arXiv subjects

Zhongxiao Jia

Publications and source records attributed to Zhongxiao Jia.

At least 19 recordsLinked to original sources

arXiv2026

Stable and effective restarting of the SS--RR and SS--RRR methods for computing partial eigenpairs of Hermitian matrices

We consider stable and effective restarts of the SS--RR and SS--RRR methods for computing the eigenpairs of a large Hermitian matrix $A\in\mathbb{C}^{n\times n}$ with the $n_{ev}$ eigenvalues in a given interval. Taking the CJ--SS--RR and CJ--SS--RRR methods for instance, we prove that the commonly used restart scheme and its orthonormalized variant, named the natural restart schemes, are not numerically stable, thereby lowering attainable accuracy of approximate eigenpairs and missing some of the $n_{ev}$ desired eigenpairs. To cure these deficiencies, we propose a new restart scheme, called the combination restart, that takes the $n\times \ell$ starting matrix to be the product of the matrix, which consists of the selected $n_{ev}$ approximate eigenvectors, and an $n_{ev}\times \ell$ Gaussian random matrix. We prove the remarkable mathematical and numerical superiority of the combination restart over the natural restarts, then present a hybrid restart scheme that fully exploits the advantages of the natural restart and combination restart, and finally develop CJ--SS--RR and CJ--SS--RRR algorithms with the hybrid restart. Numerical experiments confirm that the new algorithms are numerically backward stable to the working precision and more efficient than the CJ--SS--RRR algorithm with the natural restart. The combination restart and hybrid restart schemes can be straightforwardly extended to the SS--RR and SS--RRR methods for general standard and generalized eigenvalue problems.

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arXiv2026

The Refined Joint Bidiagonalization Method and an Implicitly Restarted Algorithm for Large GSVD Computations

We make a convergence analysis on the joint bidiagonalization (JBD) method that computes several extreme generalized singular value decomposition (GSVD) components of a regular matrix pair $\{A,L\}$, and show that the right and left Ritz vectors obtained by it may converge erratically and even may fail to converge, while Ritz values converge. These convergence results hold for a class of general Rayleigh--Ritz projection methods for the GSVD problem under the hypothesis that the deviation of a desired right generalized singular vector from the right subspace tends to zero. We prove the interlacing property of Ritz values and generalized singular values, and extend it to the generalized singular values of $\{A,L\}$ and the matrix pairs consisting of subsets of its columns. To overcome the irregular convergence or possible non-convergence of the JBD method, we nontrivially extend the refined Rayleigh--Ritz projection for the eigenvalue problem to the GSVD problem, and propose a refined JBD (RJBD) method that replaces the right Ritz vectors by new approximations, called the right refined Ritz vectors, satisfying certain residual optimality; we define new approximate left generalized singular vectors, called the left refined Ritz vectors. We prove that the left and right refined Ritz vectors unconditionally converge under the same hypothesis. We extend the implicit restarting scheme to the RJBD method, and develop an implicitly restarted RJBD algorithm with the refined shifts proposed. Numerical experiments illustrate that the new algorithm is at least competitive and often considerably more efficient than the implicitly restarted JBD algorithm.

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arXiv2026

A refined CJ--SS--RR method with a reliable removal approach of spurious Ritz values for the Hermitian eigenvalue problem

Under the hypothesis that the deviations of the desired eigenvectors of the matrix $A$ from the underlying subspace tend to zero, the Ritz vectors may not converge and have poor or little accuracy. This phenomenon is not unusual and particularly occurs when the associated Ritz values are close, which is independent of the eigenvalue distribution of $A$. For the (block) SS--RR methods, there are possibly {\em more} Ritz values that converge to the same desired eigenvalue(s) counting multiplicity in the region of interest, meaning that some of the Ritz values must be spurious and the corresponding residual norms of the Ritz pairs may not be small. Consequently, the (block) SS--RR methods including the CJ--SS--RR method cannot base on the corresponding residual norms to effectively identify if the Ritz values in the region are genuine or spurious. This paper proposes refined SS--RR, abbreviated as SS--RRR, methods based on the refined Rayleigh--Ritz projection that compute the eigenpairs of large matrices with the eigenvalues located in the given region. We present a new approach to accurately implement the RRR methods more efficiently than ever before for a general subspace.Exploiting the unconditional convergence of the refined Ritz vectors when the subspace is sufficiently accurate, we propose a tune-free removal approach to effectively remove spurious Ritz values with a rigorous theory supported, and develop a restarted CJ--SS--RRR algorithm. Numerical experiments show that the restarted CJ--SS--RRR algorithm is more efficient and effective than the restarted CJ--SS--RR algorithm.

