Stability Analysis of Matrices and Single-Parameter Matrix Families on

Special Regions

ŞERİFE YILMAZ

Department of Mathematics Education

Burdur Mehmet Akif Ersoy University

Istiklal Campus 15030Burdur

TURKEY

Abstract: This paper proposes an efficient method for determining the D-stability of matrices using Kelley’s

cutting-plane method. The study examines the D-stability of matrices in symmetric regions of the complex plane,

defined by quadratic matrix inequalities (QMI) and polynomial functions. Using Kelley’s cutting-plane method,

we present an iterative algorithm for determining the D-stability of a given matrix. Further, the robustness of

single-parameter matrix families is analyzed, and a method is proposed for considering their D-stability. Through

examples, we indicate the possible applicability of the proposed approach in addressing stability problems in

linear systems.

Key-Words: D-Stability, Single-Parameter Matrix Families, Robust Stability, QMI region, Linear Matrix

Inequality.

Received: April 19, 2024. Revised: September 13, 2024. Accepted: October 5, 2024. Published: October 25, 2024.

1 Introduction

In this study, we have developed an algorithm

based on Kelley’s cutting plane method to determine

the D-stability of a matrix. This algorithm

aims to present a different approach to stability

analysis techniques. Our method solves a matrix

inequality using simple iterative operations without

requiring direct eigenvalue calculations and the

cases of eigenvalue inclusion in set D. In recent

studies, various methods have been proposed to

address stability problems in linear systems using

advanced optimization techniques and Linear Matrix

Inequalities (LMIs), [1], [2], [3], [4], [5], [6].

The techniques discussed can also be used to solve

robust stability problems for a family of matrices.

This study focuses on the stability region defined

by second-order matrix inequalities (QMIs) and

polynomial functions. Our method addresses the

problem of determining the stability of a matrix

within these regions and the robust stability of a

matrix segment. We tested our method on matrices

and tried to illustrate the effectiveness of the proposed

algorithm using examples.

Determining whether a matrix A∈Rn×nis

D-stable is a critical problem in control theory, where

Dis a symmetric region of the complex plane C.

A matrix A∈Rn×nis said to be D-stable if all

its eigenvalues lie within a specified region D ⊂ C,

[7]. For instance, a matrix is called Hurwitz stable if

all its eigenvalues are in the open left half-plane, i.e.,

D={z∈C:Re(z)<0}. Hurwitz stability of

a matrix Aprovides that the continuous-time linear

system described by ˙x=Ax is asymptotically stable.

Similarly, a matrix is called Schur stable if all its

eigenvalues lie inside the open unit disk, i.e., D=

{z∈C:|z|<1}. This stability guarantees that

the discrete-time linear system described by xk+1 =

Axkis asymptotically stable. Likewise, a matrix is

called sector stable if all its eigenvalues lie within a

specified sector in the complex plane. Particularly,

for an angle θwith 0≤θ≤π, the sector is defined

as D={z∈C:π−θ < arg(z)< π +θ}. When the

arguments of the eigenvalues of the system matrix are

restricted, it gives rise to the sector stability problem

of that matrix.

In [8], the stability region

D={z∈C:a+b(z+ ¯z) + cz¯z < 0}

is considered, where a,b, and care real numbers,

b≥0,c≥0. The authors propose that the problem

can be reduced to the positivity of two specially

constructed determinants in the continuous case. In

contrast, in the discrete-time case, three determinants

are required.

Results on the D-stability of matrices for regions

defined by linear matrix inequalities (LMIs) are

provided in the study, [3]. Specifically, a region D

of the complex plane is defined by

D=z∈C:Q+Sz +ST¯z+Rz¯z < 0(1)

for some symmetric matrix Q∈Rm×m, a general

matrix S∈Rm×m, and a symmetric positive

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semi-definite R≥0∈Rm×m. This is called a

quadratic matrix inequality region (QMI) of order m.

QMI regions possess crucial characteristics such

as being open, convex, and symmetrical concerning

the real axis. Moreover, since the intersection of

any QMI region results in another QMI region, it

is possible to approximate practically any convex

region in the complex plane using a QMI region.

The following theorem ascertains a criterion for the

D-stability of a matrix A∈Rn×n.

Theorem 1 ([3]).A matrix A∈Rn×nis D-stable (1)

if and only if there exists a symmetric positive definite

matrix P > 0of dimension n×nsuch that

Q⊗P+S⊗(AP )+ST⊗(P AT)+R⊗(AP AT)<0.

