Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
-
Zhengge Huang
and
Jingjing Cui
Abstract
In this article, we present two new algorithms referred to as the improved modified gradient-based iterative (IMGI) algorithm and its relaxed version (IMRGI) for solving the complex conjugate and transpose (CCT) Sylvester matrix equations, which often arise from control theory, system theory, and so forth. Compared with the gradient-based iterative (GI) (A.-G. Wu, L.-L. Lv, and G.-R. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Appl. Math. Comput. 217 (2011), 8343–8353) and the relaxed GI (RGI) (W.-L. Wang, C.-Q. Song, and S.-P. Ji, Iterative solution to a class of complex matrix equations and its application in time-varying linear system, J. Appl. Math. Comput. 67 (2021), 317–341) algorithms, the proposed ones can make full use of the latest information and need less computations, which leads to higher computational efficiency. With the real representation of a complex matrix as a tool, we establish sufficient and necessary conditions for the convergence of the IMGI and the IMRGI algorithms. Finally, some numerical examples are given to illustrate the effectiveness and advantages of the proposed algorithms.
1 Introduction
Sylvester matrix equations are often encountered in many scientific and engineering fields, such as signal processes, stability theory, observes design, control theory, matrix computation, pole assignment, optimal control, prediction and stability, system theory, eigenstructure assignment, and so on, we can refer to the related references [1–7] for details. This indicates that studying the computational methods of different kinds of Sylvester matrix equations has become an important subject in the field of computational mathematics and control. This motivates us to establish some effective algorithms for some types of Sylvester matrix equations in this work.
Due to the fact that the Sylvester matrix equations widely appear in practical problems and have important applications in many fields, in the past few decades, many researchers have devoted themselves to deriving a great deal of different methods to solve the Sylvester matrix equations, such as the Krylov subspace methods, neural network methods, direct methods, and so forth. The Krylov subspace methods are one of the classical iterative methods for the matrix equations, including the conjugate gradient method, the conjugate direction method, and so on. Besides, unlike numerical methods, artificial neural networks have the property of high-speed parallel operations and potential hardware implementations. Many effective neural network methods have been developed for matrix equation problems, for example, recurrent neural network method, zeroing neural network method. Also, by making use of matrix straightening operator and Kronecker product, Sylvester matrix equations can be transformed into linear systems, and the exact solutions of them can be calculated by applying the direct methods. However, it may lead to high-scale problems, and will consume much more computer time and memory space with the dimensions increasing. Hence, the research of iterative methods for the matrix equations have attracted considerable attention from many researchers.
A large number of efficient iterative methods have been developed for solving different kinds of Sylvester matrix equations. For instance, Ding and Chen [8] proposed the gradient-based iterative (GI) algorithm for solving the Sylvester matrix equation
Note that the CCT Sylvester matrix equations are quite general and include several kinds of Sylvester matrix equations, such as the Sylvester matrix equations, the complex conjugate Sylvester matrix equations, the Sylvester transpose matrix equations, and the conjugate transpose Sylvester matrix equations. This shows that the CCT Sylvester matrix equations appear in many practical problems and applications, and the solving problem of the CCT Sylvester matrix equations is worth research. For example, according to [21], the continuous zeroing dynamics design of time-varying linear system can be defined as
where
with
where
Until now, some different algorithms for the CCT Sylvester matrix equations have been proposed. Apart from the ones in [1,14,26] mentioned earlier, Zhang and Yin [30] offered a new proof of the GI algorithm for the CCT Sylvester matrix equations by making use of the properties of the real representation of a complex matrix, and deduced the necessary and sufficient conditions for the convergence, optimal parameter, and corresponding optimal convergence factor. Then the optimal GI (OGI) algorithm is obtained. Furthermore, on the basis of the relaxation technique used in [15,17], Wang et al. [21] introduced a relaxed factor into the GI algorithm and developed the relaxed GI (RGI) algorithm, and necessary and sufficient conditions for guaranteeing the convergence of the RGI algorithm are provided. Numerical results in [21] revealed that the RGI algorithm performs better than the GI and the OGI ones.
To improve the efficiency of the GI algorithm for the Sylvester matrix equations, many improved and modified versions of the GI algorithm have been established. For instance, Wang et al. [20] proposed the modified GI (MGI) algorithm for solving the Sylvester matrix equations, which uses the latest information to compute the next result. This idea has been extended to the generalized Sylvester-transpose matrix equations and the extended Sylvester-conjugate matrix equations in [19] and [18], respectively. In addition, to reduce the computations and the storage spaces of the GI algorithm, Tian et al. [23] and Hu and Wang [31] derived the Jacobi GI (JGI) algorithm for solving the Sylvester matrix equations by replacing the coefficient matrices by their diagonal parts.