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arXiv2026

New Jacobi--Davidson type methods for the large SVD computations

In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, by fully exploiting useful information from current subspaces, a new effective correction equation is derived at each outer iteration, leading to a new variant of JDSVD, called JDSVD-V. It is proved that JDSVD-V retains the same convergence of the outer iterations as JDSVD. A substantial advantage of JDSVD-V over JDSVD is that the new correction equations in JDSVD-V are much easier to iteratively solve than the standard ones in JDSVD: the MINRES method for the new correction equations converges much faster when there is a cluster of singular values closest to a given target, a typical case in applications. A new thick-restart JDSVD-V algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets. Numerical experiments justify the theory and illustrate the considerable superiority of JDSVD-V to JDSVD, and demonstrate that a similar two-stage JDSVD-V algorithm substantially outperforms the most advanced PRIMME\_SVDS software nowadays for computing the smallest singular triplets.

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arXiv2026

A Numerical Analysis of Sketched Linear Squares Problems and Stopping Criteria for Iterative Solvers

Randomized subspace embedding methods have had a great impact on the solution of a linear least squares (LS) problem by reducing its row dimension, leading to a randomized or sketched LS (sLS) problem, and use the solution of the sLS problem as an approximate solution of the LS problem. This work makes a numerical analysis on the sLS problem, establishes its numerous theoretical properties, and show their crucial roles on the most effective and efficient use of iterative solvers. We first establish a compact bound on the norm of the residual difference between the solutions of the LS and sLS problems, which is the first key result towards understanding the rationale of the sLS problem. Then from the perspective of backward errors, we prove that the solution of the sLS problem is the one of a certain perturbed LS problem with minimal backward error, and quantify how the embedded quality affects the residuals, solution errors, and the relative residual norms of normal equations of the LS and sLS problems. These theoretical results enable us to propose new novel and reliable general-purpose stopping criteria for iterative solvers for the sLS problem, which dynamically monitor stabilization patterns of iterative solvers for the LS problem itself and terminate them at the earliest iteration. Numerical experiments justify the theoretical bounds and demonstrate that the new stopping criteria work reliably and result in a tremendous reduction in computational cost without sacrificing attainable accuracy.

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arXiv2025

A Chebyshev--Jackson series based block SS--RR algorithm for computing partial eigenpairs of real symmetric matrices

This paper considers eigenpair computations of large symmetric matrices with the desired eigenvalues lying in a given interval using the contour integral-based block SS--RR method, a Rayleigh--Ritz projection onto a certain subspace generated by moment matrices. Instead of using a numerical quadrature to approximately compute the moments by solving a number of large shifted complex linear systems at each iteration, we make use of the Chebyshev--Jackson (CJ) series expansion to approximate the moments, which only involves matrix-vector products and avoids expensive solutions of the linear systems. We prove that the CJ series expansions pointwise converge to the moments as the series degree increases, but at different convergence rates depending on point positions and moment orders. These extend the available convergence results on the zeroth moment of CJ series expansions to higher order ones. Based on the results established, we develop a CJ--SS--RR algorithm. Numerical experiments illustrate that the new algorithm is more efficient than the contour integral-based block SS--RR algorithm with the trapezoidal rule.