(2)

Here, the symbol ‘⊗’ stands for the Kronecker

product of matrices.

Another region of stability that draws attention

is the region defined by polynomials with real

coefficients in complex variables. In [9], the stability

of a matrix for such defined regions has been

characterized. In [9], the region

D:= {z∈C|Refi(z)<0, i = 1, . . . , N}(3)

is considered, where the coefficients of the

polynomial fi(z)are all real. This region is open

and symmetrical about the real axis. The following

theorem suggests a necessary and sufficient condition

for a common positive definite solution to a set of

Lyapunov inequalities for the eigenvalues of a matrix

to lie in D.

Theorem 2 ([9]).A matrix A∈Rn×nis D-stable

(3) if and only if there exists a matrix P=PT>0

satisfying

{fi(A)}TP+P fi(A)<0, i = 1, . . . , N. (4)

Consider the switched system

˙x(t) = Ax(t), A ∈ {A1, A2, . . . , AN}(5)

where x(t)∈Rnand t≥0. In (5), the

matrix Aswitches among NHurwitz stable matrices

A1, A2, . . . , AN.

A key issue is to determine the existence of a

quadratic Lyapunov function V(x) = xTP x, where

P=PT>0, such that:

iP+P Ai<0for all i∈ {1,2, . . . , N }.(6)

This function V(x), known as a common quadratic

Lyapunov function (CQLF), guarantees the uniform

asymptotic stability of the switched system.

In [10], the common P > 0solution of Lyapunov

inequalities given by equation (6) is investigated by

Kelley’s cutting-plane method.

In our research, we discuss the solution of the

inequality system (4) in Theorem 2 using the method

presented in [10]. We will use this method to

determine the D-stability of a matrix A, as given by

equation (3). The D-stability of matrix Afor the QMI

region defined by equation (1) is equivalent to the

LMI problem (2) formulated for this matrix. In our

study, we propose an algorithm to solve this problem.

2 Kelley’s Cutting-Plane Method and

LMI’s

Kelley’s cutting-plane method is an iterative

algorithm for convex optimization problems. The

method works by iteratively refining a feasible region,

using linear hyperplanes to exclude regions that do

not contain the optimal solution. At each iteration,

a subgradient of the objective function is computed,

and the algorithm checks whether the current

solution encounters the stopping criterion. If not, a

new hyperplane is added, and the process continues

until convergence. This method is particularly useful

for solving linear matrix inequalities (LMIs) in

D-stability problems.

Let x∈Rrbe xT= (x1, x2, . . . , xr)and Pbe an

n×nsymmetric matrix defined as

P=P(x) = 





x1x2··· xn

x2xn+1 ··· x2n−1

xnx2n−1··· xr







,(7)

with r=n(n+ 1)/2.

For the inequalities involving the matrix P

linearly, as in (2) or (4), let us define the symmetric

matrices

M(x) := Q⊗P+S⊗(AP ) + ST⊗(P AT)+

R⊗(AP AT),(8)

and

L(x, i) := {fi(A)}TP+P fi(A),

i= 1,2, . . . , N. (9)

The function λmax(A)for a symmetric matrix A

represents the largest eigenvalue of the matrix A.

Define

ϕM(x) = λmax(M(x))

=max

∥u∥=1

uTM(x)u. (10)

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and

ϕL(x) = max

1≤i≤Nλmax(L(x, i))

=max

1≤i≤N,∥u∥=1

uTL(x, i)u. (11)

We consider two problems:

1) Is there P > 0such that M(x)<0(referred to as

(2))? If there exists a point ˜xsuch that P(˜x)>

0and ϕM(˜x)<0, then the matrix P(˜x)is the

solution we are pursuing.

2) Is there a common matrix P > 0such that

L(x, i)<0for all i= 1,2, . . . , m (referred to as

(4))? If there exists a point ˆxsuch that P(ˆx)>0

and ϕL(ˆx)<0, then P(ˆx)satisfies the necessary

conditions.

These problems can be reduced to minimizing a

convex function under convex constraints.

Consider the following convex minimization

problem:

ϕ(x)→minimize

min

∥v∥=1

vTP(x)v > 0.(12)

Assume X⊂Rnis a convex set and F:X→R

is a convex function. We say that a vector g∈Rn

is a subgradient of F(x)at x∗∈Xif the following

inequality holds for every x∈X:

F(x)≥F(x∗) + gT(x−x∗).