It can be seen from the iterative scheme of the GI algorithm in [14,30] that it does not use the latest calculation result to compute the next result. Then the convergence speed of the GI algorithm is slow in many cases, and even the optimal parameter is adopted in GI algorithm. In addition, when the coefficient matrices of the CCT Sylvester matrix equation are large and dense, the matrix multiplication in the GI algorithm may require a large amount of computation and storage spaces, which consumes much time and reduces the computational efficiency of the GI algorithm. To overcome the aforementioned shortcomings and improve the convergence rate of the GI algorithm, we first use the hierarchical identification principle to turn CCT Sylvester matrix equation into four subsystems. Enlightened by the ideas of the JGI and the MGI algorithms, we apply the updated technique to the GI algorithm, replace the related coefficient matrices by their diagonal parts, and then construct the improved modified GI (IMGI) algorithm for the CCT Sylvester matrix equation (2). Moreover, we apply the relaxation technique to the IMGI algorithm and develop the improved modified relaxed GI (IMRGI) algorithm. We investigate some convergence properties of the IMGI and the IMRGI algorithms, which indicate that the proposed algorithms are convergent under proper restrictions. Numerical experiments are provided to demonstrate the effectiveness and superiorities of the IMGI and the IMRGI algorithms.
The rest of this article is organized as follows. In Section 2, some notations, definitions, and results, which will be used in the latter parts of this article are listed. In Section 3, we put forward two new algorithms referred to as the IMGI and the IMRGI algorithms for the CCT Sylvester matrix equation (2), and establish the convergence theorems of the two proposed algorithms. Several numerical experiments are provided to validate the effectiveness and advantages of the proposed algorithms in Section 4. Finally, some conclusions are drawn in Section 5 to end this article.
2 Preliminaries
In this section, we review and recall some notations, definitions, and lemmas, which come from References [11,21,30,32,33] and will be used throughout this article.
For a complex number
In addition, we present several useful definitions below.
Definition 2.1
[32] Let
Definition 2.2
[21] The vec-permutation matrix
where
Definition 2.3
[32] For two matrices
Definition 2.4
[32] Let
In the following, we list several useful lemmas.
Lemma 2.1
[13] (The properties of the real representation)
For
If
If
For
with
Lemma 2.3
[33] Let
Lemma 2.4
[5] Let
Lemma 2.5
[34] Let T be a normal matrix, then
3 The IMGI and the IMRGI algorithms
In this section, we will establish two new algorithms for solving the CCT Sylvester matrix equations. The proposed methods outperform the GI one in [14] and its relaxed version (RGI) [21] from the point of view of computational efficiency, which will be confirmed by the numerical results in Section 5. To this end, we first introduce a lemma as follows.
Lemma 3.1
[9] Consider the matrix equation
with
If this matrix equation has a unique solution
Let
The GI algorithm [14]:
Step 1: Given matrices
Step 2: If
Step 3: Update the sequences
Step 4: Set
Numerical experiments in [14] showed the effectiveness of the GI algorithm for the CCT Sylvester matrix equations.
According to the framework of the aforementioned GI algorithm, it can be seen that Step 3 of this algorithm may consume much time because the matrices
According to the hierarchical identification principle, we first transform the CCT Sylvester matrix equation (2) into the following four equations:
where
which yields that
Besides, Wang et al. [20] developed the modified GI (MGI) algorithm for the Sylvester equations. The information generated in the first half-iterative step is fully exploited and used to compute the result of the second half-iterative step, which leads to faster convergence rate. This motivates us to apply the updated technique in [20] to (7) and construct the following IMGI algorithm based on Lemma 3.1.
The IMGI algorithm:
Step 1: Given matrices
Step 2: If
Step 3: Update the sequences
Step 4: Set
Remark 3.1
Comparing the IMGI algorithm with the GI algorithm, it can be seen that the differences between the two algorithms are the Step 3 of them. The related coefficient matrices in the GI algorithm are replaced by their diagonal parts, and the next result is computed by the latest information. This leads to less computations of each iteration and faster convergence. So we can expect that the proposed IMGI algorithm performs better than the GI one, which will be verified by numerical results.
In the sequel, we will establish the convergence condition of the IMGI algorithm. Before that, we start with a lemma as follows.
Lemma 3.2
[21] The CCT Sylvester matrix equation (2) has a unique solution if and only if the matrix
where
And the corresponding homogeneous matrix equation
In the following, by applying the properties of the real representation of a complex matrix and the vector stretching operator, we study the necessary and sufficient condition for the convergence of the IMGI algorithm.
