arXiv subjects

Sk. Safique Ahmad

Publications and source records attributed to Sk. Safique Ahmad.

14 recordsLinked to original sources

arXiv2026

Sparsity-Preserving Structured Backward Error Analysis for Double Saddle Point Problems

Backward error (BE) analysis emerges as a powerful tool for assessing the backward stability and strong backward stability of numerical algorithms. In this paper, we explore structured BEs for a class of double saddle point problems (DSPPs) and aim to assess the strong backward stability of numerical algorithms designed to find their solutions. Our investigations preserve the inherent matrix structure and sparsity pattern in the corresponding perturbation matrices and derive explicit formulae for the structured BEs. Moreover, we provide formulae for the structure-preserving minimal perturbation matrices for which the structured BE is attained. Utilizing the relationship between the DSPP and the least squares problem with equality constraints (LSE), we derive the sparsity-preserving BE formula for LSE within our framework. Numerical experiments are performed to test the strong backward stability of various numerical algorithms.

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arXiv2025

A Class of Generalized Shift-Splitting Preconditioners for Double Saddle Point Problems

In this paper, we propose a generalized shift-splitting (GSS) preconditioner, along with its two relaxed variants to solve the double saddle point problem (DSPP). The convergence of the associated GSS iterative method is analyzed, and sufficient conditions for its convergence are established. Spectral analyses are performed to derive sharp bounds for the eigenvalues of the preconditioned matrices. Numerical experiments based on examples arising from the PDE-constrained optimization problem and the leaky lid-driven cavity problem demonstrate the effectiveness and robustness of the proposed preconditioners compared with existing state-of-the-art preconditioners.

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arXiv2025

Structured Backward Errors of Sparse Generalized Saddle Point Problems with Hermitian Block Matrices

In this paper, we derive the structured backward error (BE) for a class of generalized saddle point problems (GSPP) by preserving the sparsity pattern and Hermitian structures of the block matrices. Additionally, we construct the optimal backward perturbation matrices for which the structured BE is achieved. Our analysis also examines the structured BE in cases where the sparsity pattern is not maintained. Through numerical experiments, we demonstrate the reliability of the derived structured BEs and the corresponding optimal backward perturbations. Additionally, the derived structured BEs are used to assess the strong backward stability of numerical methods for solving the GSPP.

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arXiv2025

Partial Condition Numbers for Double Saddle Point Problems

This paper presents a unified framework for investigating the partial condition number (CN) of the solution of double saddle point problems (DSPPs) and provides closed-form expressions for it. This unified framework encompasses the well-known partial normwise CN (NCN), partial mixed CN (MCN) and partial componentwise CN (CCN) as special cases. Furthermore, we derive sharp upper bounds for the partial NCN, MCN and CCN, which are computationally efficient and free of expensive Kronecker products. By applying perturbations that preserve the structure of the block matrices of the DSPPs, we analyze the structured partial NCN, MCN and CCN when the block matrices exhibit linear structures. By leveraging the relationship between DSPP and equality constrained indefinite least squares (EILS) problems, we recover the partial CNs for the EILS problem. Numerical results confirm the sharpness of the derived upper bounds and demonstrate their effectiveness in estimating the partial CNs.

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arXiv2025

On the specific solutions of reduced biquaternion equality constrained least squares problem and their relative forward error bound

This study focuses on addressing the challenge of solving the reduced biquaternion equality constrained least squares (RBLSE) problem. We develop algebraic techniques to derive real and complex solutions for the RBLSE problem by utilizing the real and complex forms of reduced biquaternion matrices. Furthermore, we propose algorithms and provide a detailed analysis of their computational complexity for finding special solutions to the RBLSE problem. A perturbation analysis is conducted, establishing an upper bound for the relative forward error of these solutions. This analysis ensures the accuracy and stability of the solutions in the presence of data perturbations, which is crucial for practical applications where errors arising from input inaccuracies can cause deviations between computed and true solutions. Numerical examples are presented to validate the proposed algorithms, demonstrate their effectiveness, and verify the accuracy of the established upper bound for the relative forward errors. These findings lay the groundwork for exploring applications in 3D and 4D algebra such as robotics, signal, and image processing, expanding their impact on practical and emerging domains.

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arXiv2025

Structured Backward Errors for Special Classes of Saddle Point Problems with Applications

In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the coefficient matrix of the SPP. Moreover, the existing techniques are not applicable when the block matrices have circulant, Toeplitz, or symmetric-Toeplitz structures and do not even provide structure preserving minimal perturbation matrices for which the BE is attained. To overcome these limitations, we investigate the structured BEs of SPPs when the perturbation matrices exploit the sparsity pattern as well as circulant, Toeplitz, and symmetric-Toeplitz structures. Furthermore, we construct minimal perturbation matrices that preserve the sparsity pattern and the aforementioned structures. Applications of the developed frameworks are utilized to compute BEs for the weighted regularized least squares problem. Finally, numerical experiments are performed to validate our findings, showcasing the utility of the obtained structured BEs in assessing the strong backward stability of numerical algorithms.

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arXiv2024

A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems

This paper proposes a new parameterized enhanced shift-splitting (PESS) preconditioner to solve the three-by-three block saddle point problem (SPP). Additionally, we introduce a local PESS (LPESS) preconditioner by relaxing the PESS preconditioner. Necessary and sufficient criteria are established for the convergence of the proposed PESS iterative process for any initial guess. Furthermore, we meticulously investigate the spectral bounds of the PESS and LPESS preconditioned matrices. Moreover, empirical investigations have been performed for the sensitivity analysis of the proposed PESS preconditioner, which unveils its robustness. Numerical experiments are carried out to demonstrate the enhanced efficiency and robustness of the proposed PESS and LPESS preconditioners compared to the existing state-of-the-art preconditioners.

