arXiv subjects

Shanheng Han

Publications and source records attributed to Shanheng Han.

3 recordsLinked to original sources

arXiv2025

A Minimal Perturbation Approach For The Rectangular Multiparameter Eigenvalue Problem

The rectangular multiparameter eigenvalue problem (RMEP) involves rectangular coefficient matrices (usually with more rows than columns) and may potentially have no solution in its original form. A minimal perturbation framework is proposed to defines approximate solutions. Computationally, two particular scenarios are considered: computing one approximate eigen-tuple or a complete set of approximate eigen-tuples. For computing one approximate eigen-tuple, an alternating iterative scheme with proven convergence is devised, while for a complete set of approximate eigen-tuples, the framework leads to a standard MEP (RMEP with square coefficient matrices) for numerical solutions. The proposed approach is validated on RMEPs from discretizing the multiparameter Sturm-Liouville equation and the Helmholtz equations by the least-squares spectral method.

math.NA↗

arXiv2025

Rational minimax approximation of matrix-valued functions

In this paper, we present a rigorous framework for rational minimax approximation of matrix-valued functions that generalizes classical scalar approximation theory. Given sampled data $\{(x_\ell, {F}(x_\ell))\}_{\ell=1}^m$ where ${F}:\mathbb{C} \to \mathbb{C}^{s \times t}$ is a matrix-valued function, we study the problem of finding a matrix-valued rational approximant ${R}(x) = {P}(x)/q(x)$ (with ${P}:\mathbb{C} \to \mathbb{C}^{s \times t}$ a matrix-valued polynomial and $q(x)$ a nonzero scalar polynomial of prescribed degrees) that minimizes the worst-case Frobenius norm error over the given nodes: $$ \inf_{{R}(x) = {P}(x)/q(x)} \max_{1 \leq \ell \leq m} \|{F}(x_\ell) - {R}(x_\ell)\|_{\rm F}. $$ By reformulating this min-max optimization problem through Lagrangian duality, we derive a maximization dual problem over the probability simplex. We analyze weak and strong duality properties and establish a sufficient condition ensuring that the solution of the dual problem yields the minimax approximant $R(x)$. For numerical implementation, we propose an efficient method (\textsf{m-d-Lawson}) to solve the dual problem, generalizing Lawson's iteration to matrix-valued functions. Convergence analysis of \textsf{m-d-Lawson} is established. Numerical experiments are conducted and compared to state-of-the-art approaches, demonstrating its efficiency as a novel computational framework for matrix-valued rational approximation.

math.NA↗

arXiv2024

A convergence analysis of Lawson's iteration for computing polynomial and rational minimax approximations

Lawson's iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson's iteration for the rational minimax approximation problem with both computationally high efficiency and theoretical guarantee is challenging. A recent work [L.-H. Zhang, L. Yang, W. H. Yang and Y.-N. Zhang, A convex dual problem for the rational minimax approximation and Lawson's iteration, Math. Comp., 94(2025), 2457-2494.] reveals that Lawson's iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation problem, and a new type of Lawson's iteration, namely, d-Lawson, was proposed, which reduces to the classical Lawson's iteration for the linear minimax approximation problem. For the rational case, such a dual problem is guaranteed to obtain the original minimax solution under Ruttan's sufficient condition, and numerically, d-Lawson was observed to converge monotonically with respect to the dual objective function. In this paper, we present a theoretical convergence analysis of d-Lawson for both the linear and rational minimax approximation problems. In particular, we show that (i) for the linear minimax approximation problem, $\beta=1$ is a near-optimal Lawson exponent in Lawson's iteration, and (ii) for the rational minimax approximation problem, under certain conditions, d-Lawson converges monotonically with respect to the dual objective function for any sufficiently small $\beta>0$, and the limiting approximant satisfies the complementary slackness condition: any node associated with positive weight either is an interpolation point or has a constant error.

math.NA↗