Theorem 3.1
Suppose that the CCT Sylvester matrix equation (2) has a unique solution
with
Proof
Define the following error matrices:
It follows from Line 1 of Step 3 of the IMGI algorithm that
which together with the real representation and Lemma 2.1 results in
Using straightening operator on both sides of relation (12) yields that
where
From Line 3 of Step 3 of the IMGI algorithm, we have
Taking the real representation of the both sides of (14) and using Lemma 2.1 result in
Applying Kronecker product and vec operator in (15) leads to
with
According to Line 5 of Step 3 of the IMGI algorithm, it has
By making use of the real representation of the both sides of (17) and Lemma 2.1, we derive
By utilizing Kronecker product and vec operator in (18), we deduce that
with
In the same manner applied in (18), we can deduce that
by Line 7 of Step 3 of the IMGI algorithm. Using the real representation in (20) and Lemma 2.1 yields that
By utilizing Kronecker product and vec operator in (21), it holds that
with
Combining (13) with Line 8 of Step 3 of IMGI algorithm leads to
Besides, according to (16), (23), and Line 2 of Step 3 of IMGI algorithm, we obtain
It follows from (19), (23), (24), and Line 4 of Step 3 of the IMGI algorithm that
Finally, combining (22)–(25) with Line 6 of Step 3 of IMGI algorithm leads to
According to (23)–(26) and by some computations, we derive
where the matrices
Note that the order of the matrix
Corollary 3.1
Suppose that the conditions of
Theorem 3.1
are satisfied and let
with
Although the convergence conditions of the IMGI algorithm are derived in Theorem 3.1 and Corollary 3.1, they do not determine any interval of the step size factor
Theorem 3.2
Let the conditions of
Theorem 3.1
be satisfied, and S and T defined in
Theorem 3.1
be normal matrices. Denote by
Proof
Inasmuch as
This implies that
which is equivalent to
Solving the aforementioned inequalities yields that
which gives the convergence condition (28) of the IMGI algorithm.□
In the sequel, we establish the different convergence condition of the IMGI algorithm below.
Theorem 3.3
Assume that the CCT Sylvester matrix equation (2) has a unique solution
then the matrix sequence
Proof
We first define the error matrices
and
Thus,
Then, it follows from the iteration scheme of the IMGI algorithm and (32) that
By taking the Frobenious norm of each equation in (33) and using the properties of the norm, it holds that
and
In view of (34)–(37) and Cauchy inequality, it has
By the analogous methods in [19,20,24], we can obtain
Thus, if the parameter
then (38) implies that
Having in mind that the CCT Sylvester matrix equation (2) has a unique solution, then it follows from (41) and Lemma 3.2 that
In [17], Niu et al. introduced a relaxation parameter into the GI algorithm and established the RGI algorithm. And the numerical results of [17] illustrated that the RGI algorithm outperforms the GI one with proper relaxation parameter. Then it motivates many researchers to derive the relaxed versions of some existing algorithms for several kinds of matrix equations, see [15,16,19] for details. Recently, Wang et al. [21] constructed a RGI algorithm for the CCT Sylvester matrix equation (2). The framework of the RGI algorithm is as follows.
The RGI algorithm [21]:
Step 1: Given matrices
Step 2: If
Step 3: Update the sequences
Step 4: Set
Numerical results of [21] validated that the RGI algorithm with proper
The IMRGI algorithm:
Step 1: Given matrices
Step 2: If
Step 3: Update the sequences
Step 4: Set
In what follows, we establish the convergence theorem of the IMRGI algorithm.
Theorem 3.4
Assume that the CCT Sylvester matrix equation (2) has a unique solution
with
Proof
The proof of this theorem is similar to that of Theorem 3.1, and hence, we omit it here.□
Theorem 3.4 gives the convergence condition of the IMRGI algorithm, but it does not propose the intervals of the parameters
Theorem 3.5
Let the conditions of
Theorem 3.4
be satisfied, and U and V defined in
Theorem 3.4
be normal matrices. Denote by
Besides, if all the eigenvalues of the matrices
Proof
Inasmuch as
in view of
which can be equivalently transformed into the following inequalities
By solving the above inequalities, it has
Then we obtain the convergence condition (43) of the IMRGI algorithm in terms of (45).
Besides, if all the eigenvalues of the matrices
where
To minimize
Remark 3.2
Theorem 3.5 gives the convergence conditions for the parameters
In addition, we deduce another convergence theorem of the IMRGI algorithm as follows.
Theorem 3.6
Suppose that the CCT Sylvester matrix equation (2) has a unique solution
or the parameters
then the matrix sequence
Proof
Again using the notations in (30)–(31), we have
Then it follows from the IMRGI algorithm and (49) that
Taking the Frobenius norm in (50) yields that
and
By making use of Cauchy-Schwarz inequality and (51)–(54), we derive
According to the methods applied in [19,20,24], it follows that
Hence if
then (55) implies that
Inasmuch as the CCT Sylvester matrix equation (2) has a unique solution, it follows from (58) that
or
which results in
Inequality (60) holds if
4 Numerical experiments
This section gives several numerical examples to validate the effectiveness of the IMGI and the IMRGI algorithms, and compare their numerical performances with those of the GI, OGI, RGI, and MGI ones, in terms of the number of iteration steps (IT) and the elapsed time in seconds (CPU).