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arXiv2024

Condition numbers for the Moore-Penrose inverse and the least squares problem involving rank-structured matrices

Perturbation theory plays a crucial role in sensitivity analysis, which is extensively used to assess the robustness of numerical techniques. To quantify the relative sensitivity of any problem, it becomes essential to investigate structured condition numbers (CNs) via componentwise perturbation theory. This paper addresses and analyzes structured mixed condition number (MCN) and componentwise condition number (CCN) for the Moore-Penrose (M-P) inverse and the minimum norm least squares (MNLS) solution involving rank-structured matrices, which include the Cauchy-Vandermonde (CV) matrices and {1, 1}-quasiseparable (QS) matrices. A general framework has been developed to compute the upper bounds for MCN and CCN of rank deficient parameterized matrices. This framework leads to faster computation of upper bounds of structured CNs for CV and {1, 1}-QS matrices. Furthermore, comparisons of obtained upper bounds are investigated theoretically and experimentally. In addition, the structured effective CNs for the M-P inverse and the MNLS solution of {1, 1}-QS matrices are presented. Numerical tests reveal the reliability of the proposed upper bounds as well as demonstrate that the structured effective CNs are computationally less expensive and can be substantially smaller compared to the unstructured CNs.

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arXiv2024

Structured condition numbers for a linear function of the solution of the generalized saddle point problem

This paper addresses structured normwise, mixed, and componentwise condition numbers (CNs) for a linear function of the solution to the generalized saddle point problem (GSPP). We present a general framework that enables us to measure the structured CNs of the individual components of the solution. Then, we derive their explicit formulae when the input matrices have symmetric, Toeplitz, or some general linear structures. In addition, compact formulae for the unstructured CNs are obtained, which recover previous results on CNs for GSPPs for specific choices of the linear function. Furthermore, applications of the derived structured CNs are provided to determine the structured CNs for the weighted Toeplitz regularized least-squares problems and Tikhonov regularization problems, which retrieves some previous studies in the literature.

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arXiv2023

L-structure least squares solutions of reduced biquaternion matrix equations with applications

This paper presents a framework for computing the structure-constrained least squares solutions to the generalized reduced biquaternion matrix equations (RBMEs). The investigation focuses on three different matrix equations: a linear matrix equation with multiple unknown L-structures, a linear matrix equation with one unknown L-structure, and the general coupled linear matrix equations with one unknown L-structure. Our approach leverages the complex representation of reduced biquaternion matrices. To showcase the versatility of the developed framework, we utilize it to find structure-constrained solutions for complex and real matrix equations, broadening its applicability to various inverse problems. Specifically, we explore its utility in addressing partially described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our study concludes with numerical examples.

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arXiv2023

Algebraic technique for mixed least squares and total least squares problem in the reduced biquaternion algebra

This paper presents the reduced biquaternion mixed least squares and total least squares (RBMTLS) method for solving an overdetermined system $AX \approx B$ in the reduced biquaternion algebra. The RBMTLS method is suitable when matrix $B$ and a few columns of matrix $A$ contain errors. By examining real representations of reduced biquaternion matrices, we investigate the conditions for the existence and uniqueness of the real RBMTLS solution and derive an explicit expression for the real RBMTLS solution. The proposed technique covers two special cases: the reduced biquaternion total least squares (RBTLS) method and the reduced biquaternion least squares (RBLS) method. Furthermore, the developed method is also used to find the best approximate solution to $AX \approx B$ over a complex field. Lastly, a numerical example is presented to support our findings.

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arXiv2023

Special least squares solutions of the reduced biquaternion matrix equation with applications

This paper presents an efficient method for obtaining the least squares Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, F )$. The method leverages the real representation of reduced biquaternion matrices. Furthermore, we establish the necessary and sufficient conditions for the existence and uniqueness of the Hermitian solution, along with a general expression for it. Notably, this approach differs from the one previously developed by Yuan et al. $(2020)$, which relied on the complex representation of reduced biquaternion matrices. In contrast, our method exclusively employs real matrices and utilizes real arithmetic operations, resulting in enhanced efficiency. We also apply our developed framework to find the Hermitian solutions for the complex matrix equation $(AXB, CXD) = (E, F )$, expanding its utility in addressing inverse problems. Specifically, we investigate its effectiveness in addressing partially described inverse eigenvalue problems. Finally, we provide numerical examples to demonstrate the effectiveness of our method and its superiority over the existing approach.

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arXiv2016

Localization theorems for matrices and bounds for the zeros of polynomials over a quaternion division algebra

In this paper, Ostrowski and Brauer type theorems are derived for the left and right eigenvalues of a quaternionic matrix. Generalizations of Gerschgorin type theorems are discussed for the left and the right eigenvalues of a quaternionic matrix. Thereafter a sufficient condition for the stability of a quaternionic matrix is given that generalizes the stability condition for a complex matrix. Finally, a characterization of bounds for the zeros of quaternionic polynomials is presented.

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arXiv2008

The Stochastic Logarithmic Norm for Stability Analysis of Stochastic Differential Equations

To analyze the stability of Itô stochastic differential equations with multiplicative noise, we introduce the stochastic logarithmic norm. The logarithmic norm was originally introduced by G. Dahlquist in 1958 as a tool to study the growth of solutions to ordinary differential equations and for estimating the error growth in discretization methods for their approximate solutions. We extend the concept to the stability analysis of Itô stochastic differential equations with multiplicative noise. Stability estimates for linear Itô SDEs using the one, two and $\infty$-norms in the $l$-th mean, where $1 \leq l < \infty $, are derived and the application of the stochastic logarithmic norm is illustrated with examples.

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