The collection of all subgradients of F(x)at x∗is

represented by ∂F (x∗). If x∗is an interior point of X,

then the set ∂F (x∗)is guaranteed to be nonempty and

convex. The subsequent proposition follows from the

theory of nondifferentiable optimization.

Proposition 3 ([10]).Define ϕ(x)as follows:

ϕ(x) = max

y∈Yf(x, y)(13)

where Yis a compact set, and f(x, y)is continuous

and differentiable with respect to x. Then,

∂ϕ(x) = conv ∂f(x, y)

∂x :y∈Y(x)

where Y(x)is the set of all maximizing elements yin

(13), that is,

Y(x) = {y∈Y:f(x, y) = ϕ(x)}.

For a given x, if the the maximizing element

is unique, i.e. Y(x) = {y(x)}then ϕ(x)is

differentiable at xand its gradient is

∇ϕ(x) = ∂f (x, y(x))

∂x

For the function ϕM(x)as defined in (10):

∂ϕM(x) = conv ∂

∂x uTM(x)u:uis a unit

eigenvector corresponding to λmax(M(x))}.(14)

Similarly, for the function ϕL(x)as defined in (11):

∂ϕL(x) = conv ∂

∂x uTL(x, i)u:imaximizes

λmax(L(x, i)), u is a corresponding unit

eigenvector}.(15)

If the maximizing value of ifor the given xis unique

and λmax(L(x, i)) is a simple eigenvalue, then the

differentiability of ϕLat the point xis guaranteed,

[11].

We examine problem (12) using Kelley’s

cutting-plane method

Kelley’s method reformulates problem (12) as

follows:

cTz→min

c1(z)≥0,

c2(z)≥0,

−1≤xi≤1 (i= 1,2, . . . , r)

(16)

where z= (x1, x2, . . . , xr, L)T,c= (0, . . . , 0,1)T,

c1(z) = L−ϕ(x), and c2(z) = min

∥v∥=1

vTP v.

Let z0be a starting point and z0, z1, . . . , zkbe k+

1distinct points. (Here zkdenotes the point at the

k-th iteration.)

At the (k+ 1)th iteration, the cutting-plane

algorithm solves the following LP problem

minimize L

subject to −hT

1(z0)z≥ −hT

1(z0)z0−c1(z0)

−hT

2(z0)z≥ −hT

2(z0)z0−c2(z0)

−hT

1(zk)z≥ −hT

1(zk)zk−c1(zk)

−hT

2(zk)z≥ −hT

2(zk)zk−c2(zk)

−1≤xi≤1

(17)

where hj(zi)denotes a subgradient of −cj(z)at zi

(j= 1,2).

Let zk

∗be the minimizer of the problem (17).

If zk

∗satisfies the inequality

min{c1(zk

∗), c2(zk

∗)} ≥ −ε,

where εis a tolerance, then zk

∗is considered an

approximate solution to the problem in (16).

If this condition is not satisfied, we define j∗as

the index of the most negative cj(zk

∗). Update the

constraints in (17) by adding the linear constraint:

cj∗(zk+1)−hT

j∗(zk+1)(z−zk+1)≥0,(18)

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then repeat the process.

Remember, our objective is to find x∗such that

P(x∗)>0and ϕ(x∗)<0, rather than solving the

minimization problem (12), (16).

Theorem 4 ([10]).If there exists a ksuch that:

c1(zk

∗)> Lkand c2(zk

∗)>0,

where zk

∗= (xk

∗, Lk)is the minimizer of the problem

(17), then the matrix P=P(xk

∗)is a solution to (12).

This algorithm can be adapted to our problem as

follows:

Algorithm 1.

(1) Select an initial point z0= (x0, L0)T.

Compute ϕ(x0)and c2(z0).

If ϕ(x0)<0and c2(z0)>0, terminate;

Otherwise, continue to the next step.

(2) Find zk

∗by solving the LP problem in (17).

If c1(zk

∗)> Lkand c2(zk

∗)>0, terminate;

Otherwise, proceed.

Set zk+1 =zk

∗, update the constraints in (17) and

repeat the procedure.

We will present applications of Kelley’s method to

D-stability problems.

