In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed.
This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank<em>-k</em> approximation of a real <em>m</em>&...This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank<em>-k</em> approximation of a real <em>m</em>×<em>n</em> matrix, <em>A</em>. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, <em>G</em>, this method computes<em> k</em> dominant eigenvectors of <em>G</em>. To see the relation between these methods we assume that <em>G </em>=<em> A</em><sup>T</sup> <em>A</em>. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.展开更多
A new restoration algorithm based on double loops and alternant iterations is proposed to restore the object image effectively from a few frames of turbulence-degraded images, Based on the double loops, the iterative ...A new restoration algorithm based on double loops and alternant iterations is proposed to restore the object image effectively from a few frames of turbulence-degraded images, Based on the double loops, the iterative relations for estimating the turbulent point spread function PSF and object image alternately are derived. The restoration experiments have been made on computers, showing that the proposed algorithm can obtain the optimal estimations of the object and the point spread function, with the feasibility and practicality of the proposed algorithm being convincing.展开更多
In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduce...In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.展开更多
Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are presented.These approaches are on account of two-grid skill include two major phas...Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are presented.These approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution,and then locally solve linearized residual subproblems by one of three iterations(Stokes-type,Newton,and Oseen-type)on subdomains with fine grid in parallel to approximate the high frequency component.Optimal error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are given.Some numerical examples are implemented to verify the algorithm.展开更多
The simultaneous iterations rithms of the ART family. It is used reconstruction technique （SIRT） widely in tomography because of is one of several reconstruction algoits convenience in dealing with large sparse matr...The simultaneous iterations rithms of the ART family. It is used reconstruction technique （SIRT） widely in tomography because of is one of several reconstruction algoits convenience in dealing with large sparse matrices. Its theoretical background and iteration model are discussed at the beginning of this paper. Then, the implementation of the SIRT to reconstruct the three-dimensional distribution of water vapor by simulation is discussed. The results show that the SIRT can function effectively in water vapor tomography, obtain rapid convergence, and be implemented more easily than inversion.展开更多
Under the weak Lipschitz condition about the solution of the equation, convergence theorems for a family of iterations with one parameter are obtained. An estimation of the radius of the attraction ball is shown. At l...Under the weak Lipschitz condition about the solution of the equation, convergence theorems for a family of iterations with one parameter are obtained. An estimation of the radius of the attraction ball is shown. At last two examples are given.展开更多
In order to improve the performance of time difference of arrival(TDOA)localization,a nonlinear least squares algorithm is proposed in this paper.Firstly,based on the criterion of the minimized sum of square error of ...In order to improve the performance of time difference of arrival(TDOA)localization,a nonlinear least squares algorithm is proposed in this paper.Firstly,based on the criterion of the minimized sum of square error of time difference of arrival,the location estimation is expressed as an optimal problem of a non-linear programming.Then,an initial point is obtained using the semi-definite programming.And finally,the location is extracted from the local optimal solution acquired by Newton iterations.Simulation results show that when the number of anchor nodes is large,the performance of the proposed algorithm will be significantly better than that of semi-definite programming approach with the increase of measurement noise.展开更多
This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two i...This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 〈 σ =N｜｜f｜｜-1/v2≤1/√2＋1 , the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 〈 σ ≤5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory.展开更多
We introduce a general iterative method for a finite family of generalized asymptotically quasi- nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multi-st...We introduce a general iterative method for a finite family of generalized asymptotically quasi- nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multi-step iterative method of Khan et al. [1] as a special case. Our results are new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) spaces, simultaneously.展开更多
The iteration maps of Euler family for finding zeros of an operatorf in Banach spaces is defined as the partial sum of Taylor expansion of the local inversef z -1 off atz. The unified convergence theorem is establishe...The iteration maps of Euler family for finding zeros of an operatorf in Banach spaces is defined as the partial sum of Taylor expansion of the local inversef z -1 off atz. The unified convergence theorem is established for the iterations of Euler family under the assumption that $\alpha \leqslant 3 - 2\sqrt 2 $ , while the strong condition thatf is analytic in Smale’s criterion α is replaced by the weak condition thatf is of finite order derivative.展开更多
In the sense of the nonlinear multisplitting and based on the principle of suffi-ciently using the delayed information, we propose models of asynchronous parallelaccelerated overrelaxation iteration methods for solvin...In the sense of the nonlinear multisplitting and based on the principle of suffi-ciently using the delayed information, we propose models of asynchronous parallelaccelerated overrelaxation iteration methods for solving large scale system of non-linear equations. Under proper conditions, we set up the local convergence theoriesof these new method models.展开更多
