the interval [0, 1] to each element $x \in ℝ$ and the value $μ_{\tilde{u}} (x)$ shows the grade of membership of x in $\tilde{u}$ .

Definition 2.2. A fuzzy number $\tilde{u}$ is a fuzzy set as $μ_{\tilde{u}} : ℝ ⟶ [0, 1]$ which satisfies:

$μ_{\tilde{u}}$ is normal, i.e., there exists an element t₀ such that $μ_{\tilde{u}} (t_{0}) = 1$ ;

$μ_{\tilde{u}}$ is quasi-concave, i.e., $μ_{\tilde{u}} (λ t_{1} + (1 - λ) t_{2}) \geq \min {μ_{\tilde{u}} (t_{1}), μ_{\tilde{u}} (t_{2})}, \forall t_{1}, t_{2} \in ℝ, \forall λ \in [0, 1]$ ;

$μ_{\tilde{u}}$ is upper semi-continuous, i.e., $\tilde{u} (r) = {x \in ℝ : μ_{\tilde{u}} (x) \geq r}$ is a closed subset for each r ∈ (0, 1];

The 0-level set of ${\tilde{u}}_{0}$ is compact subset of $ℝ$ .

Definition 2.3. A popular fuzzy number is the triangular fuzzy number that is presented by $\tilde{u} = (a_{1}, a_{2}, a_{3})$ , where, a₁ ≤ a₂ ≤ a₃ and its membership function is as follows: $μ_{\tilde{u}} (x) = {\begin{matrix} 0 & x \leq a_{1}, \\ \frac{x - a_{1}}{a_{2} - a_{1}} & a_{1} \leq x \leq a_{2}, \\ \frac{a_{3} - x}{a_{3} - a_{2}} & a_{2} \leq x \leq a_{3}, \\ 0 & x > a_{3} . \end{matrix}$ With regard to Definition 2.2 and the above points, it is concluded that the r-cut $\tilde{u} (r)$ of $\tilde{u}$ is a compact and convex subset of $ℝ$ for each r ∈ [0, 1], i.e., $\tilde{u} (r)$ is a closed interval in $ℝ$ for each r ∈ [0, 1].

The r-cut of fuzzy number $\tilde{u}$ is as follows: $\begin{matrix} \tilde{u} (r) & = & [\underline{u} (r), \bar{u} (r)] \\ = & [a_{1} + (a_{2} - a_{1}) r, a_{3} + (a_{2} - a_{3}) r] . \end{matrix}$

Using the concept of r-level sets, the relationship between ordinary sets and fuzzy sets can be characterized by the following theorem.

Theorem 2.4. [8] (Decomposition Theorem) A fuzzy set $\tilde{u}$ can be represented by $\tilde{u} = ⋃_{r \in [0, 1]} r . \tilde{u} (r)$ denoting the algebraic product of a scaler ofrwith ther-level set of $\tilde{u} (r)$ .

The arithmetical operations on fuzzy numbers, based on Zadeh’s extension principle [15] and decomposition theorem [8], are numerically performed on level sets, i.e., r-cuts. From the interval arithmetic [8], the arithmetical operations on fuzzy numbers are written for their r-level sets as follows.

Definition 2.5. For arbitrary fuzzy numbers $\tilde{u}, \tilde{v}$ , and real number k, we have:

$\tilde{u} = \tilde{v}$ if and only if $\underline{u} (r) = \underline{v} (r)$ and $\bar{u} (r) = \bar{u} (r)$ , forall r ∈ [0, 1] ,

$u (r) + v (r) = [\underline{u} (r), \bar{u} (r)] + [\underline{v} (r), \bar{v} (r)] = [\underline{u} (r) + \underline{v} (r), \bar{u} (r) + \bar{v} (r)]$ ,

$k . u (r) = {\begin{matrix} [k . \underline{u} (r), k . \bar{u} (r)] & k \geq 0, \\ [k . \bar{u} (r), k . \underline{u} (r)] & k < 0 . \end{matrix}$

Definition 2.6. According to [3], an arbitrary fuzzy complex number may be represented as $\tilde{z} = p + iq$ , where $p = [\underline{p} (r), \bar{p} (r)]$ and $q = [\underline{q} (r), \bar{q} (r)]$ , for all 0 ≤ r ≤ 1.

As such the above can be written as $\tilde{z} = [\underline{p} (r), \bar{p} (r)] + i [\underline{q} (r), \bar{q} (r)] .$

Definition 2.7. For any two arbitrary fuzzy complex numbers ${\tilde{z}}_{1} = x_{1} + {iy}_{1}$ and ${\tilde{z}}_{2} = x_{2} + {iy}_{2}$ (where, x₁, y₁, x₂ and y₂ are fuzzy numbers), the fuzzy complex arithmetic is written for addition and multiplication as

${\tilde{z}}_{1} + {\tilde{z}}_{2} = (x_{1} + x_{2}) + i (y_{1} + y_{2})$ ,

${\tilde{z}}_{1} \times {\tilde{z}}_{2} = {(x_{1} \times x_{2}) - (y_{1} \times y_{2})} + i {(x_{1} \times y_{2}) + (y_{1} \times x_{2})}$ .

2.2 Polynomials and Gröbner basis

In this subsection, the required basic concepts related to polynomials and Gröbner basis are presented. The details can be found in [6].

Definition 2.8. [6] A monomial in x₁, ⋯ , x_n is a product of the form $x_{1}^{α_{1}} {·x}_{2}^{α_{2}} \dots x_{n}^{α_{n}}$ , where all of the exponents α₁, ⋯ , α_n are non-negative integers. The total degree of this monomial is the sum α₁ + ⋯ + α_n.

We will denote this sum by |α|. The set of all monomials in x₁, ⋯ , x_n is denoted by T. We will simplify the notation for monomials by letting α = (α₁, ⋯ , α_n) be an n-tuple of non-negative integers. Then, we write $x^{α} = x_{1}^{α_{1}} {·x}_{2}^{α_{2}} \dots x_{n}^{α_{n}} .$ When α = (0, ⋯ , 0), note that x^α = 1. Throughout this paper, $𝕂$ is a field.

Definition 2.9. [6] A polynomial f in x₁, ⋯ , x_n with coefficients in a field $𝕂$ is a finite linear combination (with coefficients in $𝕂$ ) of monomials. We will write a polynomial f in the form $f = \sum_{α} a_{α} x^{α}, a_{α} \in 𝕂,$ where the sum is over a finite number of n-tuples α = (α₁, ⋯ , α_n). The set of all polynomials in x₁, ⋯ , x_n with coefficients in $𝕂$ is denoted by $Γ = 𝕂 [x_{1}, \dots, x_{n}]$ .

