Pub Date : 2025-12-20DOI: 10.1016/j.cam.2025.117276
Hao Han , Zengqiang Tan , Xiaoqiang Yan
This paper presents a novel second-order one-parameter linearized finite difference scheme for solving the Allen-Cahn equation with the periodic boundary condition, which is derived by combining the central difference approximation in space and the one-parameter method with an extrapolation in time. It is shown that the derived fully discrete schemes are uniquely solvable and able to preserve the discrete maximum principle and energy stability under appropriate conditions. Moreover, the maximum-norm error estimate of the schemes are studied and the scheme can arrive at second-order accuracy both in time and space. Several numerical examples are conducted to validate the theoretical results.
{"title":"Second-order one-parameter maximum-principle-preserving and energy stable difference scheme for the Allen-Cahn equation","authors":"Hao Han , Zengqiang Tan , Xiaoqiang Yan","doi":"10.1016/j.cam.2025.117276","DOIUrl":"10.1016/j.cam.2025.117276","url":null,"abstract":"<div><div>This paper presents a novel second-order one-parameter linearized finite difference scheme for solving the Allen-Cahn equation with the periodic boundary condition, which is derived by combining the central difference approximation in space and the one-parameter method with an extrapolation in time. It is shown that the derived fully discrete schemes are uniquely solvable and able to preserve the discrete maximum principle and energy stability under appropriate conditions. Moreover, the maximum-norm error estimate of the schemes are studied and the scheme can arrive at second-order accuracy both in time and space. Several numerical examples are conducted to validate the theoretical results.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117276"},"PeriodicalIF":2.6,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.cam.2025.117264
Riccardo Brignone
In this paper, we propose a new approach to Asian options pricing under the Black-Scholes model that is based on the observation that, if the asset price follows a geometric Brownian motion, then the Laplace transform of the reciprocal of the arithmetic average of the asset price conditional on the price level at the endpoints of the given time period is known fully explicitly. Based on this, we propose two new option pricing formulas. In the first, we approximate the conditional distribution of the reciprocal of the arithmetic average with a gamma random variable and develop a fast and accurate approximating formula for Asian option prices. In the second, we develop a semi-analytical option pricing formula based on the numerical inversion of the aforementioned conditional Laplace transform, which provides an efficient and extremely accurate solution for the price of a large class of Asian-style derivatives, such as floating-strike Asian options and Australian options. Finally, we show how the proposed pricing approach can be applied to more complex models than Black-Scholes.
{"title":"Nested-conditional factorization approach to Asian options pricing","authors":"Riccardo Brignone","doi":"10.1016/j.cam.2025.117264","DOIUrl":"10.1016/j.cam.2025.117264","url":null,"abstract":"<div><div>In this paper, we propose a new approach to Asian options pricing under the Black-Scholes model that is based on the observation that, if the asset price follows a geometric Brownian motion, then the Laplace transform of the reciprocal of the arithmetic average of the asset price conditional on the price level at the endpoints of the given time period is known fully explicitly. Based on this, we propose two new option pricing formulas. In the first, we approximate the conditional distribution of the reciprocal of the arithmetic average with a gamma random variable and develop a fast and accurate approximating formula for Asian option prices. In the second, we develop a semi-analytical option pricing formula based on the numerical inversion of the aforementioned conditional Laplace transform, which provides an efficient and extremely accurate solution for the price of a large class of Asian-style derivatives, such as floating-strike Asian options and Australian options. Finally, we show how the proposed pricing approach can be applied to more complex models than Black-Scholes.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117264"},"PeriodicalIF":2.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.cam.2025.117265
Xiaofeng Guo , Jianyu Pan
We develop a new approximate parameterized splitting preconditioner for the Toeplitz-like discretized systems of the unsteady-state space-fractional diffusion equation to overcome the preconditioning challenge in the anisotropic case that one of the coefficients is significantly larger than the other. The construction of preconditioner is based on parameterized matrix splitting technique along with pre- and post-circulant-like approximation, which facilitates very economic implementation. Theoretical analysis shows that the preconditioner with a suitable parameter could lead to the corresponding preconditioned matrix being expressed as the sum of a matrix whose spectrum is clustered around one, a low rank matrix and a small norm matrix, which suggests that the preconditioner is effective. Numerical results of the preconditioned GMRES method indicate that the preconditioner’s practical efficiency is satisfactory.