All the computations were performed in MATLAB (version R2016b) on a personal computer with Intel (R) Pentium (R) CPU G3240T 2.870 GHz, 16.0 GB memory and Windows 10 system. For all tested algorithms, the initial matrices are taken to be
Example 4.1
[21] We consider the CCT Sylvester matrix equation (2) with the following coefficient matrices:
The unique solution of this matrix equation is
For this example, all runs are terminated once, and the number of iterations
In Table 1, we compare the proposed IMGI and IMRGI algorithms with GI, OGI, RGI, and MGI ones with respect to IT and CPU times. Here, the parameters of the GI, OGI, and RGI algorithms are taken as in [21]. And the parameters of the MGI, IMGI, and IMRGI algorithms are as follows:
According to Table 1, we see that all tested methods are convergent within the maximum number of iterations for all cases, and the IMGI and the IMRGI algorithms outperform the other ones with respect to IT and CPU times. The growth rates of the IT for the IMGI and the IMRGI algorithms are slower than the other ones, which reveals that the proposed algorithms are more stable than the other tested ones. Besides, the IMRGI algorithm performs better than the IMGI algorithm with proper relaxation factor
Numerical results of the tested iterative algorithms for Example 4.1
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 |
|---|---|---|---|---|---|---|
| IT | 250 | 819 | 1389 | 1959 | 2529 | |
| GI | CPU | 0.0265 | 0.0957 | 0.1378 | 0.2177 | 0.2299 |
| ERR |
|
|
|
|
|
|
| IT | 196 | 634 | 1,075 | 1,516 | 1,957 | |
| OGI | CPU | 0.0190 | 0.0604 | 0.1023 | 0.1319 | 0.1698 |
| ERR |
|
|
|
|
|
|
| IT | 94 | 238 | 403 | 569 | 735 | |
| RGI | CPU | 0.0157 | 0.0261 | 0.0405 | 0.0584 | 0.0714 |
| ERR |
|
|
|
|
|
|
| IT | 59 | 149 | 251 | 354 | 457 | |
| MGI | CPU | 0.0099 | 0.0166 | 0.0275 | 0.0449 | 0.0493 |
| ERR |
|
|
|
|
|
|
| IT | 19 | 42 | 70 | 98 | 127 | |
| IMGI | CPU | 0.0041 | 0.0083 | 0.0097 | 0.0133 | 0.0142 |
|
|
ERR |
|
|
|
|
|
| IT | 17 | 38 | 63 | 91 | 116 | |
| IMRGI | CPU | 0.0019 | 0.0045 | 0.0073 | 0.0089 | 0.0121 |
|
|
ERR |
|
|
|
|
|
With the parameters in Table 1, the iterative sequences
The iterative solutions for IMGI (
| Algorithm |
|
|
|
|
|
|---|---|---|---|---|---|
| 30 |
|
|
|
|
|
| 60 |
|
|
|
|
|
| IMGI | 90 |
|
|
|
|
| 120 |
|
|
|
|
|
| 150 |
|
|
|
|
|
| 30 |
|
|
|
|
|
| 60 |
|
|
|
|
|
| IMRGI | 90 |
|
|
|
|
| 120 |
|
|
|
|
|
| 150 |
|
|
|
|
|
| Solution |
|
|
|
|
To compare the tested algorithms fairly, we test the algorithms with the same values of
Numerical results of the tested iterative algorithms for Example 4.1 with three different values of
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | |
|---|---|---|---|---|---|---|---|
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMRGI | IT | 52 | 113 | 196 | 284 | 372 | |
|
|
CPU | 0.0064 | 0.0134 | 0.0224 | 0.0322 | 0.0463 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 178 | 454 | 769 | 1086 | 1403 | |
|
|
CPU | 0.0203 | 0.0501 | 0.0849 | 0.1224 | 0.1543 | |
| MGI | IT | 29 | 73 | 122 | 171 | 220 | |
| CPU | 0.0035 | 0.0089 | 0.0145 | 0.0203 | 0.0257 | ||
| IMGI | IT | 29 | 63 | 107 | 153 | 199 | |
| CPU |
|
0.0039 | 0.0129 | 0.0189 | 0.0226 | ||
| IMRGI | IT | 108 | 231 | 403 | 586 | 770 | |
|
|
CPU | 0.0124 | 0.0279 | 0.0459 | 0.0671 | 0.0868 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 118 | 300 | 507 | 716 | 924 | |
|
|
CPU | 0.0134 | 0.0336 | 0.0558 | 0.0806 | 0.1239 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | 20 | 45 | 74 | 105 | 135 | |
| CPU | 0.0025 | 0.0057 | 0.0090 | 0.0131 | 0.0152 | ||
| IMRGI | IT | 72 | 154 | 268 | 389 | 510 | |
|
|
CPU | 0.0091 | 0.0178 | 0.0324 | 0.0442 | 0.0584 |
The graphs of ERR(log10) against the IT of the tested algorithms for four different values of
The logarithmic of the ERR of the tested algorithms for Example 4.1 with
The logarithmic of the ERR of the tested algorithms for Example 4.1 with
Example 4.2
[30] We consider the CCT Sylvester matrix equation (2) with the following parametric matrices
The unique solution of this matrix equation is
In this example, we take
as the termination condition with a positive number
Numerical results for different values of
Numerical results of the tested iterative algorithms for Example 4.2
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 |
|---|---|---|---|---|---|---|
