@article{chu_lin_2024, title={On the enumeration of subcells within hypercubes and its application to the Borsuk-Ulam theorem}, volume={1}, ISSN={["1572-9265"]}, DOI={10.1007/s11075-023-01716-5}, journal={NUMERICAL ALGORITHMS}, author={Chu, Moody T. and Lin, Matthew M.}, year={2024}, month={Jan} }
 @article{chu_zhang_2023, title={An Innate Moving Frame on Parametric Surfaces: The Dynamics of Principal Singular Curves}, volume={11}, ISSN={2227-7390}, url={http://dx.doi.org/10.3390/math11153306}, DOI={10.3390/math11153306}, abstractNote={This article reports an experimental work that unveils some interesting yet unknown phenomena underneath all smooth nonlinear maps. The findings are based on the fact that, generalizing the conventional gradient dynamics, the right singular vectors of the Jacobian matrix of any differentiable map point in directions that are most pertinent to the infinitesimal deformation of the underlying function and that the singular values measure the rate of deformation in the corresponding directions. A continuous adaption of these singular vectors, therefore, constitutes a natural moving frame that carries indwelling information of the variation. This structure exists in any dimensional space, but the development of the fundamental theory and algorithm for surface exploration is an important first step for immediate application and further generalization. In this case, trajectories of these singular vectors, referred to as singular curves, unveil some intriguing patterns per the given function. At points where singular values coalesce, curious and complex behaviors occur, manifesting specific landmarks for the function. Upon analyzing the dynamics, it is discovered that there is a remarkably simple and universal structure underneath all smooth two-parameter maps. This work delineates graphs with this interesting dynamical system and the possible new discovery that, analogous to the double helix with two base parings in DNA, two strands of critical curves and eight base pairings could encode properties of a generic and arbitrary surface. This innate structure suggests that this approach could provide a unifying paradigm in functional genetics, where all smooth surfaces could be genome-sequenced and classified accordingly. Such a concept has sparked curiosity and warrants further investigation.}, number={15}, journal={Mathematics}, publisher={MDPI AG}, author={Chu, Moody T. and Zhang, Zhenyue}, year={2023}, month={Jul}, pages={3306} }
 @article{chu_2023, title={Lax dynamics for Cartan decomposition with applications to Hamiltonian simulation}, volume={4}, ISSN={0272-4979 1464-3642}, url={http://dx.doi.org/10.1093/imanum/drad018}, DOI={10.1093/imanum/drad018}, abstractNote={Abstract}, journal={IMA Journal of Numerical Analysis}, publisher={Oxford University Press (OUP)}, author={Chu, Moody T}, year={2023}, month={Apr}, pages={drad018} }
 @article{chu_lin_2022, title={A complex-valued gradient flow for the entangled bipartite low rank approximation}, volume={271}, ISSN={["1879-2944"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85116536880&partnerID=MN8TOARS}, DOI={10.1016/j.cpc.2021.108185}, abstractNote={Entanglement of quantum states in a composite system is of profound importance in many applications. With respect to some suitably selected basis, the entanglement can be mathematically characterized via the Kronecker product of complex-valued density matrices. An approximation to a mixed state can be thought of as calculating its nearest separable state. Such a task encounters several challenges in computation. First, the added twist by the entanglement via the Kronecker product destroys the multi-linearity. The popular alternating least squares techniques for tensor approximation can hardly be applied. Second, there is no clear strategy for selecting a priori a proper low rank for the approximation. Third, the conventional calculus is not enough to address the optimization of real-valued functions over complex variables. This paper proposes a dynamical system approach to tackle low rank approximation of entangled bipartite systems, which has several advantages, including 1) A gradient dynamics in the complex space can be described in a fairly concise way; 2) The global convergence from any starting point to a local solution is guaranteed; 3) The requirement that the combination coefficients of pure states must be a probability distribution can be ensured; 4) The rank can be dynamically adjusted. This paper discusses the theory, algorithms, and presents some numerical experiments.}, journal={COMPUTER PHYSICS COMMUNICATIONS}, author={Chu, Moody T. and Lin, Matthew M.}, year={2022}, month={Feb} }