{"title":"Approximate parameterized splitting preconditioning for anisotropic space-fractional diffusion equation","authors":"Xiaofeng Guo , Jianyu Pan","doi":"10.1016/j.cam.2025.117265","DOIUrl":"10.1016/j.cam.2025.117265","url":null,"abstract":"<div><div>We develop a new approximate parameterized splitting preconditioner for the Toeplitz-like discretized systems of the unsteady-state space-fractional diffusion equation to overcome the preconditioning challenge in the anisotropic case that one of the coefficients is significantly larger than the other. The construction of preconditioner is based on parameterized matrix splitting technique along with pre- and post-circulant-like approximation, which facilitates very economic implementation. Theoretical analysis shows that the preconditioner with a suitable parameter could lead to the corresponding preconditioned matrix being expressed as the sum of a matrix whose spectrum is clustered around one, a low rank matrix and a small norm matrix, which suggests that the preconditioner is effective. Numerical results of the preconditioned GMRES method indicate that the preconditioner’s practical efficiency is satisfactory.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117265"},"PeriodicalIF":2.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.cam.2025.117267
Bouharket Benaissa , Noureddine Azzouz , Fatih Hezenci
This research formulates Boole-type inequalities for h-convex functions using Riemann integrals. The subject matter is a new variant of the established Boole-type inequalities for twice differentiable functions, applying two basic identities and using simple computations based on the B-function. The first result is involving the power-mean integral inequality, and the second is via Hölder inequality. Additionally, new results on Boole-type inequalities are established for special classes of convex functions, such as s-convex functions and P-functions.
{"title":"On estimation of the boole-type inequality for twice differentiable functions","authors":"Bouharket Benaissa , Noureddine Azzouz , Fatih Hezenci","doi":"10.1016/j.cam.2025.117267","DOIUrl":"10.1016/j.cam.2025.117267","url":null,"abstract":"<div><div>This research formulates Boole-type inequalities for <em>h</em>-convex functions using Riemann integrals. The subject matter is a new variant of the established Boole-type inequalities for twice differentiable functions, applying two basic identities and using simple computations based on the <em>B</em>-function. The first result is involving the power-mean integral inequality, and the second is via Hölder inequality. Additionally, new results on Boole-type inequalities are established for special classes of convex functions, such as <em>s</em>-convex functions and <em>P</em>-functions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117267"},"PeriodicalIF":2.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.cam.2025.117250
Shailendra Singh , Abdul-Majid Wazwaz
The two versions of -dimensional variable coefficients combined pKP-BKP equations, which describe complex nonlinear wave phenomena, are examined in this work. The Painlevé analysis technique is employed to examine the integrability and nonlinear characteristics of the mentioned equations. Using this approach, both the considered equations are found completely Painlevé integrable. The auto-Bäcklund transformation technique is applied to generate various analytical solutions for both the equations including complex, rational, exponential, and linear type solutions, capturing rich nonlinear behaviors. Additionally, alternative techniques such as tan , tanh , exp type-I and type-II methods are employed to drive kink, anti-kink and anti-bell shaped solutions, which illustrate the nonlinear interaction mechanisms inherent in the system. Three-dimensional graphs illustrate the behaviour of all the obtained solutions under different coefficients and parameters, highlighting the complexity of nonlinear wave dynamics. The obtained results not only expose the complex behaviour of nonlinear wave phenomena but also confirm the effectiveness of the employed techniques in modeling a wide range of physical systems. This study also provides a meaningful contribution to ongoing research in the field of nonlinear mathematical physics and engineering.
{"title":"Auto-Bäcklund transformation, kink, anti-kink and periodic wave solutions for Painlevé integrable (3+1)-dimensional variable coefficients combined pKP-BKP equations in nonlinear waves","authors":"Shailendra Singh , Abdul-Majid Wazwaz","doi":"10.1016/j.cam.2025.117250","DOIUrl":"10.1016/j.cam.2025.117250","url":null,"abstract":"<div><div>The two versions of <span><math><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-dimensional variable coefficients combined pKP-BKP equations, which describe complex nonlinear wave phenomena, are examined in this work. The Painlevé analysis technique is employed to examine the integrability and nonlinear characteristics of the mentioned equations. Using this approach, both the considered equations are found completely Painlevé integrable. The auto-Bäcklund transformation technique is applied to generate various analytical solutions for both the equations including complex, rational, exponential, and linear type solutions, capturing rich nonlinear behaviors. Additionally, alternative techniques such as tan , tanh , exp type-I and type-II methods are employed to drive kink, anti-kink and anti-bell shaped solutions, which illustrate the nonlinear interaction mechanisms inherent in the system. Three-dimensional graphs illustrate the behaviour of all the obtained solutions under different coefficients and parameters, highlighting the complexity of nonlinear wave dynamics. The obtained results not only expose the complex behaviour of nonlinear wave phenomena but also confirm the effectiveness of the employed techniques in modeling a wide range of physical systems. This study also provides a meaningful contribution to ongoing research in the field of nonlinear mathematical physics and engineering.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"481 ","pages":"Article 117250"},"PeriodicalIF":2.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-19DOI: 10.1016/j.cam.2025.117258
Zhibo Wang, Xueyun Deng, Yan Mo
In this article, we are committed to developing a time two-grid difference scheme on nonuniform grids for the time fractional Gray-Scott model with weak singularity. Taking the initial singularity into account, a graded mesh is utilized in time. Subsequently, so as to enhance computational efficiency, the time two-grid method is designed through nonuniform L2-1σ formula. By virtue of the energy method, the stability and convergence of the numerical method are rigorously shown with convergence rate, where, NF, NC, M1 and M1 signify the number of grids, r and α denote grading parameter and fractional order, respectively. The numerical experiments ultimately confirmed the validity and accuracy of the theoretical results, ensuring the reliability of the research conclusions.