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arXiv2025

An Implicitly Restarted Joint Bidiagonalization Algorithm for Large GSVD Computations

The joint bidiagonalization (JBD) process of a regular matrix pair $\{A,L\}$ is mathematically equivalent to two simultaneous Lanczos bidiagonalization processes of the upper and lower parts of the Q-factor of QR factorization of the stacked matrix $(A^{\mathrm T},\,L^{\mathrm T})^{\mathrm T}$ when their starting vectors are closely related in a specific way. The resulting JBD method for computing extreme generalized singular values and corresponding left and right singular vectors of $\{A,L\}$ realizes the standard Rayleigh--Ritz projection of the generalized singular value decomposition (GSVD) problem of $\{A,L\}$ onto the two left and one right subspaces generated by the JBD process. In this paper, the implicit restarting technique is nontrivially and skillfully extended to the JBD process, and an implicitly restarted JBD (IRJBD) algorithm is developed with proper selection of crucial shifts proposed and a few key implementation details addressed in finite precision arithmetic. Compact upper bounds are established for the residual norm of an approximate GSVD component in both exact and finite precision arithmetic, which are used to design efficient and reliable stopping criteria and avoid the expensive computation of approximate right generalized singular vectors. Numerical experiments illustrate that IRJBD performs well and is more efficient than the thick-restart JBD algorithm.

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arXiv2024

An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems

We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(λ_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms of the deviation $\varepsilon$ of $x_{*}$ from a given subspace $\mathcal{W}$, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector. The results show that, as $\varepsilon\rightarrow 0$, there exists a Ritz value that unconditionally converges to $λ_*$ and the corresponding refined Ritz vector does so too but the Ritz vector converges conditionally and it may fail to converge and even may not be unique. We also present an error bound for the approximate eigenvector in terms of the computable residual norm of a given approximate eigenpair, and give lower and upper bounds for the error of the refined Ritz vector and the Ritz vector as well as for that of the corresponding residual norms. These results nontrivially extend some convergence results on these two methods for the linear eigenvalue problem to the NEP. Examples are constructed to illustrate the main results.

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arXiv2023

A CJ-FEAST GSVDsolver for computing a partial GSVD of a large matrix pair with the generalized singular values in a given interval

We propose a CJ-FEAST GSVDsolver to compute a partial generalized singular value decomposition (GSVD) of a large matrix pair $(A,B)$ with the generalized singular values in a given interval. The solver is a highly nontrivial extension of the FEAST eigensolver for the (generalized) eigenvalue problem and CJ-FEAST SVDsolver for the SVD problem. For a partial GSVD problem, given three left and right searching subspaces, we propose a general projection method that works on $(A,B)$ {\em directly}, and computes approximations to the desired GSVD components. For the concerning GSVD problem, we exploit the Chebyshev--Jackson (CJ) series to construct an approximate spectral projector of the generalized eigenvalue problem of the matrix pair $(A^TA,B^TB)$ associated with the generalized singular values of interest, and use subspace iteration on it to generate a right subspace. Premultiplying it with $A$ and $B$ constructs two left subspaces. Applying the general projection method to the subspaces constructed leads to the CJ-FEAST GSVDsolver. We derive accuracy estimates for the approximate spectral projector and its eigenvalues, and establish a number of convergence results on the underlying subspaces and the approximate GSVD components obtained by the CJ-FEAST GSVDsolver. We propose general-purpose choice strategies for the series degree and subspace dimension. Numerical experiments illustrate the efficiency of the CJ-FEAST GSVDsolver.

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arXiv2023

Refined and refined harmonic Jacobi--Davidson methods for computing several GSVD components of a large regular matrix pair

Three refined and refined harmonic extraction-based Jacobi--Davidson (JD) type methods are proposed, and their thick-restart algorithms with deflation and purgation are developed to compute several generalized singular value decomposition (GSVD) components of a large regular matrix pair. The new methods are called refined cross product-free (RCPF), refined cross product-free harmonic (RCPF-harmonic) and refined inverse-free harmonic (RIF-harmonic) JDGSVD algorithms, abbreviated as RCPF-JDGSVD, RCPF-HJDGSVD and RIF-HJDGSVD, respectively. The new JDGSVD methods are more efficient than the corresponding standard and harmonic extraction-based JDSVD methods proposed previously by the authors, and can overcome the erratic behavior and intrinsic possible non-convergence of the latter ones. Numerical experiments illustrate that RCPF-JDGSVD performs better for the computation of extreme GSVD components while RCPF-HJDGSVD and RIF-HJDGSVD suit better for that of interior GSVD components.