| IT | Fail | Fail | Fail | Fail | Fail | |
| GI | CPU | Fail | Fail | Fail | Fail | Fail |
|
|
RES | — | — | — | — | — |
| IT | 4,363 | 9,450 | 14,537 | 19,624 | Fail | |
| OGI | CPU | 0.4973 | 1.2514 | 1.7253 | 2.2924 | Fail |
|
|
RES |
|
|
|
|
— |
| IT | 4,363 | 9,450 | 14,537 | 19,624 | Fail | |
| RGI | CPU | 0.5793 | 1.1722 | 1.6961 | 2.2689 | Fail |
|
|
RES |
|
|
|
|
— |
| IT | 5 | 91 | 198 | 306 | 413 | |
| MGI | CPU | 0.0103 | 0.0137 | 0.0236 | 0.0343 | 0.0500 |
|
|
RES |
|
|
|
|
|
| IT | Fail | Fail | Fail | Fail | Fail | |
| IMGI | CPU | Fail | Fail | Fail | Fail | Fail |
|
|
RES | — | — | — | — | — |
| IT | 5 | 100 | 227 | 355 | 482 | |
| IMGI | CPU |
|
0.0115 | 0.0256 | 0.0373 | 0.0508 |
|
|
RES |
|
|
|
|
|
| IT | 14 | 77 | 165 | 254 | 343 | |
| IMRGI | CPU | 0.0039 | 0.0096 | 0.0185 | 0.0283 | 0.0377 |
|
|
RES |
|
|
|
|
|
Table 5 lists the same items as those in Table 2. The results in Table 5 indicate that the iterative solutions generated by the IMGI and the IMRGI algorithms are gradually tended to the exact solution
The iterative solutions for IMGI (
| Algorithm |
|
|
|
|
|
|---|---|---|---|---|---|
| IMGI | 100 |
|
|
|
|
| 200 |
|
|
|
|
|
| 300 |
|
|
|
|
|
| 400 |
|
|
|
|
|
| 500 |
|
|
|
|
|
| IMRGI | 100 |
|
|
|
|
| 200 |
|
|
|
|
|
| 300 |
|
|
|
|
|
| 400 |
|
|
|
|
|
| 500 |
|
|
|
|
|
| Solution | 3 |
|
|
|
Furthermore, to further show the advantages of the IMGI and the IMRGI algorithms over the other tested ones, the performances of the tested algorithms with three different values of
Numerical results of the tested iterative algorithms for Example 4.2 with three different values of
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | |
|---|---|---|---|---|---|---|---|
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 89 | 230 | 481 | 741 | 1,005 | |
|
|
CPU | 0.0118 | 0.0299 | 0.0612 | 0.0968 | 0.1284 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMRGI | IT | 4 | 189 | 439 | 689 | 938 | |
|
|
CPU |
|
0.0251 | 0.0587 | 0.0922 | 0.1005 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 15 | 356 | 769 | 1,186 | 1,608 | |
|
|
CPU | 0.0021 | 0.0450 | 0.0994 | 0.1139 | 0.2084 | |
| MGI | IT | 51 | 803 | 1998 | 3161 | 4356 | |
| CPU | 0.0071 | 0.1022 | 0.2652 | 0.3000 | 0.5491 | ||
| IMGI | IT | 5 | 105 | 239 | 373 | 507 | |
| CPU |
|
0.0137 | 0.0321 | 0.0388 | 0.0645 | ||
| IMRGI | IT | 7 | 394 | 923 | 1455 | 1986 | |
|
|
CPU | 0.0013 | 0.0549 | 0.1221 | 0.1413 | 0.2534 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | 14 | 105 | 306 | 397 | 605 | |
| CPU | 0.0019 | 0.0107 | 0.0386 | 0.0506 | 0.0765 | ||
| IMRGI | IT | 5 | 238 | 554 | 871 | 1188 | |
|
|
CPU |
|
0.0245 | 0.0706 | 0.1114 | 0.1514 |
Figures 3 and 4 display the convergence curves of the GI, OGI, RGI, MGI, IMGI, and IMRGI algorithms for four different values of
The logarithmic of the RES of the tested algorithms for Example 4.2 with
The logarithmic of the RES of the tested algorithms for Example 4.2 with
Example 4.3
[30] Consider the CCT Sylvester matrix equation (2) with
In this example, we adopt the termination criterion as in Example 4.1, i.e.,
The numerical results of the tested algorithms for Example 4.3 are shown in Table 7. It follows from Table 7 that the GI algorithm has the slowest convergence speed, and the IMGI and the IMRGI algorithms have advantages over the other ones except for the case of
Numerical results of the tested iterative algorithms for Example 4.3
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 |
|---|---|---|---|---|---|---|
| IT | 27 | 138 | 489 | 841 | 1194 | |
| GI | CPU | 0.0390 | 0.2016 | 0.7073 | 1.3760 | 1.6753 |
|
|
ERR |
|
|
|
|
|
| IT | 24 | 115 | 205 | 296 | 387 | |
| OGI | CPU | 0.0304 | 0.1610 | 0.3625 | 0.4663 | 0.6565 |
|
|
ERR |
|
|
|
|
|
| IT | 24 | 115 | 205 | 296 | 387 | |
| RGI | CPU | 0.0805 | 0.2041 | 0.3431 | 0.4895 | 0.8332 |
|
|
ERR |
|
|
|
|
|
| IT | 11 | 77 | 189 | 306 | 425 | |
| MGI | CPU | 0.0188 | 0.1115 | 0.2632 | 0.4301 | 0.6852 |
|
|
ERR |
|
|
|
|
|
| IT | 47 | 99 | 132 | 194 | 221 | |
| IMGI | CPU | 0.0052 | 0.0116 | 0.0141 | 0.0194 | 0.0954 |
|
|
ERR |
|
|
|
|
|
| IT | 43 | 90 | 130 | 176 | 204 | |
| IMRGI | CPU | 0.0054 | 0.0091 | 0.0125 | 0.0163 | 0.0178 |
|
|
ERR |
|
|
|
|
|
In Table 8, we list the same items as those in Tables 2 and 5. And we can obtain the similar conclusions from Table 8 as in Tables 2 and 5.