{"title":"Nonuniform L2-1σ method combined with time two-grid algorithm for the time fractional Gray-Scott model with initial singularity","authors":"Zhibo Wang, Xueyun Deng, Yan Mo","doi":"10.1016/j.cam.2025.117258","DOIUrl":"10.1016/j.cam.2025.117258","url":null,"abstract":"<div><div>In this article, we are committed to developing a time two-grid difference scheme on nonuniform grids for the time fractional Gray-Scott model with weak singularity. Taking the initial singularity into account, a graded mesh is utilized in time. Subsequently, so as to enhance computational efficiency, the time two-grid method is designed through nonuniform <em>L</em>2-1<sub><em>σ</em></sub> formula. By virtue of the energy method, the stability and convergence of the numerical method are rigorously shown with <span><math><mrow><mi>O</mi><mo>(</mo><msubsup><mi>N</mi><mi>F</mi><mrow><mo>−</mo><mi>min</mi><mo>{</mo><mi>r</mi><mi>α</mi><mo>,</mo><mn>2</mn><mo>}</mo></mrow></msubsup><mo>+</mo><msubsup><mi>N</mi><mi>C</mi><mrow><mo>−</mo><mi>min</mi><mo>{</mo><mn>2</mn><mi>r</mi><mi>α</mi><mo>,</mo><mn>4</mn><mo>}</mo></mrow></msubsup><mo>+</mo><msubsup><mi>M</mi><mn>1</mn><mrow><mo>−</mo><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mi>M</mi><mn>2</mn><mrow><mo>−</mo><mn>2</mn></mrow></msubsup><mo>)</mo></mrow></math></span> convergence rate, where, <em>N<sub>F</sub></em>, <em>N<sub>C</sub></em>, <em>M</em><sub>1</sub> and <em>M</em><sub>1</sub> signify the number of grids, <em>r</em> and <em>α</em> denote grading parameter and fractional order, respectively. The numerical experiments ultimately confirmed the validity and accuracy of the theoretical results, ensuring the reliability of the research conclusions.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117258"},"PeriodicalIF":2.6,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is aiming at delving on a generalized system of triple fractional stochastic differential variational inequalities carrying Lévy jumps (GSTFSDVI carrying Lévy jumps), that comprises of two systems, i.e., a generalized system of triple stochastic variational inequalities (GSTSVI) and a generalized system of triple fractional stochastic differential equations (GSTFSDE) carrying Lévy jumps. With the help of Picard’s successive iteration method and the projection technique, it is proven that there holds the unique existence of solutions to the GSTFSDVI carrying Lévy jumps under some appropriate restrictions. In addition, the main outcomes are exploited to establish the unique existence of solutions to the spatial-price equilibria general system in stochastic circumstances.