We use the following terminology in our discussion of polynomials:

Definition 2.10. [6] Let f = ∑_αa_αx^α be a polynomial in Γ.

We call a_α the coefficient of the monomial x^α.

If a_α ≠ 0, then we call a_αx^α a term of f.

The total degree of f, denoted deg (f), is the maximum |α| such that the coefficient a_α is nonzero.

It is clear that, under addition and multiplication, the set of polynomials Γ is a ring. We call it a polynomial ring. The polynomial ring Γ is also commutative under multiplication, and so it is a commutative ring.

Now we introduce the notion of affine space.

Definition 2.11. [6] We define the n-dimensional affine space over $𝕂$ to be the set $𝕂^{n} = {(a_{1}, \dots, a_{n}) : a_{1}, \dots, a_{n} \in 𝕂} .$

We will now consider the relationship between the polynomials and affine space. A polynomial can be regarded as a function from $𝕂^{n}$ to $𝕂$ . We can see that the polynomial f = ∑_αa_αx^α ∈ Γ gives a function as $f : 𝕂^{n} \to 𝕂$ . We now define a geometric object based on these ideas.

Definition 2.12. [6] Let f₁, ⋯ , f_s be polynomials in Γ. Then, we set $\begin{matrix} V (f_{1}, . . ., f_{s}) \\ = {(a_{1}, . . ., a_{n}) \in 𝕂^{n} : f_{i} (a_{1}, . . ., a_{n}) \\ = 0 \forall 1 \leq i \leq s} . \end{matrix}$

We call V (f₁, ⋯ , f_s) the affine variety defined by f₁, ⋯ , f_s.

With regard to Definition 2.12, an affine variety $V (f_{1}, \dots, f_{n}) \subset 𝕂^{n}$ is the set of all solutions of the system of polynomial equations f₁ (x₁, ⋯ , x_n) = ⋯ = f_s (x₁, ⋯ , x_n) =0.

Definition 2.13. [6] Let f₁,. . . , f_s be polynomials in Γ. Then, we set $〈f_{1}, \dots, f_{s}〉 = \{\sum_{i = 1}^{s} h_{i} f_{i} : h_{1}, \dots, h_{s} ∊ Γ\} .$

We call $〈f_{1}, \dots, f_{s}〉$ the ideal generated by f₁, ⋯ , f_s. We now establish an ordering ≻ on the space $ℤ_{\geq 0}^{n}$ , which will give us an ordering on monomials in the following definition.

Definition 2.14. [6] A monomial ordering on Γ is any relation ≻ on $ℤ_{\geq 0}^{n}$ satisfying:

≻ is a total ordering on $ℤ_{\geq 0}^{n}$ .

If α ≻ β and $γ \in ℤ_{\geq 0}^{n}$ , then α + γ ≻ β + γ.

≻ is a well-ordering on $ℤ_{\geq 0}^{n}$ . This means that every non-empty subset of $ℤ_{\geq 0}^{n}$ has a smallest element under ≻.

With attention to Definition 2.14, we construct a monomial $x^{α} = x_{1}^{α_{1}} x_{2}^{α_{2}} \dots x_{n}^{α_{n}}$ with the n-tuple α as the exponent. Hence, there is a one-to-one correspondence between T and $ℤ_{\geq 0}^{n}$ . Therefore, the ordering ≻ on $ℤ_{\geq 0}^{n}$ gives us an ordering on the monomials in Γ. Hence, we can present Definition 2.14 for the monomials as follows: $x^{α} ≻ x^{β} if and only if α ≻ β .$

Also, note that in $ℤ_{\geq 0}^{n}$ , addition is done component-wise. This means that, if we have (a₁, ⋯ , a_n) and $(b_{1}, \dots, b_{n}) \in ℤ_{\geq 0}^{n}$ , then (a₁, ⋯ , a_n) + (b₁, ⋯ , b_n) = (a₁ + b₁, ⋯ , a_n + b_n).

Throughout this paper, we use degree reverse lexicographical orderings which is defined as following:

Definition 2.15. [6] Let $α, β \in ℤ_{\geq 0}^{n}$ . We say that α ≻ β if |α| > |β| or |α| = |β| and in $α - β \in ℤ_{\geq 0}^{n}$ the right-most nonzero entry is negative.

For any nonzero polynomial f = ∑_αc_αx^α, the leading term of f (with respect to ≻) is the product c_αx^α, where x^α is the largest monomial appearing in f with respect to ≻. We will use the notation LT (f) for the leading term of f. Furthermore, if LT (f) = cx^α, then LC (f) = c is the leading coefficient of f and LM (f) = x^α is the leading monomial of f. Finally, the total degree of the monomial x^α is called the multidegree of f denoted by multideg (f).

For an ideal in Γ, we denote the set of leading terms of I by LT (I) and the ideal generated by the elements of LT (I) by $〈L T (I)〉$ respectively. Assume that finite set F is a generator for ideal I. In a general case, the ideals $〈L T (F)〉$ and $〈L T (I)〉$ are not equal to each other. Hence, a generator like G for ideal I where $〈L T (G)〉$ = $〈L T (I)〉$ have an special role in the resolution of polynomial systems.

Definition 2.16. A finite subset G = {g₁, ⋯ , g_s} of an ideal I is said to be Gröbner basis if $〈L T (I)〉 = 〈L T (g_{1}), \dots, L T (g_{s})〉$ . A Gröbner basis is called a reduced Gröbner basis for an ideal I if for any g_i, the coefficient of LT (g_i) is 1 and no monomial of g_i lies in $〈L T (G - \{g_{i}\})〉$ .

Now, we present the following theorem to compute the Gröbner basis.

Theorem 2.17. [6] (Division Algorithm) Fix a monomial order ≻ on $ℤ_{\geq 0}^{n}$ and letF = (f₁,. . . , f_n) be an ordereds-tuple of polynomials. Then, everyf ∈ Γ can be written asf = a₁f₁ + . . . + a_sf_s + r, wherea_i, r ∈ Γ and either r = 0 or r is linear combination, with coefficients in $𝕂$ , of monomials, none of which is divisible by anyLT (f₁) ,. . . , LT (f_s). Furthermore, ifa_if_i ≠ 0, then we havemultideg (f) ≥ multideg (a_if_i). We callra reminder offon division byF.