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arXiv2023

An augmented matrix-based CJ-FEAST SVDsolver for computing a partial singular value decomposition with the singular values in a given interval

The cross-product matrix-based CJ-FEAST SVDsolver proposed previously by the authors is shown to compute the left singular vector possibly much less accurately than the right singular vector and may be numerically backward unstable when a desired singular value is small. In this paper, an alternative augmented matrix-based CJ-FEAST SVDsolver is considered to compute the singular triplets of a large matrix $A$ with the singular values in an interval $[a,b]$ contained in the singular spectrum. The new CJ-FEAST SVDsolver is a subspace iteration applied to an approximate spectral projector of the augmented matrix $[0, A^T; A, 0]$ associated with the eigenvalues in $[a,b]$, and constructs approximate left and right singular subspaces with the desired singular values independently, onto which $A$ is projected to obtain the Ritz approximations to the desired singular triplets. Compact estimates are given for the accuracy of the approximate spectral projector, and a number of convergence results are established. The new solver is proved to be always numerically backward stable. A convergence comparison of the cross-product and augmented matrix-based CJ-FEAST SVDsolvers is made, and a general-purpose choice strategy between the two solvers is proposed for the robustness and overall efficiency. Numerical experiments confirm all the results.

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arXiv2022

A skew-symmetric Lanczos bidiagonalization method for computing several largest eigenpairs of a large skew-symmetric matrix

The spectral decomposition of a real skew-symmetric matrix $A$ can be mathematically transformed into a specific structured singular value decomposition (SVD) of $A$. Based on such equivalence, a skew-symmetric Lanczos bidiagonalization (SSLBD) method is proposed for the specific SVD problem that computes extreme singular values and the corresponding singular vectors of $A$, from which the eigenpairs of $A$ corresponding to the extreme conjugate eigenvalues in magnitude are recovered pairwise in real arithmetic. A number of convergence results on the method are established, and accuracy estimates for approximate singular triplets are given. In finite precision arithmetic, it is proven that the semi-orthogonality of each set of basis vectors and the semi-biorthogonality of two sets of basis vectors suffice to compute the singular values accurately. A commonly used efficient partial reorthogonalization strategy is adapted to maintaining the needed semi-orthogonality and semi-biorthogonality. For a practical purpose, an implicitly restarted SSLBD algorithm is developed with partial reorthogonalization. Numerical experiments illustrate the effectiveness and overall efficiency of the algorithm.

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arXiv2022

A FEAST SVDsolver based on Chebyshev--Jackson series for computing partial singular triplets of large matrices

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of $A^TA$ corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev--Jackson polynomial series, which are symmetric positive semi-definite with the eigenvalues in $[0,1]$. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, establish numerous convergence results on the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and for reliably determining the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that our FEAST SVDsolver is at least competitive with and is much more efficient than the contour integral-based FEAST SVDsolver when the desired singular values are extreme and interior ones, respectively, and it is also more robust than the latter.

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arXiv2022

Two harmonic Jacobi--Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair

Two harmonic extraction based Jacobi--Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair. They are called cross product-free (CPF) and inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD algorithm, the harmonic extraction based algorithms converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some deflation and purgation techniques are developed to compute more than one GSVD components. Numerical experiments confirm the superiority of CPF-HJDGSVD and IF-HJDGSVD to the standard extraction based JDGSVD algorithm.

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arXiv2022

The joint bidiagonalization method for large GSVD computations in finite precision

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair $\{A,L\}$, where we propose three approaches to compute approximate generalized singular values and vectors. We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of $\{A,L\}$. The results show that, under some mild conditions, the semiorthogonality of Lanczos type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We also establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.