The iterative solutions for IMGI (
| Method |
|
|
|
|
|
|---|---|---|---|---|---|
| IMGI | 50 |
|
|
|
|
| 100 |
|
|
|
|
|
| 150 |
|
|
|
|
|
| 200 |
|
|
|
|
|
| 250 |
|
|
|
|
|
| IMRGI | 50 |
|
|
|
|
| 100 |
|
|
|
|
|
| 150 |
|
|
|
|
|
| 200 |
|
|
|
|
|
| 250 |
|
|
|
|
|
| Solution |
|
|
1 | 3i |
In addition, in Table 9, we list the same items as those in Tables 3 and 6. As observed in Table 9, besides the RGI and the IMRGI algorithms, the other tested ones are not convergent for
Numerical results of the tested iterative algorithms for Example 4.3 with three different values of
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | |
|---|---|---|---|---|---|---|---|
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 17 | 86 | 304 | 522 | 740 | |
|
|
CPU | 0.0068 | 0.0084 | 0.0217 | 0.0474 | 0.0733 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMRGI | IT | 68 | 144 | 205 | 282 | 322 | |
|
|
CPU | 0.0122 | 0.0111 | 0.0161 | 0.0223 | 0.0240 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 27 | 139 | 492 | 846 | 1,200 | |
|
|
CPU | 0.0028 | 0.0113 | 0.0453 | 0.0621 | 0.1513 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | 30 | 62 | 89 | 120 | 145 | |
| CPU | 0.0031 | 0.0073 | 0.0099 | 0.0155 | 0.0347 | ||
| IMRGI | IT | 108 | 231 | 328 | 451 | 519 | |
|
|
CPU | 0.0111 | 0.0229 | 0.0327 | 0.0464 | 0.0697 | |
|
|
GI | IT | 9 | 39 | 134 | 229 | 324 |
| CPU | 0.0010 | 0.0036 | 0.0114 | 0.0185 | 0.0246 | ||
| OGI | IT | 9 | 39 | 134 | 229 | 324 | |
| CPU |
|
0.0029 | 0.0103 | 0.0183 | 0.0249 | ||
| RGI | IT | 29 | 150 | 534 | 919 | 1,305 | |
|
|
CPU | 0.0027 | 0.0119 | 0.0383 | 0.0682 | 0.1095 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | 32 | 67 | 96 | 130 | 154 | |
| CPU | 0.0026 | 0.0057 | 0.0080 | 0.0103 | 0.0113 | ||
| IMRGI | IT | 118 | 251 | 355 | 489 | 564 | |
|
|
CPU | 0.0110 | 0.0182 | 0.0252 | 0.0405 | 0.0396 |
The graphs of ERR(log10) against the IT of the tested algorithms for four different values of
Comparison of convergence curves for Example 4.3 with
Comparison of convergence curves for Example 4.3 with
Example 4.4
[18] Consider the CCT Sylvester matrix equation (2) with
Let
to be the termination condition. And the prescribed maximum iterative number is 20,000 as in Examples 4.1–4.3.