When f is divided by a Gröbner basis G, we denote its reminder as $r = {\bar{f}}^{G}$ .

Since the polynomial ring Γ is Noetherian, with attention to Hilbert Basis Theorem, it is easily concluded that any ideal has a Gröbner basis with respect to any monomial order. Gröbner bases are computed using various algorithms, the most famous of which is the Buchberger’s algorithm. In fact, the ideal of this algorithm is to find the characterization of Gröbner bases that involves finitely many tests.

Definition 2.18. [6] The S-polynomials of f₁, f₂ ∈ Γ is defined as $S (f_{1}, f_{2}) = \frac{x^{γ}}{LT (f_{1})} f_{1} - \frac{x^{γ}}{LT (f_{2})} f_{2},$ where x^γ = lcm (LM (f₁) , LM (f₂)) is the least common multiple of the leading monomials LM (f₁) and LM (f₂).

We can now express a sufficient and necessary condition where a generator of an ideal is a Gröbner basis.

Theorem 2.19. [6] (Buchberger Theorem) A generator {g₁, ⋯ , g_s} forIis a Gröbner basis ofIif and only if ${\bar{S (f_{i}, f_{j})}}^{G} = 0$ , for all 1 ≤ i < j ≤ n.

The above provides an algorithm to determine Gröbner bases.

Algorithm 2.20. [6] (Buchberger’s Algorithm)

Input:F a generator for I and ≺, a term order

Output:G a Gröbner basis for I

G : = F, P : = {(p, q) : p, q ∈ G, p ≠ q}.

Choose a pair of polynomial (p, q) in G and put P : = P − {(p, q)}.

$h : = {\bar{S (p, q)}}^{G}$ .

If h ≠ 0 then

add (h, g) to P for all g ∈ G.

add h to G.

If P≠ ∅ then go to 2 else return G.

We can now express the relationship between Gröbner basis and variety in terms of the following theorem.

Theorem 2.21. [6] LetIbe an ideal in $ℝ [x_{1}, \dots, x_{n}]$ andGbe a reduced Gröbner basis forIwith respect to any monomial order. IfG ={ 1 }, thenV (I) =∅.

Let G ={ g₁,. . . , g_s } be a Gröbner basis for an ideal I. By dividing f ∈ Γ by G, we can consider the remainder ${\bar{f}}^{G}$ as a standard representative of its costs $[f] \in A = \frac{Γ}{I}$ . Remainders are the linear combinations of the monomials x^α, where $x^{α} \notin 〈L T (I)〉$ in this vector space A. Since this set of monomials is linearly independent in A, it can be regarded as a basis of A. In other words, the set of monomials $B = \{x^{α} : x^{α} \notin 〈L T (I)〉\}$ , forms a basis of A (more precisely, their costs are a basis). The set B is called the set of standard monomials. We now express some equivalent conditions for the finiteness of basis B in terms of the following theorem.

Theorem 2.22. [5] (Finiteness Theorem) Let $𝕂 \subset ℂ$ be a field and $I \subset 𝕂 [x_{1}, \dots, x_{n}]$ be an ideal. Then, the following conditions are equivalent:

The vector space $\frac{𝕂 [x_{1}, \dots, x_{n}]}{I}$ is finite dimensional over $𝕂$ .

The variety $V (I) \subset ℂ^{n}$ is a finite set.

IfGis a Gröbner basis for I, then for eachi, 1 ≤ i ≤ n, there is anm_i ≥ 0 such that $LT (g) = x_{i}^{m_{i}}$ for someg ∈ G.

We recall that an ideal satisfying any of the above conditions of Finiteness Theorem is said to be a zero- dimensional ideal. The required tools have now been provided to solve a system of polynomial equations via eigenvalues.

2.3 Solving a system of polynomial equations via eigenvalue method

The main problem of this subsection is to find the solutions of a system of polynomial equations f₁ = ⋯ = f_s = 0 over $ℝ$ , i.e., the points of the variety V (I), where I is the ideal generated by f₁, ⋯ , f_s. Let I be a zero-dimensional ideal of $ℝ [x_{1}, \dots, x_{n}]$ . If f is a polynomial in $ℝ [x_{1}, \dots, x_{n}]$ and θ_f is a map on $A^{'} = \frac{ℝ [x_{1}, \dots, x_{n}]}{I}$ such that θ_f ([g]) = [f . g] then θ_f has following properties:

Proposition 2.23. [5] Let $f \in ℝ [x_{1}, . . ., x_{n}]$ , then

The mapθ_fis a linear mapping fromA′ to A′.

We haveθ_f = θ_gexactly whenf − g ∈ I. Thus, two polynomials give the same linear map if and only if they differ by an element ofI. In particular, θ_fis the zero map whenf ∈ I.

Let B be a basis for A′. Then, the matrix representation θ_f with respect to B is denoted by m_f. Proposition 2.23 implies that $θ_{f} = θ_{{\bar{f}}^{G}}$ , therefore, f can be considered as a reminder.

Theorem 2.24. [5] (Stickelberger’s Theorem) LetIbe a zero-dimensional ideal in $ℝ [x_{1}, . . ., x_{n}]$ and $f \in ℝ [x_{1}, . . ., x_{n}]$ . Then, the following statements are equivalent:

λis an eigenvalue of the matrixm_f.

λis a value of the functionfon V (I).

The following corollary is a direct result of Theorem 2.24.

Corollary 2.25.LetIbe a zero-dimensional ideal in $ℝ [x_{1}, . . ., x_{n}]$ . Then, the eigenvalues of the multiplication operatorm_{x
_i}onAcoincide withx_i- coordinates of the points ofV (I).

The above corollary leads to the following algorithm to compute real solutions of a zero-dimensional ideal I.

Algorithm 2.26. (EigMethod)

Input: A generator set for the ideal I

Output: The real solution set of I.

Compute a standard monomial set for I.

For 1 ≤ i ≤ n do

Compute multiplication matrix m_{x
_i}.

Put Θ_i : = the real eigenvalues set of m_{x
_i}.

Return({(λ₁, ⋯ , λ_n) ∈ Θ₁ × ⋯ × Θ_n : g (λ₁, ⋯ , λ_n) =0 for all g ∈ G}).

Now, we will illustrate the eigenvalue method to find the real solutions of a polynomial system by the following example.

