在矩阵（广义）特征值问题、（广义）奇异值分解问题和一般非线性矩阵特征值问题的数值解法的理论和算法领域、离散不适定和反问题的正则化理论和数值解法领域等做出了系统的、基础性的、有重要国际影响的研究成果。所提出的精化Rayleigh-Ritz方法与传统的标准Rayleigh-Ritz方法和调和Rayleigh-Ritz方法一道，自2000年以来被公认为是求解这大规模矩阵特征值问题和奇异值分解问题的三类投影方法之一。对于非对称情形的特征值问题，首次建立了这三类方法的普适性收敛性理论，适用于它们所包含的所有具体的方法。国际计算数学界权威Stewart的经典专著“Matrix Algorithms: Vol. II Eigensystems, SIAM, Philadelphia, 2001”（470页）和国际著名计算数学家van der Vorst的专著“Computational Methods for Large Eigenvalue Problems, North-Holland (Elsevier), 2002”（177页）分别用10页多和4页多的篇幅系统描述和讨论贾仲孝的精化投影方法。此外，在线性最小二乘和总体最小二乘问题的扰动理论、信頼域子问题的数值求解方法研究、稀疏线性方程组的迭代法和有效预处理技术等领域均做出国际水平的研究成果，解决了若干关键问题。1995-2022年期间，在Inverse Problems, Mathematics of Computation, Numerische Mathematik, SIAM Journal on Matrix Analysis and Applications, SIAM Journal on Optimization, SIAM Journal on Scientific Computing等国际顶尖和著名知名杂志上发表论文70余篇。研究成果受到国际学术界的广泛关注，引发了大量的后续研究，被43个国家和地区的1200余名专家和研究人员在19部经典著作、专著和教材及1000篇论文中引用1500多篇次。引用的书目包括 Bai、Demmel、Dongarra、Ruhe、van der Vorst 等五人编辑的 “Templates for the Solution of Algebraic Eigenvalue Problems: a Practical Guide ”(2000)，Golub & van Loan 的经典著作“Matrix Computations” 第三、第四版 (1996, 2013)，Stewart 的经典著作“Matrix Algorithms II: Eigensystems ”(2001)，Bjorck 的专著 Numerical Methods in Matrix Computations (2015)，van der Vorst 的专著 “Computational Methods for Large Eigenvalue Problems” (2002)，Trefethen & Embree 的专著“Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators” (2005)，Meurant & Tebbens 的专著 Krylov Methods for Nonsymmetric Linear Systems ”(2020)，Quarteroni、Sacco & Saleri 的专著 Numerical Mathematics (2000)，Brezinski、Meurant & Revido-Zaglia 的著作 “A Journey Through the History of Numerical Linear Algebra” (2022).
主要论著：
[1] The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM Journal on Matrix Analysis and Applications, 16 (3) (1995), 843--862.
[2] A block incomplete orthogonalization method for large nonsymmetric eigenproblems, BIT, 34 (4) (1995), 516--539.
[3] On IOM(q): the incomplete orthogonalization method for large unsymmetric linear systems, Numerical Linear Algebra with Applications, 3 (6) (1996), 491--512.
[4] Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems, Linear Algebra and Its Applications,259 (1997), 1--23.
[5] A refined iterative algorithm based on the block Arnoldi process for large unsymmetric eigenproblems, Linear Algebra and Its Applications, 270 (1998), 171--189.
[6] Generalized block Lanczos methods for large unsymmetric eigenproblems, Numerische Mathematik, 80 (2) (1998), 239--266.
[7] On IGMRES: an incomplete generalized minimal residual method for large unsymmetric linear systems, Science in China (Series A), 41 (12)(1998), 1178--1188.
[8] A variation on the block Arnoldi method for large unsymmetric eigenproblems, Acta Mathematica Applicatae Sinica, 14 (4) (1998), 425--432.
[9] The convergence of Krylov subspace methods for large unsymmetric linear systems, Acta Mathematica Sinica-New Series, 14 (4) (1998), 507--518.