For Example 4.4, we compare the IMGI and the IMRGI algorithms with the GI, OGI, RGI, and MGI ones in Table 10. The parameters adopted in the tested algorithms are obtained by utilizing the methods in Example 4.1. From the results listed in Table 10, we can conclude some observations: First, the IT of the OGI and the RGI algorithms are the same, and they perform better than the GI one. Second, the MGI algorithm outperforms the GI, OGI, and RGI ones except for the case of
Numerical results of the tested iterative algorithms for Example 4.4
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 |
|---|---|---|---|---|---|---|
| IT | 53 | 255 | 726 | 1362 | 1,998 | |
| GI | CPU | 0.0766 | 0.4319 | 1.1939 | 1.9604 | 2.7920 |
|
|
ERR |
|
|
|
|
|
| IT | 46 | 269 | 499 | 729 | 959 | |
| OGI | CPU | 0.0737 | 0.4681 | 0.8855 | 1.1386 | 1.4615 |
|
|
ERR |
|
|
|
|
|
| IT | 46 | 269 | 499 | 729 | 959 | |
| RGI | CPU | 0.0702 | 0.4569 | 0.8565 | 1.1687 | 1.4533 |
|
|
ERR |
|
|
|
|
|
| IT | 26 | 126 | 360 | 677 | 994 | |
| MGI | CPU | 0.0433 | 0.2135 | 0.6016 | 1.2261 | 1.6008 |
|
|
ERR |
|
|
|
|
|
| IT | 56 | 127 | 194 | 262 | 329 | |
| IMGI | CPU | 0.0072 | 0.0158 | 0.0189 | 0.0235 | 0.0332 |
|
|
ERR |
|
|
|
|
|
| IT | 36 | 80 | 122 | 164 | 205 | |
| IMRGI | CPU | 0.0037 | 0.0071 | 0.0129 | 0.0175 | 0.0171 |
|
|
ERR |
|
|
|
|
|
In Table 11, the iterative solutions
The iterative solutions for IMGI (
| Method |
|
|
|
|
|
|---|---|---|---|---|---|
| IMGI | 100 |
|
|
|
|
| 200 |
|
|
|
|
|
| 300 |
|
|
|
|
|
| 400 |
|
|
|
|
|
| 500 |
|
|
|
|
|
| IMRGI | 100 |
|
|
|
|
| 200 |
|
|
|
|
|
| 300 |
|
|
|
|
|
| 400 |
|
|
|
|
|
| 500 |
|
|
|
|
|
| Solution |
|
|
|
|
Furthermore, the same items as those in Tables 3, 6 and 9 are exhibited in Table 12. From the results of
Numerical results of the tested iterative algorithms for Example 4.4 with three different values of
| Method |
|
0.1 | 0.01 | 0.001 | 0.0001 | 0.00001 | |
|---|---|---|---|---|---|---|---|
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMRGI | IT | 40 | 90 | 138 | 186 | 233 | |
|
|
CPU | 0.0053 | 0.0169 | 0.0156 | 0.0195 | 0.0242 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 30 | 143 | 408 | 764 | 1121 | |
|
|
CPU | 0.0034 | 0.0156 | 0.0492 | 0.0803 | 0.1694 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMRGI | IT | 87 | 198 | 304 | 410 | 515 | |
|
|
CPU | 0.0109 | 0.0204 | 0.0444 | 0.0477 | 0.0750 | |
|
|
GI | IT | Fail | Fail | Fail | Fail | Fail |
| CPU | — | — | — | — | — | ||
| OGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| RGI | IT | 22 | 105 | 298 | 558 | 819 | |
|
|
CPU | 0.0024 | 0.0107 | 0.0297 | 0.0555 | 0.0795 | |
| MGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMGI | IT | Fail | Fail | Fail | Fail | Fail | |
| CPU | — | — | — | — | — | ||
| IMRGI | IT | 64 | 145 | 222 | 299 | 376 | |
|
|
CPU | 0.0069 | 0.0150 | 0.0229 | 0.0312 | 0.0384 |
In Figures 7 and 8, we depict the error curves of the tested algorithms for four different values of
Comparison of convergence curves for Example 4.4 with
Comparison of convergence curves for Example 4.4 with
5 Conclusions
In this article, by replacing the coefficient matrices with their diagonal parts and making use of the updated technique, we first construct the IMGI algorithm for the CCT Sylvester matrix equations. Then we further apply the relaxation technique to the IMGI algorithm and establish the IMRGI algorithm. By making use of the real representation of a complex matrix and the vector stretching operator, we give the convergence conditions of the IMGI and the IMRGI algorithms. Some numerical examples are presented to validate the effectiveness and superiorities of the IMGI and the IMRGI algorithms. Compared with the GI, OGI, RGI, and MGI algorithms, the proposed IMGI and IMRGI algorithms can own higher computational efficiencies by adjusting the values of the relaxation factors and the step size factors. Then it is more conducive to solving Sylvester matrix equations in pole assignment, control theory, signal processes, prediction and stability, and so forth.
It is noteworthy that only one step size factor
Acknowledgements
We thank the editor and anonymous reviewers for their careful reading of the manuscript and for the helpful comments.
-
Funding information: This work was supported by the National Natural Science Foundation of China (No. 12361078), and the Guangxi Natural Science Foundation (No. 2024GXNSFAA010498).
-
Author contributions: All authors contributed to the study conception and design. All authors performed material preparation, data collection, and analysis. The authors read and approved the final manuscript.
-
Conflict of interest: The authors declare that they have no competing interests.
-
Ethical approval: The conducted research is not related to either human or animal use.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.