Example 2.27. Consider the following system of equations: ${\begin{matrix} x^{2} + y^{2} + z^{2} = 6, \\ x^{3} + y^{3} + z^{3} - xyz = - 4, \\ xy + xz + yz = - 3 . \end{matrix}$

Let I be the ideal generated by ${\begin{matrix} f_{1} = x^{2} + y^{2} + z^{2} - 6, \\ f_{2} = x^{3} + y^{3} + z^{3} - xyz + 4, \\ f_{3} = xy + xz + yz + 3 . \end{matrix}$

We first compute a Gröbner basis for I with respect to the reverse graded lexicographic order in Maple by using the command $G = Basis ([f_1, f_2, f_3], tdeg (x, y, z)) .$

So, we obtain the following standard monomials: $B = {1, z, y, x, z^{2}, yz, xz, y^{2}, z^{3}, {yz}^{2}, y^{2} z, {yz}^{3}}$

By the following command: $NormalSet (G, tdeg (x, y, z)) .$

Now we obtain the matrix representation of θ_x respect to B as follows: $m_{x} = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ - 3 & 0 & 0 & 0 & 0 & - 1 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 6 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 \\ 2 & \frac{3}{2} & \frac{9}{2} & \frac{9}{2} & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \\ - 2 & - \frac{9}{2} & - \frac{9}{2} & - \frac{9}{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & - 1 & 0 \\ 2 & \frac{9}{2} & \frac{3}{2} & \frac{9}{2} & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 2 & 0 & 0 & \frac{3}{2} & \frac{9}{2} & \frac{9}{2} & 0 & 0 & 0 & 0 & - 1 \\ 0 & - 2 & 0 & 0 & - \frac{9}{2} & - \frac{9}{2} & - \frac{9}{2} & 0 & 0 & 0 & 0 & 0 \\ \frac{27}{2} & 0 & - 2 & 0 & 0 & 0 & \frac{9}{2} & - \frac{9}{2} & 0 & 0 & 0 & 0 \\ - 9 & - \frac{27}{4} & - \frac{81}{4} & - \frac{81}{4} & - 2 & 0 & 0 & 0 & - \frac{9}{2} & 0 & 0 & 0 \end{matrix}]$

This matrix can also be computed using the $“ {MultiplicationMatrix}^{″}$ command in Maple. Now, applying eigenvalue routine “Eigenvalues” in Maple, we obtain two real eigenvalues {1, − 2}. Similar to the above method, we can compute the values y and z. The eigenvalues of m_y, m_z are {1, − 2} and {1, − 2}, respectively. Finally we arrive at the following solutions: $V (I) = {(1, 1, - 2), (1, - 2, 1), (- 2, 1, 1)} .$

3 Fuzzy complex system of linear equations

The n × n fuzzy complex system of linear equations is written as ${\begin{matrix} c_{11} {\tilde{z}}_{1} + c_{12} {\tilde{z}}_{2} + . . . + c_{1 n} {\tilde{z}}_{n} = {\tilde{w}}_{1}, \\ c_{21} {\tilde{z}}_{1} + c_{22} {\tilde{z}}_{2} + . . . + c_{2 n} {\tilde{z}}_{n} = {\tilde{w}}_{2}, \\ ⋮ \\ c_{n 1} {\tilde{z}}_{1} + c_{n 2} {\tilde{z}}_{2} + . . . + c_{nn} {\tilde{z}}_{n} = {\tilde{w}}_{n} . \end{matrix}$ (1)

In matrix notation, we may then write the above as $[C] {\tilde{Z}} = {\tilde{W}}$ , where the coefficient matrix [C] = (c_kj) , 1 ≤ k ≤ n, j ≤ n is a complex n × n matrix, ${\tilde{W}} = {{\tilde{w}}_{k}}$ , 1 ≤ k is a column vector of fuzzy complex number and ${\tilde{Z}} = {{\tilde{z}}_{j}}$ is the vector of fuzzy complex unknown. System (1) can be represented as $\sum_{j = 1}^{n} c_{kj} {\tilde{z}}_{j} = {\tilde{w}}_{k}, k = 1, 2, . . ., n .$ (2)

The complex coefficient matrix, fuzzy complex unknown and the right hand fuzzy complex number vector may be written, respectively, as $c_{kj} = a_{kj} + {ib}_{kj},$ $\tilde{Z} = {\tilde{z}}_{j} = p_{j} + i q_{j} = [\underline{p_{j}} (r), \bar{p_{j}} (r)] + i [\underline{q_{j}} (r), \bar{q_{j}} (r)]$ and $\begin{matrix} \tilde{W} & = & {\tilde{w}}_{k} = u_{k} + i v_{k} \\ = & [\underline{u_{k}} (r), \bar{u_{k}} (r)] + i [\underline{v_{k}} (r), \bar{v_{k}} (r)] . \end{matrix}$

Next, the following equation is obtained by substituting $\tilde{Z}$ and $\tilde{W}$ in Equation (2)

$\begin{matrix} \sum_{j = 1}^{n} (a_{kj} + i b_{kj}) ([\underline{p_{j}} (r), \bar{p_{j}} (r)] + i [\underline{q_{j}} (r), \bar{q_{j}} (r)]) \\ = [\underline{u_{k}} (r), \bar{u_{k}} (r)] + i [\underline{v_{k}} (r), \bar{v_{k}} (r)], \\ k = 1, 2, . . ., n . \end{matrix}$ (3)

Equation (3) can now be written as

$\begin{matrix} \sum_{j = 1}^{n} a_{kj} ([\underline{p_{j}} (r), \bar{p_{j}} (r)] + i [\underline{q_{j}} (r), \bar{q_{j}} (r)]) + i \\ \sum_{j = 1}^{n} b_{kj} ([\underline{p_{j}} (r), \bar{p_{j}} (r)] + i [\underline{q_{j}} (r), \bar{q_{j}} (r)]) \\ = [\underline{u_{k}} (r), \bar{u_{k}} (r)] + i [\underline{v_{k}} (r), \bar{v_{k}} (r)], \end{matrix}$ (4) for k = 1, 2,. . . , n. Then, Equation (4) is equivalent to the following equation:

$\begin{matrix} {\sum_{j = 1}^{n} a_{kj} [\underline{p_{j}} (r), \bar{p_{j}} (r)] + \sum_{j = 1}^{n} - b_{kj} [\underline{q_{j}} (r), \bar{q_{j}} (r)]} + i \\ {\sum_{j = 1}^{n} a_{kj} [\underline{q_{j}} (r), \bar{q_{j}} (r)] + \sum_{j = 1}^{n} b_{kj} [\underline{p_{j}} (r), \bar{p_{j}} (r)]} \\ = [\underline{u_{k}} (r), \bar{u_{k}} (r)] + i [\underline{v_{k}} (r), \bar{v_{k}} (r)], \end{matrix}$ (5) for k = 1, 2, ⋯ , n. Here, a_kj and b_kj both may be positive and/or negative. To handle the positive and negative values of a_kj and b_kj, the above equation is written as below $\begin{matrix} {\sum_{a_{kj} \geq 0} a_{kj} [\underline{p_{j}} (r), \bar{p_{j}} (r)] + \sum_{b_{kj} < 0} - b_{kj} [\underline{q_{j}} (r), \bar{q_{j}} (r)] + \\ \sum_{a_{kj} < 0} a_{kj} [\bar{p_{j}} (r), \underline{p_{j}} (r)] + \sum_{b_{kj} \geq 0} - b_{kj} [\bar{q_{j}} (r), \underline{q_{j}} (r)]} + i \\ {\sum_{a_{kj} \geq 0} a_{kj} [\underline{q_{j}} (r), \bar{q_{j}} (r)] + \sum_{b_{kj} \geq 0} b_{kj} [\underline{p_{j}} (r), \bar{p_{j}} (r)] + \\ \sum_{a_{kj} < 0} a_{kj} [\bar{q_{j}} (r), \underline{q_{j}} (r)] + \sum_{b_{kj} < 0} b_{kj} [\bar{p_{j}} (r), \underline{p_{j}} (r)]} \\ = [\underline{u_{k}} (r), \bar{u_{k}} (r)] + i [\underline{v_{k}} (r), \bar{v_{k}} (r)], \end{matrix}$ (6)

for k = 1, 2,. . . , n. So, with regard to Definition 2.5, the parametric form of system (1) can be presented in the following form: ${\begin{matrix} \underline{u_{k}} (r) = \sum_{a_{kj} \geq 0} a_{kj} \underline{p_{j}} (r) + \sum_{b_{kj} < 0} - b_{kj} \underline{q_{j}} (r) + \\ \sum_{a_{kj} < 0} a_{kj} \bar{p_{j}} (r) + \sum_{b_{kj} \geq 0} - b_{kj} \bar{q_{j}} (r), \\ \bar{u_{k}} (r) = \sum_{a_{kj} \geq 0} a_{kj} \bar{p_{j}} (r) + \sum_{b_{kj} < 0} - b_{kj} \bar{q_{j}} (r) + \\ \sum_{a_{kj} < 0} a_{kj} \underline{p_{j}} (r) + \sum_{b_{kj} \geq 0} - b_{kj} \underline{q_{j}} (r), \\ \underline{v_{k}} (r) = \sum_{a_{kj} \geq 0} a_{kj} \underline{q_{j}} (r) + \sum_{b_{kj} \geq 0} b_{kj} \underline{p_{j}} (r) + \\ \sum_{a_{kj} < 0} a_{kj} \bar{q_{j}} (r) + \sum_{b_{kj} < 0} b_{kj} \bar{p_{j}} (r), \\ \bar{v_{k}} (r) = \sum_{a_{kj} \geq 0} a_{kj} \bar{q_{j}} (r) + \sum_{b_{kj} \geq 0} b_{kj} \bar{p_{j}} (r) + \\ \sum_{a_{kj} < 0} a_{kj} \underline{q_{j}} (r) + \sum_{b_{kj} < 0} b_{kj} \underline{p_{j}} (r) . \end{matrix}$ (7)

Theorem 3.1. The systems of (1) and (7) have the same solutions.

Proof. Since all of the coefficients and all of the unknowns in the polynomial functions of $f_{k} ({\tilde{z}}_{1}, \dots, {\tilde{z}}_{n}) = \sum_{j = 1}^{n} c_{kj} {\tilde{z}}_{j}$ , for k = 1, ⋯ , n, are fuzzy complex numbers, according to Definition (2.7), the values of $f_{k} ({\tilde{z}}_{1}, . . ., {\tilde{z}}_{n})$ , for k = 1, ⋯ , n, are fuzzy complex numbers. With regard to Definition (2.5), two fuzzy complex numbers of $f_{k} ({\tilde{z}}_{1}, \dots, {\tilde{z}}_{n})$ and ${\tilde{w}}_{k}$ , for k = 1, ⋯ , n, are equal when Equation (7) holds for 1 ≤ k ≤ n, j ≤ n.

4 Resolution of a fuzzy complex system of linear equations using eigenvalue method

In this section we propose a new method to solve a fuzzy complex system of linear equations. we consider the fuzzy complex system of linear equations as (1), where p_j and q_j in the fuzzy complex unknown ${\tilde{z}}_{j} = p_{j} + i q_{j}$ and u_k and v_k in the right hand fuzzy complex number vector ${\tilde{w}}_{k} = u_{k} + i v_{k}$ , 1 ≤ k ≤ n, j ≤ n are represented by triangular fuzzy numbers (p_j1, p_j2, p_j3), (q_j1, q_j2, q_j3), (u_k1, u_k2, u_k3) and (v_k1, v_k2, v_k3). Then, with regard to Definition 2.3, we have

$\begin{matrix} \underline{p_{j}} (r) = p_{j 1} + (p_{j 2} - p_{j 1}) r, \\ \bar{p_{j}} (r) = p_{j 3} + (p_{j 2} - p_{j 3}) r, \\ \underline{q_{j}} (r) = q_{j 1} + ((p_{j 2} - p_{j 1}) r, \\ \bar{q_{j}} (r) = q_{j 3} + (q_{j 2} - q_{j 3}) r \end{matrix}$ and

$\begin{matrix} \underline{u_{k}} (r) = u_{k 1} + (u_{k 2} - u_{k 1}) r, \\ \bar{u_{k}} (r) = u_{k 3} + (u_{k 2} - u_{k 3}) r, \\ \underline{v_{k}} (r) = v_{k 1} + ((v_{k 2} - v_{k 1}) r, \\ \bar{v_{k}} (r) = v_{k 3} + (v_{k 2} - v_{k 3}) r . \end{matrix}$

Now, if we substitute $\underline{p_{j}} (r)$ , $\bar{p_{j}} (r)$ , $\underline{q_{j}} (r)$ , $\bar{q_{j}} (r),$ $\underline{u_{k}} (r)$ , $\bar{u_{k}} (r)$ , $\underline{v_{k}} (r)$ and $\bar{v_{k}} (r)$ in (7), with attention to Definition 2.5, system (1), may be converted into a crisp system which is a polynomial system with 8n equations and 6n unknowns. Now, we compute the Gröbner basis G for the ideal generated by the crisp system in the ring $\begin{matrix} R & = & ℝ [p_{11}, p_{12}, p_{13}, p_{21}, p_{22}, p_{32}, . . . p_{n 1}, \\ p_{n 2}, p_{n 3}, q_{11}, q_{12}, q_{13}, q_{21}, q_{22}, q_{32}, \\ . . . q_{n 1}, q_{n 2}, q_{n 3}], \end{matrix}$ with respect to the reverse graded lexicographic order. Then, by using eigenvalue method all of the solutions of system (1) can be obtained.