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arXiv2022

Theoretical and Computable Optimal Subspace Expansions for Matrix Eigenvalue Problems

Consider the optimal subspace expansion problem for the matrix eigenvalue problem $Ax=λx$: Which vector $w$ in the current subspace $\mathcal{V}$, after multiplied by $A$, provides an optimal subspace expansion for approximating a desired eigenvector $x$ in the sense that $x$ has the smallest angle with the expanded subspace $\mathcal{V}_w=\mathcal{V}+{\rm span}\{Aw\}$, i.e., $w_{opt}=\arg\max_{w\in\mathcal{V}}\cos\angle(\mathcal{V}_w,x)$? This problem is important as many iterative methods construct nested subspaces that successively expand $\mathcal{V}$ to $\mathcal{V}_w$. An expression of $w_{opt}$ by Ye (Linear Algebra Appl., 428 (2008), pp. 911--918) for $A$ general, but it could not be exploited to construct a computable (nearly) optimally expanded subspace. He turns to deriving a maximization characterization of $\cos\angle(\mathcal{V}_w,x)$ for a {\em given} $w\in \mathcal{V}$ when $A$ is Hermitian. We generalize Ye's maximization characterization to the general case and find its maximizer. Our main contributions consist of explicit expressions of $w_{opt}$, $(I-P_V)Aw_{opt}$ and the optimally expanded subspace $\mathcal{V}_{w_{opt}}$ for $A$ general, where $P_V$ is the orthogonal projector onto $\mathcal{V}$. These results are fully exploited to obtain computable optimally expanded subspaces within the framework of the standard, harmonic, refined, and refined harmonic Rayleigh--Ritz methods. We show how to efficiently implement the proposed subspace expansion approaches. Numerical experiments demonstrate the effectiveness of our computable optimal expansions.

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arXiv2021

A cross-product free Jacobi-Davidson type method for computing a partial generalized singular value decomposition (GSVD) of a large matrix pair

A Cross-Product Free (CPF) Jacobi-Davidson (JD) type method is proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair $(A,B)$. It implicitly solves the mathematically equivalent generalized eigenvalue problem of $(A^TA,B^TB)$ but does not explicitly form the cross-product matrices and thus avoids the possible accuracy loss of the computed generalized singular values and generalized singular vectors. The method is an inner-outer iteration method, where the expansion of the right searching subspace forms the inner iterations that approximately solve the correction equations involved and the outer iterations extract approximate GSVD components with respect to the subspaces. Some convergence results are established for the inner and outer iterations, based on some of which practical stopping criteria are designed for the inner iterations. A thick-restart CPF-JDGSVD algorithm with deflation is developed to compute several GSVD components. Numerical experiments illustrate the efficiency of the algorithm.

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arXiv2021

A comparison of eigenvalue-based algorithms and the generalized Lanczos trust-region algorithm for Solving the trust-region subproblem

Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. Based on a fundamental result that the solution of TRS of size $n$ is mathematically equivalent to finding the rightmost eigenpair of a certain matrix pair of size $2n$, eigenvalue-based methods are promising due to their simplicity. For $n$ large, the implicitly restarted Arnoldi (IRA) and refined Arnoldi (IRRA) algorithms are well suited for this eigenproblem. For a reasonable comparison of overall efficiency of the algorithms for solving TRS directly and eigenvalue-based algorithms, a vital premise is that the two kinds of algorithms must compute the approximate solutions of TRS with (almost) the same accuracy, but such premise has been ignored in the literature. To this end, we establish close relationships between the two kinds of residual norms, so that, given a stopping tolerance for IRA and IRRA, we are able to determine a reliable one that GLTR should use so as to ensure that GLTR and IRA, IRRA deliver the converged approximate solutions with similar accuracy. We also make a convergence analysis on the residual norms by the Generalized Lanczos Trust-Region (GLTR) algorithm for solving TRS directly, the Arnoldi method and the refined Arnoldi method for the equivalent eigenproblem. A number of numerical experiments are reported to illustrate that IRA and IRRA are competitive with GLTR and IRRA outperforms IRA.

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