The fast solution of linear equations has always been one of the hot spots in scientific computing.A kind of the diagonal matrix splitting iteration methods are provided,which is different from the classical matrix sp...The fast solution of linear equations has always been one of the hot spots in scientific computing.A kind of the diagonal matrix splitting iteration methods are provided,which is different from the classical matrix splitting methods.Taking the decomposition of the diagonal elements for coefficient matrix as the key point,some new preconditioners are constructed.Taking the tri-diagonal coefficient matrix as an example,the convergence domains and optimal relaxation factor of the new method are analyzed theoretically.The presented new iteration methods are applied to solve linear algebraic equations,even 2D and 3D diffusion problems with the fully implicit discretization.The results of numerical experiments are matched with the theoretical analysis,and show that the iteration numbers are reduced greatly.The superiorities of presented iteration methods exceed some classical iteration methods dramatically.展开更多
The main purpose of this paper is to investigate the singularities of solutions to the single Tricomi equation with derivative term and combined memory term.In addition,the blow-up of the solution to the weakly couple...The main purpose of this paper is to investigate the singularities of solutions to the single Tricomi equation with derivative term and combined memory term.In addition,the blow-up of the solution to the weakly coupled system with memory term is also considered,where one is a power nonlinear term and the other is a derivative nonlinear term.Upper bound lifespan estimates of solution are obtained in the sub-critical by utilizing the test function method and iteration technique.The innovation of this paper focuses on the lifespan estimates of the solutions,which extends the well-known Strauss and Glassey conjectures.展开更多
In this paper,we consider a Schr¨odinger-Poisson system with sublinear nonlinearity.The growth of nonlinearity depends on potential function and a bounded function.We first obtain the existence of nontrivial solu...In this paper,we consider a Schr¨odinger-Poisson system with sublinear nonlinearity.The growth of nonlinearity depends on potential function and a bounded function.We first obtain the existence of nontrivial solution sequence with negative energy for the system via a variant Clark’s theorem.Then we get the asymptotical property of the solution sequence by L∞norm.展开更多
In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to ...In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.展开更多
Presents the first estimation conditions for Nourein iterations for simultaneous finding all zeros of a polynomial under which the iteration processes are guaranteed to converge. Computational formulas; Theorems and p...Presents the first estimation conditions for Nourein iterations for simultaneous finding all zeros of a polynomial under which the iteration processes are guaranteed to converge. Computational formulas; Theorems and proofs.展开更多
Deep learning methods have achieved significant progress in solving partial differential equations.However,when applied to the widely used anisotropic scattering neutron transport equations in reactor engineering,thes...Deep learning methods have achieved significant progress in solving partial differential equations.However,when applied to the widely used anisotropic scattering neutron transport equations in reactor engineering,these encounter significant challenges.To address this issue,this study introduces a multi-antiderivative transformation alternating iterative deep learning method(M-AIM).This method transforms the integral terms of the scattering and fission sources in the transport equation into multiple antiderivative functions corresponding to the integrand,converts the differential-integral form of the transport equation into an exact differential equation,and establishes the necessary constraints for a unique solution.The M-AIM uses multiple deep neural networks to map the unknown angular flux density of transport equations and represents various forms of antiderivative functions.It constructs the corresponding weighted loss functions.By alternating iterative training with deep learning methods applied to these neural networks,the loss is reduced gradually.When the loss decreases to a preset minimum,the neural network approaches a numerical solution for both angular flux density and antiderivative functions.This paper presents a numerical verification of geometries such as flat plates and spheres.It verifies the validity of the theoretical framework and associated methods.The study contributes to the development of novel technical approaches for applying deep learning to solve anisotropic scattering neutron transport equations in reactor engineering.展开更多
The iterative continuation task(ICT)requires English as a foreign language(EFL)learners to read a segment and write a continuation that aligns with the preceding segment of an English novel with successive turns,offer...The iterative continuation task(ICT)requires English as a foreign language(EFL)learners to read a segment and write a continuation that aligns with the preceding segment of an English novel with successive turns,offering exposure to diverse grammatical structures and opportunities for contextualized usage.Given the importance of integrating technology into second language(L2)writing and the critical role that grammar plays in L2 writing development,automated written corrective feedback provided by Grammarly has gained significant attention.This study investigates the impact of Grammarly on grammar learning strategies,grammar grit,and grammar competence among EFL college students engaged in ICT.This study employed a mixed-methods sequential exploratory design;56 participants were divided into an experimental group(n=28),receiving Grammarly feedback for ICT,and a control group(n=28),completing ICT without Grammarly feedback.Quantitative results revealed that both groups showed improvements in L2 grammar learning strategies,grit and competence.For the experimental group,significant differences were observed across all variables of L2 grammar learning strategies,grit,and competence between pre-and post-tests.For the control group,significant differences were only observed in the affective dimension of grammar learning strategies,Consistency of Interest(COI)of grammar grit,and grammar competence.However,the control group presented a significantly higher improvement in grammar competence.Qualitative analysis showed both positive and negative perceptions of Grammarly.The pedagogical implications of integrating Grammarly and ICT for L2 grammar development are discussed.展开更多
文摘In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed.