{"title":"A generalized system of triple fractional stochastic differential variational inequalities with Lévy jumps","authors":"Lu-Chuan Ceng , Yue Zhang , Jinxia Cen , Jen-Chih Yao , Yunshui Liang","doi":"10.1016/j.cam.2025.117259","DOIUrl":"10.1016/j.cam.2025.117259","url":null,"abstract":"<div><div>This article is aiming at delving on a generalized system of triple fractional stochastic differential variational inequalities carrying Lévy jumps (GSTFSDVI carrying Lévy jumps), that comprises of two systems, i.e., a generalized system of triple stochastic variational inequalities (GSTSVI) and a generalized system of triple fractional stochastic differential equations (GSTFSDE) carrying Lévy jumps. With the help of Picard’s successive iteration method and the projection technique, it is proven that there holds the unique existence of solutions to the GSTFSDVI carrying Lévy jumps under some appropriate restrictions. In addition, the main outcomes are exploited to establish the unique existence of solutions to the spatial-price equilibria general system in stochastic circumstances.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117259"},"PeriodicalIF":2.6,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-14DOI: 10.1016/j.cam.2025.117254
Zaheen A-Rahman , Luca di Mare
A novel algorithm for the fast parallel iterative solution of large narrow band matrices on shared memory systems is presented. For this, a new parallel many-block factorization is developed for block tridiagonal systems. In the process, several new results are obtained for relationships between norms, antinorms, and spectral radii of block matrices. Using these, it is shown that under very mild assumptions, for all iterative solvers such that the spectral radius of the iteration matrix varies monotonically with that of the Jacobi matrix, iterations on a system preconditioned using the proposed factorization converges faster than on the original system. In particular, under some mild assumptions, preconditioned Jacobi iterations converge about four times faster in terms of number of iterations. To achieve near-optimal parallel performance, our factorization is coupled with the block stair SOR algorithm to develop a new algorithm, each iteration of which is roughly comparable to an iteration of the former in computational complexity. A minimal workspace is required and our algorithm converges asymptotically optimally in time, and has better parallel performance than block stair SOR for large matrices. Moreover, it is observed numerically that small deviations of the overrelaxation parameter from its optimal value only have a relatively small effect on the number of iterations needed to converge. Finally, numerical data is presented that demonstrates a reduction in the condition number of our test system post-factorization. To conclude, possible methods to further speed up performance and a potential extension to distributed memory systems are discussed.
{"title":"Parallel solution of large narrow band linear systems","authors":"Zaheen A-Rahman , Luca di Mare","doi":"10.1016/j.cam.2025.117254","DOIUrl":"10.1016/j.cam.2025.117254","url":null,"abstract":"<div><div>A novel algorithm for the fast parallel iterative solution of large narrow band matrices on shared memory systems is presented. For this, a new parallel many-block factorization is developed for block tridiagonal systems. In the process, several new results are obtained for relationships between norms, antinorms, and spectral radii of block matrices. Using these, it is shown that under very mild assumptions, for all iterative solvers such that the spectral radius of the iteration matrix varies monotonically with that of the Jacobi matrix, iterations on a system preconditioned using the proposed factorization converges faster than on the original system. In particular, under some mild assumptions, preconditioned Jacobi iterations converge about four times faster in terms of number of iterations. To achieve near-optimal parallel performance, our factorization is coupled with the block stair SOR algorithm to develop a new algorithm, each iteration of which is roughly comparable to an iteration of the former in computational complexity. A minimal workspace is required and our algorithm converges asymptotically optimally in time, and has better parallel performance than block stair SOR for large matrices. Moreover, it is observed numerically that small deviations of the overrelaxation parameter from its optimal value only have a relatively small effect on the number of iterations needed to converge. Finally, numerical data is presented that demonstrates a reduction in the condition number of our test system post-factorization. To conclude, possible methods to further speed up performance and a potential extension to distributed memory systems are discussed.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117254"},"PeriodicalIF":2.6,"publicationDate":"2025-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Model order reduction has become an essential technique for reducing the complexity of large-scale dynamical systems. In many practical applications, it is desirable to achieve high accuracy within a specified finite frequency range, while allowing errors outside this band. Frequency-limited balanced truncation is a well-established technique for this purpose. For single-input single-output and symmetric multi-input multi-output systems, the two Lyapunov equations involved can be equivalently replaced by a single Sylvester equation. However, in cross Gramian-based frequency-limited balanced truncation, solving the resulting high-dimensional Sylvester equation and computing the matrix-valued logarithm are computationally expensive for large-scale systems, motivating the use of low-rank approximation techniques.
The alternating direction implicit (ADI) method is widely recognized for its efficiency in computing low-rank solutions of large-scale Sylvester equations. This work shows that, when applied to the Sylvester equation corresponding to the standard cross Gramian, the low-rank ADI method satisfies a subset of the -optimality conditions. The frequency-limited -optimality conditions are then derived based on the frequency-limited cross Gramian, and it is demonstrated that none of these conditions are satisfied by the ADI method, resulting in a notable decrease in accuracy compared with the standard case. To address this issue, a modification to the ADI method is proposed, in which a subset of the frequency-limited -optimality conditions is enforced. The original ADI iteration structure and computational efficiency are preserved, while accuracy is improved through a small-scale correction step. The resulting algorithm achieves the same high accuracy in the frequency-limited case as in the standard case. Additionally, an efficient method for computing the matrix-valued logarithm product, which arises in most frequency-limited model reduction algorithms, is proposed. The performance of the proposed approach is evaluated on benchmark examples. Numerical results confirm that the frequency-limited balanced truncation algorithm, incorporating the modified ADI method and the proposed matrix-valued logarithm product method, delivers significantly improved accuracy compared to the standard ADI method.