[10] Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm, Linear Algebra and Its Applications, 287 (1999), 191--214.
[11] Composite orthogonal projection methods for large matrix eigenproblems, Science in China (Series A), 42 (6) (1999), 577--585.
[12] Arnoldi type algorithms for large unsymmetric multiple eigenvalue problems, Journal of Computational Mathematics，17 (3) (1999), 257--274.
[13] A refined subspace iteration algorithm for large sparse eigenproblems, Applied Numerical Mathematics，32(1)(2000), 35--52.
[14] Some recursions on Arnoldi's method and IOM for large non-Hermitian linear systems, Computers and Mathematics with Applications, 39 (3/4) (2000), 125--129.
[15] Jia Z. and Elsner L., Improving eigenvectors in Arnoldi's method, Journal of Computational Mathematics, 18 (3) (2000), 365--376.
[16] Jia Z. and Stewart G.W., An analysis of the Rayleigh-Ritz method for approximating eigenspaces, Mathematics of Computation，70(234) (2001), 637--647.
[17] On residuals of refined projection methods for large matrix eigenproblems, Computers and Mathematics with Applications, 41 (7/8) (2001), 813--820.
[18] The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices, Applied Numerical Mathematics, 42 (4) (2002), 489--512.
[19] Chen G. and Jia Z. A reverse order implicit Q-theorem and the Arnoldi process, Journal of Computational Mathematics, 20 (5) (2002), 519--524.
[20] Jia Z. and Zhang Y., A refined invert-and-shift Arnoldi algorithm for large generalized unsymmetric eigenproblems, Computers and Mathematics with Applications, 44 (8/9) (2002), 1117--1127.
[21] Jia Z. and Niu D., An implicitly restarted refined bidiagonalization Lanczos method for computing a partial singular value decomposition, SIAM Journal on Matrix Analysis and Applications, 25(1) (2003), 246--265.
[22] Chen G and Jia Z, Theoretical and numerical comparisons of GMRES and WZ-GMRES, Computers and Mathematics with Applications, 47 (8/9) (2004), 1335-1350.
[23] Chen G and Jia Z., An analogue of the results of Saad and Stewart for harmonic Ritz vectors, Journal of Computational and Applied Mathematics,167 (2004), 493--498.
[24] Some theoretical comparisons of refined Ritz vectors and Ritz vectors, Science in China, Series A, 47 (Suppl.) (2004), 222--233.
[25] Feng S. and Jia Z., A refined Jacobi-Davidson method and its correction equation, Computers and Mathematics with Applications, 49 (2/3) (2005), 417--427.
[26] The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors, Mathematics of Computation, 74 (251) (2005), 1441--1456.
[27] Chen G. and Jia Z., A refined harmonic Rayleigh-Ritz procedure and an explicitly restarted refined harmonic Arnoldi algorithm, Mathematical and Computer Modelling, 41 (2005), 615--627.
[28] Using cross-product matrices to compute the SVD, Numerical Algorithms, 42 (1) (2006), 31--61.
[29] Jia Z. and Sun Y., A QR decomposition based solver for the least squares problem from the minimal residual method, Journal of Computational Mathematics, 25 (5) (2007), 531--542.
[30] 贾仲孝，王震，非精确Rayleigh商迭代和非精确的简化Jacobi-Davidson方法的收敛性分析，中国科学,A辑，38 (4) (2008): 365-376. Jia Z. and Wang Z., A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem, Science in China Series A, 51 (12) (2008), 2205--2216.
[31] Jia Z. and Zhu B., A power sparse approximate inverse preconditioning procedure for large linear systems, Numerical Linear Algebra with Applications, 16 (4) (2009), 259--299.
[32] Applications of the Conjugate Gradient (CG) method in optimal surface parameterizations, International Journal of Computer Mathematics, 87 (5) (2010), 1032--1039.
[33] Jia Z. and Niu D., A refined harmonic Lanczos bidiagonalization method and an implicitly restarted algorithm for computing the smallest singular triplets of large matrices, SIAM Journal on Scientific Computing, 32 (2) (2010), 714--744.