Theorem 4.1. The proposed approach determines the existence, or lack of existence, of a solution for system (1).

Proof. According to Theorem 3.1, both systems (1) and (7) have the same solutions. Therefore, with regard to Theorem (2.21), the sufficient and necessary condition for lack of existence of a solution is G = {1}.

Now, we express the resolution processes of a fuzzy complex system of linear equations using the eigenvalue method in terms of the followingalgorithm.

Algorithm 4.2. (Main Algorithm)

Input: The fuzzy complex system of linear equations (1)

Output: The set of solutions, i.e., S for system (1)

Step 1.Compute the parametric form of system (1).

Step 2.Compute the crisp form of system (1) from the parametric system.

Step 3.Compute the Gröbner basis G for the ideal generated by the crisp form in R with respect to the reverse graded lexicographic order..

Step 4.S : = EigMethod (G).

5 Applications and numerical examples

In this section, we study some examples of the fuzzy complex systems of linear equations and solve them with the proposed approach in this paper. Then, their results are compared with other methods. The applied fuzzy numbers in the following examples are the triangular fuzzy numbers with the linear membership function (see Definition 2.3 inSubsection 2.1).

Example 5.1. [17] Let us consider 2 × 2 complex fuzzy linear system as ${\begin{matrix} {\tilde{z}}_{1} - {\tilde{z}}_{2} = [r, 2 - r] + i [1 + r, 3 - r], \\ {\tilde{z}}_{1} + 3 {\tilde{z}}_{2} = [4 + r, 7 - 2 r] + i [r - 4, - 1 - 2 r] . \end{matrix}$

We solve this system by our algorithm. Let ${\tilde{z}}_{1} = p_{1} + {iq}_{1} = (p_{11}, p_{12}, p_{13}) + i (q_{11}, q_{12}, q_{13})$ and ${\tilde{z}}_{2} = p_{2} + {iq}_{2} = (p_{21}, p_{22}, p_{23}) + i (q_{21}, q_{22}, q_{33})$ . Now, the above system may be converted into the following equivalent system that is called parametric system: ${\begin{matrix} ([p_{11} + (p_{12} - p_{11}) r, p_{13} + (p_{12} - p_{13}) r] \\ + i [q_{11} + (q_{12} - q_{11}) r, q_{13} + (q_{12} - q_{13}) r]) \\ - ([p_{21} + (p_{22} - p_{21}) r, p_{23} + (p_{22} - p_{23}) r] \\ + i [q_{21} + (q_{22} - q_{21}) r, q_{23} + (q_{22} - q_{23}) r]) \\ = [r, 2 - r] + i [1 + r, 3 - r], \\ ([p_{11} + (p_{12} - p_{11}) r, p_{13} + (p_{12} - p_{13}) r] + \\ i [q_{11} + (q_{12} - q_{11}) r, q_{13} + (q_{12} - q_{13}) r]) + 3 \\ ([p_{21} + (p_{22} - p_{21}) r, p_{23} + (p_{22} - p_{23}) r] + \\ i [q_{21} + (q_{22} - q_{21}) r, q_{23} + (q_{22} - q_{23}) r]) \\ = [4 + r, 7 - 2 r] + i [r - 4, - 1 - 2 r] . \end{matrix}$

The above system can be rewritten as follows: ${\begin{matrix} ([p_{11} + (p_{12} - p_{11}) r, p_{13} + (p_{12} - p_{13}) r] \\ + i [q_{11} + (q_{12} - q_{11}) r, q_{13} + (q_{12} - q_{13}) r]) \\ + ([- p_{23} + (- p_{22} + p_{23}) r, - p_{21} \\ + (- p_{22} + p_{21}) r] + i [- q_{23} + (- q_{22} + q_{23}) r, \\ - q_{21} + (- q_{22} + q_{21}) r]) \\ = [r, 2 - r] + i [1 + r, 3 - r], \\ ([p_{11} + (p_{12} - p_{11}) r, p_{13} + (p_{12} - p_{13}) r] \\ + i [q_{11} + (q_{12} - q_{11}) r, q_{13} + (q_{12} - q_{13}) r]) \\ ([3 p_{21} + (3 p_{22} - 3 p_{21}) r, 3 p_{23} + (3 p_{22} - 3 p_{23}) r] \\ + i [3 q_{21} + (3 q_{22} - 3 q_{21}) r, 3 q_{23} + (3 q_{22} - 3 q_{23}) r]) \\ = [4 + r, 7 - 2 r] + i [r - 4, - 1 - 2 r] . \end{matrix}$

The above system can be rewritten as follows: ${\begin{matrix} ([p_{11} - p_{23} + (p_{12} - p_{11} - p_{22} + p_{23}) r, \\ p_{13} - p_{21} + (p_{12} - p_{13} - p_{22} + p_{21}) r] \\ + i [q_{11} - q_{23} + (q_{12} - q_{11} - q_{22} + q_{23}) r, \\ q_{13} - q_{21} + (q_{12} - q_{13} - q_{22} + q_{21}) r]) + \\ = [r, 2 - r] + i [1 + r, 3 - r], \\ ([p_{11} + 3 p_{21} + (p_{12} - p_{11} + 3 p_{22} - 3 p_{21}) r, \\ p_{13} + 3 p_{23} + (p_{12} - p_{13} + 3 p_{22} - 3 p_{23}) r] \\ + i [q_{11} + 3 q_{21} + (q_{12} - q_{11} + 3 q_{22} - 3 q_{21}) r, \\ q_{13} + 3 q_{23} + (q_{12} - q_{13} + 3 q_{22} - 3 q_{23}) r]) \\ = [4 + r, 7 - 2 r] + i [r - 4, - 1 - 2 r] . \end{matrix}$