3 Examples and Applications

We demonstrate the D-stability of a matrix Ausing

Algorithm 1, where the set Dis defined as a QMI

region given by equation (1) and a region defined

by equation (3), respectively. In the proposed

method, we calculate the subgradient of the convex

function by using the maximal eigenvalue and its

corresponding eigenvector. We then update the set of

constraints (17) for the linear programming problem.

This approach requires fewer operations than other

methods, reducing computational complexity.

Example 1. Let us consider the set Ddefined by

equation (2) with the following matrices (see, [3]):

Q=





−1000

0 1 0 0

0 0 0 0





, R =





1000

0000







S=





0 0 0 0

0 1 0 0

0 0 1/√2 1/√2

0 0 −1/√2 1/√2





,

This D ⊂ Cregion is as shown in Fig. 1. Given the

0.5

1.0

−0.5

−1.0

0.5 1.0−0.5−1.0

Fig1: Region Din Example 1.

matrix Aas:

A=−2.369 5.297 5.225

−1.351 2.481 3.445

0.684 −1.148 −2.112.

If we take the initial point as

z0= (x0

1, x0

2, x0

3, x0

4, x0

5, x0

6, L0)T

= (1,0,0,1,0,1,1)T,

then

P(x0) = 100

010

001.

Applying Algorithm 1, in the first step, we

find c1(z0) = −84.342,c2(z0) = 1, and

ϕM(x0) = λmax(M(x0)) = 85.342 >0.

Solving the LP problem (17) yields z1=

(−1,1,1,−1,−1,0,−191.062)T. Calculations

give the following Table 1.

Table 1Iterative Results of Kelley’s Method for

Example 1

kLkc1(zk)c2(zk)

1 -19.772 -220.596 -2.732

2 -2.693 -21.071 -2.125

3 -1.353 -10.277 -0.361

32 -0.000826 -0.001011 0.000587

33 -0.000826 -0.000802 0.000526

z33 = (x33, L33)T

= (1,0.322,0.018,1,−0.849,0.818,−0.000826)T.

Since ϕM(x33) = L33 −c1(z33) = −0.000024 <0

and λmax(P(x0)) = c2(z33)=0.000526 >0, the

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inequality (2) in Theorem 1 is satisfied for the matrix

P(x33) = 1 0.322 0.018

0.322 1 −0.849

0.018 −0.849 0.818 ,

which means M(x33)<0. Therefore, all the

eigenvalues of the matrix Aare within the Ddefined

here, indicating that the matrix Ais D-stable.

Example 2 ([9]).Let f1(z) = z,f2(z) = −z2and

f3(z) = −z3. Define

D1={z∈C:Ref1(z)<0,Re f2(z)<0},

D2={z∈C:Ref1(z)<0,Re f3(z)<0}.

Consider stability region D=D1∩ D2in Fig. 2.

−1

−2

−3

1−1−2

−1

−2

−3

1−1−2

−1

−2

−3

1−1−2

D=D1∩ D2

Fig2: Regions: D1,D2and D.

In [9], the D-stability of

A=





−94.7−47.1−41.1−2.3

15.2−46.9 3.0−7.6

121.0 77.9 46.3 9.1

−116.9 65.2−54.6−4.7







is considered. It has been solved with an algorithm

based on the Schur decomposition method. Here we

solve the D-stability of this matrix using the Kelley’s

method. The polynomials

f1(z) = z, f2(z) = −z2, f3(z) = −z3

define the boundaries of the Dregion, ensuring that

the eigenvalues of the matrix Alie within this sector.

According to Theorem 2 in [9], to ensure the D

stability of the matrix A, we need to find a symmetric

positive definite matrix Psuch that the following

matrix inequalities are satisfied:

f1(A)TP+P f1(A)<0,

f2(A)TP+P f2(A)<0,

f3(A)TP+P f3(A)<0,

(19)

that is,

ATP+P A < 0

(−A2)TP+P(−A2)<0,

(−A3)TP+P(−A3)<0.

These inequalities ensure that the matrix Ais D-stable

with a common positive definite matrix P.