文摘This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank<em>-k</em> approximation of a real <em>m</em>×<em>n</em> matrix, <em>A</em>. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, <em>G</em>, this method computes<em> k</em> dominant eigenvectors of <em>G</em>. To see the relation between these methods we assume that <em>G </em>=<em> A</em><sup>T</sup> <em>A</em>. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.
文摘A new restoration algorithm based on double loops and alternant iterations is proposed to restore the object image effectively from a few frames of turbulence-degraded images, Based on the double loops, the iterative relations for estimating the turbulent point spread function PSF and object image alternately are derived. The restoration experiments have been made on computers, showing that the proposed algorithm can obtain the optimal estimations of the object and the point spread function, with the feasibility and practicality of the proposed algorithm being convincing.
文摘In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.
基金Project supported by the National Natural Science Foundation of China(Nos.11971410 and12071404)the Natural Science Foundation of Hunan Province of China(No.2019JJ40279)+2 种基金the Excellent Youth Program of Scientific Research Project of Hunan Provincial Department of Education(Nos.18B064 and 20B564)the China Postdoctoral Science Foundation(Nos.2018T110073 and 2018M631402)the International Scientific and Technological Innovation Cooperation Base of Hunan Province for Computational Science(No.2018WK4006)。
文摘Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are presented.These approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution,and then locally solve linearized residual subproblems by one of three iterations(Stokes-type,Newton,and Oseen-type)on subdomains with fine grid in parallel to approximate the high frequency component.Optimal error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are given.Some numerical examples are implemented to verify the algorithm.
基金supported by the National Natural Science Foundation of China(40974018)Nationa l863 Plan Projects(2009AA12Z307)
文摘The simultaneous iterations rithms of the ART family. It is used reconstruction technique （SIRT） widely in tomography because of is one of several reconstruction algoits convenience in dealing with large sparse matrices. Its theoretical background and iteration model are discussed at the beginning of this paper. Then, the implementation of the SIRT to reconstruct the three-dimensional distribution of water vapor by simulation is discussed. The results show that the SIRT can function effectively in water vapor tomography, obtain rapid convergence, and be implemented more easily than inversion.
基金Supported by Shanghai Municipal Foundation of Selected Academic Research and the National Natural Science Foundation of China（10571059,10571060）.
文摘Under the weak Lipschitz condition about the solution of the equation, convergence theorems for a family of iterations with one parameter are obtained. An estimation of the radius of the attraction ball is shown. At last two examples are given.
基金This study was supported by the“High level research and training project for professional leaders of teachers in Higher Vocational Colleges in Jiangsu Province”.
文摘In order to improve the performance of time difference of arrival(TDOA)localization,a nonlinear least squares algorithm is proposed in this paper.Firstly,based on the criterion of the minimized sum of square error of time difference of arrival,the location estimation is expressed as an optimal problem of a non-linear programming.Then,an initial point is obtained using the semi-definite programming.And finally,the location is extracted from the local optimal solution acquired by Newton iterations.Simulation results show that when the number of anchor nodes is large,the performance of the proposed algorithm will be significantly better than that of semi-definite programming approach with the increase of measurement noise.
基金supported by the National Natural Science Foundation of China(No.11271298)
文摘This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 〈 σ =N｜｜f｜｜-1/v2≤1/√2＋1 , the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 〈 σ ≤5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory.
文摘We introduce a general iterative method for a finite family of generalized asymptotically quasi- nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multi-step iterative method of Khan et al. [1] as a special case. Our results are new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) spaces, simultaneously.
文摘The iteration maps of Euler family for finding zeros of an operatorf in Banach spaces is defined as the partial sum of Taylor expansion of the local inversef z -1 off atz. The unified convergence theorem is established for the iterations of Euler family under the assumption that $\alpha \leqslant 3 - 2\sqrt 2 $ , while the strong condition thatf is analytic in Smale’s criterion α is replaced by the weak condition thatf is of finite order derivative.
文摘In the sense of the nonlinear multisplitting and based on the principle of suffi-ciently using the delayed information, we propose models of asynchronous parallelaccelerated overrelaxation iteration methods for solving large scale system of non-linear equations. Under proper conditions, we set up the local convergence theoriesof these new method models.