The equivalent crisp polynomial equation system is obtained from the above system as follows: ${\begin{matrix} p_{11} - p_{23} = 0, \\ p_{12} - p_{11} - p_{22} + p_{23} = 1, \\ p_{13} - p_{21} = 2, \\ p_{12} - p_{13} - p_{22} + p_{21} = - 1, \\ q_{11} - q_{23} = 1, \\ q_{12} - q_{11} - q_{22} + q_{23} = 1, \\ q_{13} - q_{21} = 3, \\ q_{12} - q_{13} - q_{22} + q_{21} = - 1, \\ p_{11} + 3 p_{21} = 4, \\ p_{12} - p_{11} + 3 p_{22} - 3 p_{21} = 1, \\ p_{13} + 3 p_{23} = 7, \\ p_{12} - p_{13} + 3 p_{22} - 3 p_{23} = - 2, \\ q_{11} + 3 q_{21} = - 4, \\ q_{12} - q_{11} + 3 q_{22} - 3 q_{21} = 1, \\ q_{13} + 3 q_{23} = - 1, \\ q_{12} - q_{13} + 3 q_{22} - 3 q_{23} = - 2 . \end{matrix}$

Then, a Gröbner basis for the ideal generated by the above system with respect to the lexicographic order is as: G = {p₁₁ − p₂₃, p₁₂ − p₁₁ + p₂₃ − p₂₂ − 1, p₁₃ − p₂₁ − 2, p₁₂ − p₁₃ + p₂₁ − p₂₂ + 1, q₁₁ − q₂₃ − 1, q₁₂ − q₁₁ + q₂₃ − q₂₂ − 1, q₁₃ − q₂₁ − 3, q₁₂ − q₁₃ + q₂₁ − q₂₂ + 1, p₁₁ + 3p₂₁ − 4, p₁₂ − p₁₁ + 3p₂₂ − 3p₂₁ − 1, p₁₃ + 3p₂₃ − 7, p₁₂ − p₁₃ + 3p₂₂ − 3p₂₃ + 2, q₁₁ + 3q₂₁ + 4, q₁₂ − q₁₁ + 3q₂₂ − 3q₂₁ − 1, q₁₃ + 3q₂₃ + 1, q₁₂ − q₁₃ + 3q₂₂ − 3q₂₃ + 2}. We obtain the following standard monomial: B = {1}. Using the monomial basis B, the eigenvalues of the matrices of the full multiplication operator m_{p
₁₁},m_{p
₁₂}, m_{p
₁₃}, m_{q
₁₁}, m_{q
₁₂}, m_{q
₁₃}, m_{p
₂₁}, m_{p
₂₂}, m_{p
₂₃}, m_{q
₂₁}, m_{q
₂₂} and m_{q
₂₃} can be obtained as $\frac{11}{8}$ , 2, $\frac{23}{8}$ , $\frac{1}{8}$ , $\frac{3}{4}$ , $\frac{13}{8}$ , $\frac{7}{8}$ , 1, $\frac{11}{8}$ , $- \frac{11}{8}$ , $- \frac{5}{4}$ and $- \frac{7}{8}$ , respectively. Finally, the solution of the system is as $\begin{matrix} {\tilde{z}}_{1} = p_{1} + {iq}_{1} \\ = (1.375, 2, 2.875) + i (0.125, 0.75, 1.625) \\ = [1.375 + 0.625 r, 2.875 - 0.875 r] + i \\ [0.125 + 0.625 r, 1.625 - 0.875 r] \end{matrix}$ and $\begin{matrix} {\tilde{z}}_{2} = p_{2} + {iq}_{2} \\ = (0.875, 1, 1.375) + i (- 1.375, - 1.25, 0.875) \\ = [0.875 + 0.125 r, 1.375 - 0.375 r] + i \\ [- 1.375 + 0.125 r, - 0.875 - 0.375 r] . \end{matrix}$

Finally, we should mention that the proposed method and the presented method in [4] find the same solution. But, the presented method in [17] finds the solution ${\tilde{z}}_{1} = [1.375 + 0.625 r, 2.875 - 0.875 r] + i [0.125 + 0.625 r, 1.625 - 0.875 r]$ and ${\tilde{z}}_{2} = [0.875 + 0.125 r, 1.375 - 0.375 r] + i [1.375 + 0.125 r, 0.875 - 0.375 r]$ .

Example 5.2. A linear time invariant electric circuit with complex coefficient and fuzzy complex sources may be modeled as fuzzy complex system of linear equations (Equation (1)). Here, (c_kj) is complex coefficient matrix, ${{\tilde{w}}_{k}}$ is a fuzzy complex source and ${{\tilde{z}}_{j}}$ may be current or voltage in the system.

An example problem of a simple RLC circuit [20] with fuzzy current and fuzzy source as shown in Fig. 1 is considered. Corresponding fuzzy complex system of linear equations for this circuit problem may be represented as ${\begin{matrix} (10 - i 7.5) {\tilde{z}}_{1} - (6 - i 5) {\tilde{z}}_{2} \\ = [4 + r, 6 - r] + i [- 1 + r, 1 - r], \\ - (6 - i 5) {\tilde{z}}_{1} + (16 + i 3) {\tilde{z}}_{2} \\ = [- 2 + r, - r] + i [- 3 + r, - 1 - r] . \end{matrix}$

Let ${\tilde{z}}_{1} = p_{1} + {iq}_{1} = (p_{11}, p_{12}, p_{13}) + i (q_{11}, q_{12}, q_{13})$ and ${\tilde{z}}_{2} = p_{2} + {iq}_{2} = (p_{21}, p_{22}, p_{23}) + i (q_{21}, q_{22}, q_{33})$ . If we apply the proposed method in this paper, the parametric form of the original system is as follows: ${\begin{matrix} (10 - i 7.5) ([p_{11} + (p_{12} - p_{11}) r, p_{13} \\ + (p_{12} - p_{13}) r] + i [q_{11} + (q_{12} - q_{11}) r, q_{13} \\ + (q_{12} - q_{13}) r]) - (6 - i 5) ([p_{21} + (p_{22} - p_{21}) r, \\ p_{23} + (p_{22} - p_{23}) r] + i [q_{21} + (q_{22} - q_{21}) r, \\ q_{23} + (q_{22} - q_{23}) r]) \\ = [4 + r, 6 - r] + i [- 1 + r, 1 - r], \\ - (6 - i 5) ([p_{11} + (p_{12} - p_{11}) r, p_{13} \\ + (p_{12} - p_{13}) r] + i [q_{11} + (q_{12} - q_{11}) r, \\ q_{13} + (q_{12} - q_{13}) r]) + (16 + i 3) ([p_{21} \\ + (p_{22} - p_{21}) r, p_{23} + (p_{22} - p_{23}) r] + \\ i [q_{21} + (q_{22} - q_{21}) r, q_{23} + (q_{22} - q_{23}) r]) \\ = [4 + r, 6 - r] + i [- 1 + r, 1 - r] . \end{matrix}$