[34] Some properties of LSQR for large sparse linear least squares problems, Journal of Systems Science and Complexity, 23 (4) (2010), 815--821.
[35] Duan C. and Jia Z., A global harmonic Arnoldi method for large non-Hermitian eigenproblems with an application to multiple eigenvalue problems, Journal of Computational and Applied Mathematics, 234 (2010), 845--860.
[36] Chu, E K.-W., Fan H.Y., Jia Z., Li T. and Lin W.-W, The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs, Journal of Computational and Applied Mathematics, 235 (2011), 2626--2639.
[37] Duan D and Jia Z., A global Arnoldi method for large non-Hermitian eigenproblems with special applications to multiple eigenproblems, Taiwanese Journal of Mathematics, 15 (4) (2011), 1497--1525.
[38] Li B. and Jia Z., Some results on condition numbers of the scaled total least squares problems, Linear Algebra and Its Applications, 435 (3) (2011), 674—686.
[39] On convergence of the inexact Rayleigh quotient iteration with MINRES, Journal of Computational and Applied Mathematics, 236 (2012): 4276—4295.
[40] Jia Z. and Sun Y., SHIRRA: A refined variant of SHIRA for the Skew-Hamiltonian/Hamiltonian (SHH) pencil eigenvalue problem, Taiwanese Journal of Mathematics, 17 (1) (2013): 259--274. (SCI)
[41] On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems, Science China Mathematics, 56 (10) (2013): 2145--2160.
[42] Jia Z. and Li B., On the condition number of the total least squares problem, Numerische Mathematik, 125 (1) (2013), 61--87.
[43] Jia Z. and Zhang Q., An approach to making SPAI and PSAI preconditioning effective for large irregular sparse linear systems, SIAM Journal on Scientific Computing, 35 (4) (2013), A1903--A1927.
[44] Huang T-M, Jia Z. and Lin W-W., On the convergence of Ritz pairs and refined Ritz vectors for quadratic eigenvalue problems, BIT Numerical Mathematics, 53 (4) (2013), 941--958.
[45] Jia Z. and Zhang Q., Robust dropping criteria for F-norm minimization based sparse approximate inverse preconditioning, BIT Numerical Mathematics, 53 (4) (2013), 959--985.
[46] Jia Z. and Li C., Inner iterations in the shift-invert residual Arnoldi method and the Jacobi--Davidson method, Science China Mathematics, 57 (8) (2014), 1733--1752.
[47] Jia Z. and Li C., Harmonic and refined harmonic shift-invert residual Arnoldi and Jacobi--Davidson methods for interior eigenvalue problems, Journal of Computational and Applied Mathematics, 282 (2015), 83--97.
[48] Jia Z. and Sun Y., Implicitly restarted generalized second-order Arnoldi type algorithms for the quadratic eigenvalue problem, Taiwanese Journal of Mathematics, 19 (1) (2015), 1-30.
[49] Jia Z. and Lv H., A posteriori error estimates of Krylov subspace approximations to matrix functions, Numerical Algorithms, 69 (1) (2015), 1--28.
[50] Jia Z., Lin W.-W and Liu C.-S. A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible M-matrix, Numerische Mathematik, 130 (4) (2015), 645--679.
[51] Huang Y. and Jia Z., Some results on regularization of LSQR for large-scale discrete ill-posed problems, Science China Mathematics, 60 (4) (2017), 701--718.
[52] Jia Z. and Kang WJ., A residual based sparse approximate inverse preconditioning procedure for large sparse linear systems, Numerical Linear Algebra with Applications, 24 (2) (2017), 1-13.
[53] Huang Y. and Jia Z., On regularizing effects of MINRES and MR-II for large-scale symmetric discrete ill-posed problems, Journal of Computational and Applied Mathematics, 320 (2017), 145--163.
[54] Jia Z. and Yang Y., Modified truncated randomized singular value decomposition (MTRSVD) algorithms for large scale discrete ill-posed problems with general-form regularization, Inverse Problems, 34 (2018), article no. 055013 (28pp).
