arXiv subjects

Shao-Liang Zhang

Publications and source records attributed to Shao-Liang Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Factorized Krylov subspace methods for solving large Sylvester equations

Krylov subspace methods, such as the Conjugate Gradient (CG) and BiCGSTAB methods, are widely used in scientific computing for solving linear systems. In this study, we propose a new framework for solving large Sylvester equations in a low-rank format by reconstructing matrix-oriented Krylov subspace methods. The framework realizes efficient algorithms that are mathematically equivalent to the matrix-oriented Krylov subspace methods by exploiting the mathematical properties of the Sylvester operator and the low-rank structure of the right-hand side. Specifically, by leveraging these properties, approximate solutions can be expressed in a low-rank factorized form, enabling efficient computation and reduced memory requirements. The effectiveness of our algorithms is demonstrated through numerical experiments.

math.NA↗

arXiv2026

Error control technique of quadrature-based algorithms for the action of real powers of a Hermitian positive-definite matrix

This study considers quadrature-based algorithms to compute $A^α\boldsymbol{b}$, the action of a real power of a Hermitian positive-definite matrix $A$ on a vector $ \boldsymbol{b}$. In these algorithms, the computation of an integral representation of $A^α \boldsymbol{b}$ is reduced to solving several tens or hundreds of shifted linear systems. Current approaches usually analyze the quadrature discretization error, but rarely take into account the additional error introduced by solving these shifted linear systems with iterative solvers. Here, we bound this error with the residual of the approximated solution of these linear systems. This allows the derivation of a stopping criterion for iterative solvers to keep the error of $A^α\boldsymbol{b}$ below a prescribed error tolerance. Numerical results demonstrate that the proposed criterion enables the computation of $A^α\boldsymbol{b}$ within prescribed tolerance limits.

math.NA↗

arXiv2025

A preconditioning technique of Gauss--Legendre quadrature for the logarithm of symmetric positive definite matrices

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between $130$ and $3.0\times 10^5$.

math.NA↗

arXiv2024

A tensor Alternating Anderson-Richardson method for solving multilinear systems with M-tensors

It is well-known that a multilinear system with a nonsingular M-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson-Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson-Richardson method, finally, we periodically employ the tensor Anderson-Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.

math.NA↗

arXiv2023

Matrix equation representation of the convolution equation and its unique solvability

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.

math.NA↗

arXiv2023

Computing the matrix exponential with the double exponential formula

This paper considers the computation of the matrix exponential $\mathrm{e}^A$ with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore it is necessary to truncate the infinite interval. In this paper, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates $\mathrm{e}^A$ with the fixed mesh size which is a parameter in the DE formula affecting the accuracy. Second one computes $\mathrm{e}^A$ based on the first one with automatic selection of the mesh size depending on the given error tolerance.

math.NA↗

arXiv2021

Computing the matrix fractional power with the double exponential formula

Two quadrature-based algorithms for computing the matrix fractional power $A^α$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^α$, is proposed. Then, two algorithms are presented -- one computes $A^α$ with a fixed number of abscissas, and the other computes $A^α$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $α$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.

math.NA↗

arXiv2020

Emergent periodic and quasiperiodic lattices on surfaces of synthetic Hall tori and synthetic Hall cylinders

Synthetic spaces allow physicists to bypass constraints imposed by certain physical laws in experiments. Here, we show that a synthetic torus, which consists of a ring trap in the real space and internal states of ultracold atoms cyclically coupled by Laguerre-Gaussian Raman beams, could be threaded by a net effective magnetic flux through its surface---an impossible mission in the real space. Such synthetic Hall torus gives rise to a periodic lattice in the real dimension, in which the periodicity of density modulation of atoms fractionalizes that of the Hamiltonian. Correspondingly, the energy spectrum is featured by multiple bands grouping into clusters with nonsymmorphic symmetry protected band crossings in each cluster, leading to braidings of wavepackets in Bloch oscillations. Our scheme allows physicists to glue two synthetic Hall tori such that localization may emerge in a quasicrystalline lattice. If the Laguerre-Gaussian Raman beams and ring traps were replaced by linear Raman beams and ordinary traps, a synthetic Hall cylinder could be realized and deliver many of the aforementioned phenomena.

cond-mat.quant-gas↗

arXiv2019

Modified Strang splitting for semilinear parabolic problems

We consider applying the Strang splitting to semilinear parabolic problems. The key ingredients of the Strang splitting are the decomposition of the equation into several parts and the computation of approximate solutions by combining the time evolution of each split equation. However, when the Dirichlet boundary condition is imposed, order reduction could occur due to the incompatibility of the split equations with the boundary condition. In this paper, to overcome the order reduction, a modified Strang splitting procedure is presented for the one-dimensional semilinear parabolic equation with first-order spatial derivatives, like the Burgers equation.

math.NA↗

arXiv2019

A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation

We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.

math.NA↗

arXiv2019

Algorithms for the computation of the matrix logarithm based on the double exponential formula