As mentioned in Section 3, the crisp polynomial system can be written as ${\begin{matrix} 10 p_{11} + 7.5 q_{11} - 6 p_{23} - 5 q_{23} = 4, \\ 10 p_{12} - 10 p_{11} + 7.5 q_{12} - 7.5 q_{11} \\ + 6 p_{23} - 6 p_{22} + 5 q_{23} - 5 q_{22} = 1, \\ 10 p_{13} + 7.5 q_{13} - 6 p_{21} - 5 q_{21} = 6, \\ 10 p_{12} - 10 p_{13} + 7.5 q_{12} - 7.5 q_{13} \\ + 6 p_{21} - 6 p_{22} + 5 q_{21} - 5 q_{22} = - 1, \\ 10 q_{11} - 7.5 p_{13} - 6 q_{23} + 5 p_{21} = - 1, \\ 10 q_{12} - 10 q_{11} + 7.5 p_{13} - 7.5 p_{12} \\ + 6 q_{23} - 6 q_{22} + 5 p_{22} - 5 p_{21} = 1, \\ 10 q_{13} - 7.5 p_{11} - 6 q_{21} + 5 p_{23} = 1, \\ 10 q_{12} - 10 q_{13} + 7.5 p_{11} - 7.5 p_{12} \\ + 6 q_{21} - 6 q_{22} + 5 p_{22} - 5 p_{23} = - 1, \\ - 6 p_{13} - 5 q_{13} + 16 p_{21} - 3 q_{23} = - 2, \\ 6 p_{13} - 6 p_{12} + 5 q_{13} - 5 q_{12} + 16 p_{22} \\ - 16 p_{21} + 3 q_{23} - 3 q_{22} = 1, \\ - 6 p_{11} - 5 q_{11} + 16 p_{23} - 3 q_{21} = 0, \\ 6 p_{11} - 6 p_{12} + 5 q_{11} - 5 q_{12} + 16 p_{22} \\ - 16 p_{23} + 3 q_{21} - 3 q_{22} = - 1, \\ - 6 q_{13} + 5 p_{11} + 16 q_{21} + 3 p_{21} = - 3, \\ 6 q_{13} - 6 q_{12} + 5 p_{12} - 5 p_{11} + 16 q_{22} \\ - 16 q_{21} + 3 p_{22} - 3 p_{21} = 1, \\ - 6 q_{11} + 5 p_{13} + 16 q_{23} + 3 p_{23} = - 1, \\ 6 q_{11} - 6 q_{12} + 5 p_{12} - 5 p_{13} + 16 q_{22} \\ - 16 q_{23} + 3 p_{22} - 3 p_{23} = - 1 . \end{matrix}$

Then, a Gröbner basis for the ideal generated by the above system with respect to the lexicographic order is as: $\begin{matrix} G & = & {51288327 q_{21} + 12204112, 121249 q_{22} \\ + 25125, 51288327 q_{23} + 9051638, \\ 51288327 p_{21} - 1778153, 121249 p_{22} \\ - 7930, 51288327 p_{23} - 4930627, 51288327 \\ q_{11} - 3630080, 121249 q_{12} - 13168, \\ 51288327 q_{13} - 7510048, 51288327 \\ p_{11} - 16225328, 121249 p_{12} - 42944, \\ 51288327 p_{13} - 20105296} . \end{matrix}$

We obtain the following standard monomial: B = {1}. Using the monomial basis B, the eigenvalues of the matrices of the full multiplication operator m_{p
₁₁},m_{p
₁₂}, m_{p
₁₃}, m_{q
₁₁}, m_{q
₁₂}, m_{q
₁₃}, m_{p
₂₁}, m_{p
₂₂}, m_{p
₂₃}, m_{q
₂₁}, m_{q
₂₂} and m_{q
₂₃} can be obtained as $\frac{16225328}{51288327}$ , $\frac{42944}{121249}$ , $\frac{20105296}{51288327}$ , $\frac{3630080}{51288327}$ , $\frac{13168}{121249}$ , $\frac{7510048}{51288327}$ , $\frac{1778153}{51288327}$ , $\frac{7930}{121249}$ , $\frac{4930627}{51288327}$ , $- \frac{12204112}{51288327}$ , $- \frac{25125}{121249}$ and $- \frac{9051638}{51288327}$ , respectively. Finally, the solution of the system is as $\begin{matrix} {\tilde{z}}_{1} = p_{1} + {iq}_{1} = (0.3164, 0.3542, 0.3920) + i \\ (0.0708, 0.1086, 0.1464) \\ = [0.3164 + 0.0378 r, 0.3920 - 0.0378 r] + i \\ [0.0708 + 0.0378 r, 0.1464 - 0.0378 r] \end{matrix}$ and $\begin{matrix} {\tilde{z}}_{2} = p_{2} + {iq}_{2} = (0.0347, 0.0654, 0.0961) + i \\ (- 0.2379, - 0.2072, - 0.1765,) \\ = [0.0347 + 0.0307 r, 0.0961 - 0.0307 r] + i \\ [- 0.2379 + 0.0307 r, - 0.1765 - 0.0307 r] . \end{matrix}$

It follows that the proposed method and the presented methods in [4, 20] find the same solution.

6 Conclusion

In this work, an innovative approach to find solution of a fuzzy complex system of linear equations is presented. Then, a criterion for existence of solution of the system was discussed. Also, an algorithm was designed to find the solution based on the eigenvalue method. Example problems are solved by the present method and are compared with known solutions to show efficiency and effectiveness of the methodology. The proposed method has the following advantages:

The proposed approach finds all the solutions of the fuzzy complex system of linear equations.

The proposed approach determines the existence, or lack of existence, of a solution for the system.

In this approach, we do not require a suitable initial point.

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