基金The National Natural Science Foundations of China (12202219)the Natural Science Foundations of Ningxia (2024AAC02009, 2023AAC05001)the Ningxia Youth Top Talents Training Project。
文摘The fast solution of linear equations has always been one of the hot spots in scientific computing.A kind of the diagonal matrix splitting iteration methods are provided,which is different from the classical matrix splitting methods.Taking the decomposition of the diagonal elements for coefficient matrix as the key point,some new preconditioners are constructed.Taking the tri-diagonal coefficient matrix as an example,the convergence domains and optimal relaxation factor of the new method are analyzed theoretically.The presented new iteration methods are applied to solve linear algebraic equations,even 2D and 3D diffusion problems with the fully implicit discretization.The results of numerical experiments are matched with the theoretical analysis,and show that the iteration numbers are reduced greatly.The superiorities of presented iteration methods exceed some classical iteration methods dramatically.
基金Supported by National Natural Science Foundation of China Under Grant(12401647)Supported by Fundamental Research Program of Shanxi Province(202203021212336)+2 种基金Taiyuan Institute of Technology Scientific Research Initial Funding(2023KJ057,2024KJ007,2024LJ005)Supported by Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi(2024L358)Youth Program of Taiyuan University(24TYQN10)。
文摘The main purpose of this paper is to investigate the singularities of solutions to the single Tricomi equation with derivative term and combined memory term.In addition,the blow-up of the solution to the weakly coupled system with memory term is also considered,where one is a power nonlinear term and the other is a derivative nonlinear term.Upper bound lifespan estimates of solution are obtained in the sub-critical by utilizing the test function method and iteration technique.The innovation of this paper focuses on the lifespan estimates of the solutions,which extends the well-known Strauss and Glassey conjectures.
基金Supported by the National Natural Science Foundation of China(12226412)Natural Science Foundation of Jiangsu Province(BK20221339)。
文摘In this paper,we consider a Schr¨odinger-Poisson system with sublinear nonlinearity.The growth of nonlinearity depends on potential function and a bounded function.We first obtain the existence of nontrivial solution sequence with negative energy for the system via a variant Clark’s theorem.Then we get the asymptotical property of the solution sequence by L∞norm.
基金Supported in part by Natural Science Foundation of Guangxi(2023GXNSFAA026246)in part by the Central Government's Guide to Local Science and Technology Development Fund(GuikeZY23055044)in part by the National Natural Science Foundation of China(62363003)。
文摘In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.
基金National Natural Science Foundation of China Natural ScienceFoundation of Zhejiang Province.
文摘Presents the first estimation conditions for Nourein iterations for simultaneous finding all zeros of a polynomial under which the iteration processes are guaranteed to converge. Computational formulas; Theorems and proofs.
基金supported by the National Natural Science Foundation of China(No.12575189)。
文摘Deep learning methods have achieved significant progress in solving partial differential equations.However,when applied to the widely used anisotropic scattering neutron transport equations in reactor engineering,these encounter significant challenges.To address this issue,this study introduces a multi-antiderivative transformation alternating iterative deep learning method(M-AIM).This method transforms the integral terms of the scattering and fission sources in the transport equation into multiple antiderivative functions corresponding to the integrand,converts the differential-integral form of the transport equation into an exact differential equation,and establishes the necessary constraints for a unique solution.The M-AIM uses multiple deep neural networks to map the unknown angular flux density of transport equations and represents various forms of antiderivative functions.It constructs the corresponding weighted loss functions.By alternating iterative training with deep learning methods applied to these neural networks,the loss is reduced gradually.When the loss decreases to a preset minimum,the neural network approaches a numerical solution for both angular flux density and antiderivative functions.This paper presents a numerical verification of geometries such as flat plates and spheres.It verifies the validity of the theoretical framework and associated methods.The study contributes to the development of novel technical approaches for applying deep learning to solve anisotropic scattering neutron transport equations in reactor engineering.
文摘The iterative continuation task(ICT)requires English as a foreign language(EFL)learners to read a segment and write a continuation that aligns with the preceding segment of an English novel with successive turns,offering exposure to diverse grammatical structures and opportunities for contextualized usage.Given the importance of integrating technology into second language(L2)writing and the critical role that grammar plays in L2 writing development,automated written corrective feedback provided by Grammarly has gained significant attention.This study investigates the impact of Grammarly on grammar learning strategies,grammar grit,and grammar competence among EFL college students engaged in ICT.This study employed a mixed-methods sequential exploratory design;56 participants were divided into an experimental group(n=28),receiving Grammarly feedback for ICT,and a control group(n=28),completing ICT without Grammarly feedback.Quantitative results revealed that both groups showed improvements in L2 grammar learning strategies,grit and competence.For the experimental group,significant differences were observed across all variables of L2 grammar learning strategies,grit,and competence between pre-and post-tests.For the control group,significant differences were only observed in the affective dimension of grammar learning strategies,Consistency of Interest(COI)of grammar grit,and grammar competence.However,the control group presented a significantly higher improvement in grammar competence.Qualitative analysis showed both positive and negative perceptions of Grammarly.The pedagogical implications of integrating Grammarly and ICT for L2 grammar development are discussed.