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- Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
- Fractional Sturm-Liouville operators on compact star graphs
- Exact controllability for nonlinear thermoviscoelastic plate problem
- Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
- Superposition operator problems of Hölder-Lipschitz spaces
- A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
- Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
- A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
- Fixed point results for generalized convex orbital Lipschitz operators
- Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
- Multiplicity of k-convex solutions for a singular k-Hessian system
- Poisson C*-algebra derivations in Poisson C*-algebras
- Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
- Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
- Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
- Pseudo compact almost automorphic solutions to a family of delay differential equations
- Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
- Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
- Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
- Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
- Review Articles
- Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
- Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
- Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
- Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
- Binary relations applied to the fuzzy substructures of quantales under rough environment
- Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
- A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
- Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
- Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
- A study on a type of degenerate poly-Dedekind sums
- Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
- Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
- Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
- The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
- Neural network quaternion-based controller for port-Hamiltonian system
- Matching ontologies with kernel principle component analysis and evolutionary algorithm
- Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
- Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
- Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
- Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
- Some fixed point results on ultrametric spaces endowed with a graph
- On the generalized Mellin integral operators
- On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
- Approximation process of a positive linear operator of hypergeometric type
- On Kantorovich variant of Brass-Stancu operators
- A higher-dimensional categorical perspective on 2-crossed modules
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
- On parameterized inequalities for fractional multiplicative integrals
- On inverse source term for heat equation with memory term
- On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
- New extensions related to Fejér-type inequalities for GA-convex functions
- Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
- Some Hardy's inequalities on conformable fractional calculus
- The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
- A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
- Some new fixed point theorems of α-partially nonexpansive mappings
- Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
- Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
- Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
- Topological structure of the solution sets to neutral evolution inclusions driven by measures
- (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
-
Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
- Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
- Klein-Gordon potential in characteristic coordinates
- Asymptotic analysis of Leray solution for the incompressible NSE with damping
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
- Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
- Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
- Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
- Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
- A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
- The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
- Absence of global solutions to wave equations with structural damping and nonlinear memory
- Special Issue on Differential Equations and Numerical Analysis - Part I
- Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
- Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
- A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Articles in the same Issue
- Regular Articles
- On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
- L-Fuzzy fixed point results in ℱ -metric spaces with applications
- Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
- Nonparametric methods of statistical inference for double-censored data with applications
- LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
- Existence of projected solutions for quasi-variational hemivariational inequality
- Spectral collocation method for convection-diffusion equation
- New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
- On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
- On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
- The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
- Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
- Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
- An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
- An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
- On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
- Some properties of a class of holomorphic functions associated with tangent function
- The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
- On the continuity in q of the family of the limit q-Durrmeyer operators
- Results on solutions of several systems of the product type complex partial differential difference equations
- On Berezin norm and Berezin number inequalities for sum of operators
- Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
- On a generalization of the Opial inequality
- A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
- Holomorphic curves into projective spaces with some special hypersurfaces
- On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
- Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
- Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
- Jordan left derivations in infinite matrix rings
- Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
- Invariant means and lacunary sequence spaces of order (α, β)
- Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
- Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
- Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
- Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
- On three-dimensional q-Riordan arrays
- Some aspects of normal curve on smooth surface under isometry
- Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
- Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
- Remark on the Daugavet property for complex Banach spaces
- Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
- Uniqueness of meromorphic functions concerning small functions and derivatives-differences
- Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
- Hyers-Ulam stability of Davison functional equation on restricted domains
- Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
- Composition of some positive linear integral operators
- On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
- Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
- Online makespan minimization for MapReduce scheduling on multiple parallel machines
- The sequential Henstock-Kurzweil delta integral on time scales
- On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
- Existence of three solutions for two quasilinear Laplacian systems on graphs
- Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
- Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
- Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
- Fractional Sturm-Liouville operators on compact star graphs
- Exact controllability for nonlinear thermoviscoelastic plate problem
- Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
- Superposition operator problems of Hölder-Lipschitz spaces
- A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
- Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
- A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
- Fixed point results for generalized convex orbital Lipschitz operators
- Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
- Multiplicity of k-convex solutions for a singular k-Hessian system
- Poisson C*-algebra derivations in Poisson C*-algebras
- Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
- Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
- Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
- Pseudo compact almost automorphic solutions to a family of delay differential equations
- Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
- Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
- Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
- Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
- Review Articles
- Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
- Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
- Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
- Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
- Binary relations applied to the fuzzy substructures of quantales under rough environment
- Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
- A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
- Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
- Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
- A study on a type of degenerate poly-Dedekind sums
- Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
- Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
- Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
- The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
- Neural network quaternion-based controller for port-Hamiltonian system
- Matching ontologies with kernel principle component analysis and evolutionary algorithm
- Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
- Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
- Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
- Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
- Some fixed point results on ultrametric spaces endowed with a graph
- On the generalized Mellin integral operators
- On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
- Approximation process of a positive linear operator of hypergeometric type
- On Kantorovich variant of Brass-Stancu operators
- A higher-dimensional categorical perspective on 2-crossed modules
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
- On parameterized inequalities for fractional multiplicative integrals
- On inverse source term for heat equation with memory term
- On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
- New extensions related to Fejér-type inequalities for GA-convex functions
- Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
- Some Hardy's inequalities on conformable fractional calculus
- The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
- A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
- Some new fixed point theorems of α-partially nonexpansive mappings
- Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
- Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
- Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
- Topological structure of the solution sets to neutral evolution inclusions driven by measures
- (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
-
Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
- Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
- Klein-Gordon potential in characteristic coordinates
- Asymptotic analysis of Leray solution for the incompressible NSE with damping
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
- Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
- Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
- Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
- Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
- A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
- The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
- Absence of global solutions to wave equations with structural damping and nonlinear memory
- Special Issue on Differential Equations and Numerical Analysis - Part I
- Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
- Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
- A comparison of two nonconforming finite element methods for linear three-field poroelasticity