We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss--Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine a suitable finite integration interval, which provides the required accuracy and efficiency. In this paper, we present a method for selecting a suitable finite interval based on an error analysis as well as two algorithms, and one of these algorithms addresses error control.

math.NA↗

arXiv2019

Adaptive SOR methods based on the Wolfe conditions

Because the expense of estimating the optimal value of the relaxation parameter in the successive over-relaxation (SOR) method is usually prohibitive, the parameter is often adaptively controlled. In this paper, new adaptive SOR methods are presented that are applicable to a variety of symmetric positive definite linear systems and do not require additional matrix-vector products when updating the parameter. To this end, we regard the SOR method as an algorithm for minimising a certain objective function, which yields an interpretation of the relaxation parameter as the step size following a certain change of variables. This interpretation enables us to adaptively control the step size based on some line search techniques, such as the Wolfe conditions. Numerical examples demonstrate the favourable behaviour of the proposed methods.

math.NA↗

arXiv2019

Relation between the T-congruence Sylvester equation and the generalized Sylvester equation

The T-congruence Sylvester equation is the matrix equation $AX+X^{\mathrm{T}}B=C$, where $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times m}$ are given, and $X\in\mathbb{R}^{n\times m}$ is to be determined. Recently, Oozawa et al. discovered a transformation that the matrix equation is equivalent to one of the well-studied matrix equations (the Lyapunov equation); however, the condition of the transformation seems to be too limited because matrices $A$ and $B$ are assumed to be square matrices ($m=n$). In this paper, two transformations are provided for rectangular matrices $A$ and $B$. One of them is an extension of the result of Oozawa et al. for the case $m\ge n$, and the other is a novel transformation for the case $m\le n$.

math.NA↗

arXiv2018

Techniques for Accelerating the Convergence of Restarted GMRES Based on the Projection

In this paper, we study the restarted Krylov subspace method, which is typically represented by the GMRES(m) method. Our work mainly focused on the amount of change in the iterative solution of GMRES(m) at each restart. We propose an extension of the GMRES(m) method based on the idea of projection. The algorithm is named as LGMRES. In addition, LLBGMRE method is also obtained by adding backtracking restart technology to LGMRES. Theoretical analysis and numerical experiments show that LGMRES and LLBGMRES have better convergence than traditional restart GMRES(m) method.

math.NA↗

arXiv2018

Solution of the $k$-th eigenvalue problem in large-scale electronic structure calculations

We consider computing the $k$-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of $n\times n$ large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index $k.$ We present a three-stage algorithm for computing the $k$-th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the $k$-th eigenvalue $(1 \ll k \ll n)$ with a non-standard application of the Lanczos method. In the second stage, spectral bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the $k$-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the $k$-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm.

math.NA↗

arXiv2017

On the equivalence between SOR-type methods for linear systems and discrete gradient methods for gradient systems

The iterative nature of many discretisation methods for continuous dynamical systems has led to the study of the connections between iterative numerical methods in numerical linear algebra and continuous dynamical systems. Certain researchers have used the explicit Euler method to understand this connection, but this method has its limitation. In this study, we present a new connection between successive over-relaxation (SOR)-type methods and gradient systems; this connection is based on discrete gradient methods. The focus of the discussion is the equivalence between SOR-type methods and discrete gradient methods applied to gradient systems. The discussion leads to new interpretations for SOR-type methods. For example, we found a new way to derive these methods; these methods monotonically decrease a certain quadratic function and obtain a new interpretation of the relaxation parameter. We also obtained a new discrete gradient while studying the new connection.

math.NA↗

arXiv2016

A cost-efficient variant of the incremental Newton iteration for the matrix $p$th root

Incremental Newton (IN) iteration, proposed by Iannazzo, is stable for computing the matrix $p$th root, and its computational cost is $\mathcal{O}(n^3p)$ flops per iteration. In this paper, a cost-efficient variant of IN iteration is presented. The computational cost of the variant well agrees with $\mathcal{O} (n^3 \log p)$ flops per iteration, if $p$ is up to at least 100.

math.NA↗

arXiv2016

Weyl points and topological nodal superfluids in a face-centered cubic optical lattice

We point out that a face-centered cubic (FCC) optical lattice, which can be realised by a simple scheme using three lasers, provides one a highly controllable platform for creating Weyl points and topological nodal superfluids in ultracold atoms. In non-interacting systems, Weyl points automatically arise in the Floquet band structure when shaking such FCC lattices, and sophisticated design of the tunnelling is not required. More interestingly, in the presence of attractive interaction between two hyperfine spin states, which experience the same shaken FCC lattice, a three-dimensional topological nodal superfluid emerges, and Weyl points show up as the gapless points in the quasiparticle spectrum. One could either create a double Weyl point of charge 2, or split it to two Weyl points of charge 1, which can be moved in the momentum space by tuning the interactions. Correspondingly, the Fermi arcs at the surface may be linked with each other or separated as individual ones.

cond-mat.quant-gas↗