[55] Jia Z. and Kang WJ., A transformation approach that makes SPAI, PSAI and RSAI procedures efficient for large double irregular nonsymmetric sparse linear systems, Journal of Computational and Applied Mathematics, 384 (2019), 200—213.
[56] Huang J. and Jia Z., On inner iterations of Jacobi-Davidson type methods for large SVD computations, SIAM Journal on Scientific Computing, 41 (3) (2019), A1574—A1603.
[57] Approximation accuracy of the Krylov subspaces for linear discrete ill-posed problems, Journal of Computational and Applied Mathematics, 374 (2020), article no. 112786.
[58] The low rank approximations and Ritz values in LSQR for linear discrete ill-posed problems, Inverse Problems, 36 (4) (2020), article no. 045013 (32pp).
[59] Regularization properties of the Krylov iterative solvers CGME and LSMR for linear discrete ill-posed problems with an application to truncated randomized SVDs, Numerical Algorithms, 85 (4) (2020), 1281--1310.
[60] Regularization properties of LSQR for linear discrete ill-posed problems in the multiple singular value and best, near best and general low rank approximations, Inverse Problems, 36 (8) (2020), article no. 085009 (38pp).
[61] Jia Z. and Yang Y., A joint bidiagonalization based algorithm for large scale general-form Tikhonov regularization, Applied Numerical Mathematics, 157 (2020), 159--177.
[62] Huang J. and Jia Z., On choices of formulations of computing the generalized singular value decomposition of a matrix pair, Numerical Algorithms, 87 (2021), 689—718.
[63] Jia Z. and Lai. F., A convergence analysis on the iterative trace ratio algorithm and its refinements,CSIAM Transactions on Applied Mathematics, 2 (2) (2121), 297–312.
[64] Jia Z. and Wang F., The convergence of the generalized Lanczos trust-region method for the trust-region subproblem, SIAM Journal on Optimization, 31 (1) (2021), 887—914.
[65] Jia Z. and Li H., The joint bidiagonalization process with partial reorthogonalization, Numerical Algorithms, 88 (2021), 965—992.
[66] Theoretical and computable optimal subspace expansions for matrix eigenvalue problems, SIAM Journal on Matrix Analysis and Applications, 43 (2) (2022), 584—604.
[67] Huang J. and Jia Z., Two harmonic Jacobi--Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair, Journal of Scientific Computing, 93 (2022), article no. 41. (29pp).
[68] Huang J. and Jia Z., A cross-product free Jacobi--Davidson type method for computing a partial generalized singular value decomposition of a large matrix pair, Journal of Scientific Computing, 94 (2023), article no. 3 (32pp).
[69] Jia Z. and Li H., The joint bidiagonalization method for large GSVD computations in finite precision, SIAM Journal on Matrix Analysis and Applications, 44 (1) (2023), 382—407.
[70] Jia Z. and Zhang K., A FEAST SVDsolver for the computation of singular value decompositions of large matrices based on the Chebyshev—Jackson series expansion, Journal of Scientific Computing, 97 (2023), article no. 21 (36pp).
[71] Jia Z. and Zhang K., An augmented matrix-based CJ-FEAST SVDsolver for computing a partial singular value decomposition with the singular values in a given interval, SIAM Journal on Matrix Analysis and Applications, 45 (1) (2023), 24--58.
[72] Huang J. and Jia Z., A skew-symmetric Lanczos bidiagonalization method for computing several extremal eigenpairs of a large skew-symmetric matrix, SIAM Journal on Matrix Analysis and Applications, 45 (2) (2024), 1114—1147.
[73] Jia Z. and Zheng QQ., An analysis of the Rayleigh-Ritz and refined Rayleigh-Ritz methods for regular nonlinear eigenvalue problems, SIAM Journal on Matrix Analysis and Applications, in revision, 2024.
[74] Jia Z. and Zhang K., A CJ-FEAST GSVDsolver for computing a partial GSVD of a large matrix pair with the generalized singular values in a given interval, Numerische Mathematik, in revision, 2024.