Using the Kelley’s method, we iterate to find a

common positive definite matrix Pthat satisfies the

stability conditions for the given matrices. Starting

z0= (1,0,0,0,1,0,0,1,0,1,1), that is P(x0)is

a identity matrix, after 111 iterations, we find the

solution (see Table 2). Calculations give

Table 2Iterative Results of Kelley’s Method for

Example 2

kLkc1(zk)c2(zk)

0−1.53 ×106−3.603 ×1051

1−3.94 ×105−2.087 ×106-3.402

2−2.34 ×105−5.218 ×105-2

3−31924.888 −2.586 ×105-0.523

110 -11.732012 -18.400783 0.006

111 -11.732012 -11.301216 0.007

z111 = (x111, L111)T

= (1,−0.305,0.563,−0.081,1,−0.091,

0.240,0.343,−0.009,0.099,−11.732012)T.

Since ϕL(x111) = L111 −c1(z111) = −0.430796 <

0and c2(z111)=0.007 >0, the positive definite

matrix

P(x111) = 





1−0.305 0.563 −0.081

−0.305 1 −0.091 0.240

0.563 −0.091 0.343 −0.009

−0.081 0.240 −0.009 0.099







is a common solution to (19) and L(x111, i)<0for

i= 1,2,3.

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Using Kelley’s method, we obtained a common

positive definite matrix Pthat solves the matrix

inequalities derived from A. As a result, matrix A

is D-stable. The results shown in Fig. 2 demonstrate

the robustness of our solution by verifying that the

eigenvalues of the matrix are within the defined

stability region D.

The examples above show how effective Kelley’s

method is in solving stability problems for matrices

within complex Dregions. Our proposed algorithm

also has an advantage over other optimization

methods in determining the common positive solution

(if it exists) for the matrix inequalities. Kelley’s

cutting-plane method found a solution matrix that

confirmed the stability of the given matrices within

the defined Dregion. This was affirmed through

multiple iterations, where the eigenvalues of the

matrix remained within the defined stability region.

4 The D-stability of One-parameter

Matrix Families

Let A1, A2∈Rnbe D-stable matrices. The set of

their convex combinations is defined as

A={A(α) = (1−α)A1+αA2:α∈[0,1]}.(20)

If A(α)is D-stable for every α∈[0,1], then the

matrix family Ais called robustly D-stable.

In this section, we will handle the problem of the

robust stability of the one-parameter matrix family A.

Theorem 5. The family of matrices Ain equation

(20) is robustly D-stable if and only if there exists a

symmetric positive definite matrix P(α)>0for each

α∈[0,1], such that

Q⊗P(α)+S⊗(A(α)P(α))+ST⊗(P(α)A(α)T)+

R⊗(A(α)P(α)A(α)T)<0.(21)

Determining the D-stability of the matrix family

A(20) involves finding a matrix P(α)>0that

satisfies the matrix inequality in the Theorem 5 for

every α∈[0,1], which can be a challenging task.

Despite producing conservative results, we present

the following theorem because of its applicability and

relevance to our problem.

Theorem 6. Let Abe a given matrix family. If there

exists a symmetric positive definite matrix P > 0of

dimension n×nsuch that for each α∈[0,1],

Q⊗P+S⊗(A(α)P)+

ST⊗(P A(α)T)+

R⊗(A(α)P A(α)T)<0,(22)

then the matrix family Ais robustly D-stable.

In the Theorem 6, the requirement for the

existence of a positive definite matrix P > 0, which

satisfies the matrix inequality (22) for every α∈

[0,1], introduces a level of conservatism. However,

this theorem can still be applied to solve the robust

D-stability problem of a one-parameter matrix family

Let

M(x, α) := Q⊗P+S⊗(A(α)P)+

ST⊗(P A(α)T) + R⊗(A(α)P A(α)T).

Define

ϕM(x) = max

0≤α≤1λmax(M(x, α))

=max

0≤α≤1,∥u∥=1

uTM(x, α)u. (23)

If there exists a point ˜xsuch that P(˜x)>0and

ϕM(˜x)<0, then Ais robustly D-stable.

To overcome the computational challenge of

calculating

max

0≤α≤1λmax(M(x, α))

for α∈[0,1], we divide the interval [0,1] into kparts

and let

αi=i

k(i= 0,1,2, . . . , k)

then

ϕM(x)≈FM(x) := max

0≤i≤k,∥u∥=1

uTM(x, αi)u.

For this value of k, if there exists a point ˜xsuch that

P(˜x)>0and FM(˜x)<0, and if the inequality

ϕM(˜x)<0is satisfied, then we have obtained the

solution to our problem. If the inequality is not

satisfied, we proceed with a larger value of kand

continue this procedure.

Example 3. Consider the set Dgiven in Example 1.