{"title":"An ADI-variant for low-rank cross Gramian-based frequency-limited balanced truncation","authors":"Zhong-Yi Huang , Zhi-Yuan Gao , Qiu-Yan Song , Umair Zulfiqar","doi":"10.1016/j.cam.2025.117256","DOIUrl":"10.1016/j.cam.2025.117256","url":null,"abstract":"<div><div>Model order reduction has become an essential technique for reducing the complexity of large-scale dynamical systems. In many practical applications, it is desirable to achieve high accuracy within a specified finite frequency range, while allowing errors outside this band. Frequency-limited balanced truncation is a well-established technique for this purpose. For single-input single-output and symmetric multi-input multi-output systems, the two Lyapunov equations involved can be equivalently replaced by a single Sylvester equation. However, in cross Gramian-based frequency-limited balanced truncation, solving the resulting high-dimensional Sylvester equation and computing the matrix-valued logarithm are computationally expensive for large-scale systems, motivating the use of low-rank approximation techniques.</div><div>The alternating direction implicit (ADI) method is widely recognized for its efficiency in computing low-rank solutions of large-scale Sylvester equations. This work shows that, when applied to the Sylvester equation corresponding to the standard cross Gramian, the low-rank ADI method satisfies a subset of the <span><math><msub><mi>H</mi><mn>2</mn></msub></math></span>-optimality conditions. The frequency-limited <span><math><msub><mi>H</mi><mn>2</mn></msub></math></span>-optimality conditions are then derived based on the frequency-limited cross Gramian, and it is demonstrated that none of these conditions are satisfied by the ADI method, resulting in a notable decrease in accuracy compared with the standard case. To address this issue, a modification to the ADI method is proposed, in which a subset of the frequency-limited <span><math><msub><mi>H</mi><mn>2</mn></msub></math></span>-optimality conditions is enforced. The original ADI iteration structure and computational efficiency are preserved, while accuracy is improved through a small-scale correction step. The resulting algorithm achieves the same high accuracy in the frequency-limited case as in the standard case. Additionally, an efficient method for computing the matrix-valued logarithm product, which arises in most frequency-limited model reduction algorithms, is proposed. The performance of the proposed approach is evaluated on benchmark examples. Numerical results confirm that the frequency-limited balanced truncation algorithm, incorporating the modified ADI method and the proposed matrix-valued logarithm product method, delivers significantly improved accuracy compared to the standard ADI method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"479 ","pages":"Article 117256"},"PeriodicalIF":2.6,"publicationDate":"2025-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-13DOI: 10.1016/j.cam.2025.117246
A. Cantón, L. Fernández-Jambrina, M.J. Vázquez-Gallo
In this paper we develop the formalism of rational complex Bézier curves. This framework is a simple extension of the CAD paradigm, since it describes arcs of curves in terms of control polygons and weights, which are extended to complex values. One of the major advantages of this extension is that we may make use of two different groups of projective transformations. Besides the group of projective transformations of the real plane, we have the group of complex projective transformations. This allows us to apply useful transformations like the geometric inversion to curves in design. In addition to this, the use of the complex formulation allows to lower the degree of the curves in some cases. This can be checked using the resultant of two polynomials and provides a simple formula for determining whether a rational cubic curve is a conic or not. Examples of application of the formalism to classical curves are included.
{"title":"Rational complex Bezier curves","authors":"A. Cantón, L. Fernández-Jambrina, M.J. Vázquez-Gallo","doi":"10.1016/j.cam.2025.117246","DOIUrl":"10.1016/j.cam.2025.117246","url":null,"abstract":"<div><div>In this paper we develop the formalism of rational complex Bézier curves. This framework is a simple extension of the CAD paradigm, since it describes arcs of curves in terms of control polygons and weights, which are extended to complex values. One of the major advantages of this extension is that we may make use of two different groups of projective transformations. Besides the group of projective transformations of the real plane, we have the group of complex projective transformations. This allows us to apply useful transformations like the geometric inversion to curves in design. In addition to this, the use of the complex formulation allows to lower the degree of the curves in some cases. This can be checked using the resultant of two polynomials and provides a simple formula for determining whether a rational cubic curve is a conic or not. Examples of application of the formalism to classical curves are included.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"480 ","pages":"Article 117246"},"PeriodicalIF":2.6,"publicationDate":"2025-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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