Let

A1=−2.369 5.297 5.225

−1.351 2.481 3.445

0.684 −1.148 −2.112

and

A2=−0.83 0.02 −0.01

−0.07 −0.78 −0.01

0.06 −0.01 −0.82

be D-stable matrices (Fig. 3).

Starting from the initial point

z0= (x0

1, x0

2, x0

3, x0

4, x0

5, x0

6, L0)T

= (1,0,0,1,0,1,1)T,

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E-ISSN: 2224-2880

736

Volume 23, 2024

0.5

1.0

−0.5

−1.0

0.5 1.0−0.5−1.0

Fig3: The eigenvalues of matrix A1are shown

with circles, and the eigenvalues of matrix A2are

shown with points.

and applying Algorithm 1, in the first step we obtain

FM(x0) = max

0≤i≤k,∥u∥=1

uTM(x, αi)u= 85.342 >0.

When the LP problem (17) is solved, the point

z1= (−1,1,1,−1,−1,0,−191.062)Tis obtained.

Continuing this process, after 55 iterations we find

z55 = (x55

1, x55

2, x55

3, x55

4, x55

5, x55

6, L55)T

= (1,−0.0072,0.3432,0.2489,−0.2329,

0.3331,−0.0003)T.

For this point, P(x55)>0and FM(x55) =

−0.000025 <0.

By examining the signs of the principal minors of

M(x55, α)for each α∈[0,1], it is concluded that

M(x55, α)<0. Therefore, according to Theorem

5.2 the matrix family A={(1 −α)A1+αA2:α∈

[0,1]}is robustly D-stable (Fig. 4).

5 Conclusion

This study inspected the stability of a matrix within

symmetric regions of the complex plane defined by

quadratic matrix inequalities (QMI) and polynomial

functions using Kelley’s cutting-plane method. We

proposed an algorithm to efficiently determine the

D-stability of matrices, with examples demonstrating

its effectiveness.

We also addressed the robust D-stability of

one-parameter matrix families, providing theoretical

results and computational techniques. To address the

inherent conservatism in the D-stability analysis, we

0.5

1.0

−0.5

−1.0

0.5 1.0−0.5−1.0

Fig4: The set of eigenvalues of the matrices in the

set A.

plan to consider in future work the matrix P(α)as

P(α) = (1 −α)P1+αP2, where P1and P2are

the matrices to be determined. This approach has

the potential to reduce the conservatism associated

with stability conditions and produce more accurate

results.

References:

[1] S. Gutman and E. Jury, A general theory

for matrix root-clustering in subregions of the

complex plane, IEEE Transactions on Automatic

Control, V.26, N.4, 1981, pp. 853–863.

[2] M. Chilali and P. Gahinet, H∞design with pole

placement constraints: an LMI approach, IEEE

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[3] J. Zrida, M.H. Bouazizi, Exact robust

D-stability analysis for linear dynamical

systems with polynomial parameter perturbation,

International Journal of Control, V.95, N.11,

2022, pp.2885–2899.

[4] J. Pätzold, A. Schöbel, Approximate cutting

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[5] A.M. Souza, R.C.L.F. Oliveira, P.L.D. Peres,

Computation of stable convex combinations of a

set of matrices through LMIs with application in

H∞control of switched linear systems, Systems

&Control Letters, V.189, 2024.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.75

Şeri

fe Yilmaz

E-ISSN: 2224-2880

737

Volume 23, 2024

[6] B. Sereni, M.A.L. Beteto, E. Assunçõ, M.C.M.

Teixeira, Pole placement LMI constraints for

stability and transient performance of LPV

systems with incomplete state measurement,

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[7] B.R. Barmish, New Tools for Robustness of

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[8] T. Büyükköroğlu, G. Çelebi, V. Dzhafarov, On

the robust stability of polynomial matrix families,

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[9] T. Mori, H. Kokame, A parameter-dependent

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[11] R.A. Horn and C.R. Johnson, Matrix Analysis,

Cambridge University Press, Cambridge, 1985.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The entire research, including the formulation of the

problem, development of the theoretical framework,

implementation of the algorithms, and writing of

the manuscript, was conducted by Şerife Yılmaz.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

The author has no conflicts of interest to

declare that are relevant to the content of this

article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.75

Şeri

fe Yilmaz

E-ISSN: 2224-2880

738

Volume 23, 2024