 @article{lin_chu_2022, title={Low-rank approximation to entangled multipartite quantum systems}, volume={21}, ISSN={["1573-1332"]}, url={https://doi.org/10.1007/s11128-022-03467-z}, DOI={10.1007/s11128-022-03467-z}, number={4}, journal={QUANTUM INFORMATION PROCESSING}, publisher={Springer Science and Business Media LLC}, author={Lin, Matthew M. and Chu, Moody T.}, year={2022}, month={Mar} }
 @article{lin_chu_2022, title={Rank-1 Approximation for Entangled Multipartite Real Systems}, volume={91}, ISSN={["1573-7691"]}, url={http://dx.doi.org/10.1007/s10915-022-01805-y}, DOI={10.1007/s10915-022-01805-y}, number={1}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, publisher={Springer Science and Business Media LLC}, author={Lin, Matthew M. and Chu, Moody T.}, year={2022}, month={Apr} }
 @article{chu_lin_2021, title={NONLINEAR POWER-LIKE AND SVD-LIKE ITERATIVE SCHEMES WITH APPLICATIONS TO ENTANGLED BIPARTITE RANK-1 APPROXIMATION}, volume={43}, ISSN={["1095-7197"]}, url={http://dx.doi.org/10.1137/20m1336059}, DOI={10.1137/20M1336059}, abstractNote={Gauging the distance between a mixed state and the convex set of separable states in a bipartite quantum mechanical system over the complex field is an important but challenging task. As a first st...}, number={5}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Lin, Matthew M.}, year={2021}, pages={S448–S474} }
 @article{dong_jiang_chu_2020, title={Nonlinear power-like iteration by polar decomposition and its application to tensor approximation}, volume={144}, ISSN={["0945-3245"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85079242500&partnerID=MN8TOARS}, DOI={10.1007/s00211-020-01100-8}, number={4}, journal={NUMERISCHE MATHEMATIK}, author={Dong, Bo and Jiang, Nan and Chu, Moody T.}, year={2020}, month={Apr}, pages={729–749} }
 @article{jiang_chu_shen_2019, title={Structure-preserving isospectral transformation for total or partial decoupling of self-adjoint quadratic pencils}, volume={449}, ISSN={["1095-8568"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85062395924&partnerID=MN8TOARS}, DOI={10.1016/j.jsv.2019.01.009}, abstractNote={Quadratic pencils λ2M + λC + K, where M, C, and K are n × n real matrices, arise in a broad range of important applications. Its spectral properties affect the vibration behavior of the underlying system which often consists of many elements coupled together through an intricate network of inter-connectivities. It is known that an n-degree-of-freedom system with semi-simple eigenvalues can be reduced to, without tampering with the innate vibration properties, n mutually independent single-degree-of-freedom subsystems, referred to as total decoupling. This paper revisits the problem with the additional constraint that the masses should stay invariant throughout the reduction process. Rescaling the masses if necessary, M is assumed to be the identity matrix. Isospectral flows are derived to either totally or partially decouple C and K to independent units of modules. Indeed, the same framework can be tailored to handle any kinds of desired structure. Two new results are obtained. First, the global convergence is guaranteed by using the Łojasiewicz gradient inequality. Second, bounds on errors due to numerical integration and floating-point arithmetic calculation are derived, which can be used for assessing the quality of the transformation. Numerical experiments on four distinct scenarios are given to demonstrate the capabilities of the framework of handling the decoupling problem.}, journal={JOURNAL OF SOUND AND VIBRATION}, author={Jiang, Nan and Chu, Moody T. and Shen, Jihong}, year={2019}, month={Jun}, pages={157–171} }
 @book{chu_guan_jiang_dong_2018, title={A theoretical consideration of matrix-operative alternating least squares methods for orthogonal CP tensor approximation}, author={Chu, Moody T. and Guan, Y. and Jiang, N. and Dong, B.}, year={2018} }
 @book{chu_guan_dong_jiang_2018, title={Convergence analysis of alternating direction methods: A general framework and its applications to tensor approximations}, author={Chu, Moody T. and Guan, Y. and Dong, B. and Jiang, N.}, year={2018} }
 @article{guan_chu_chu_2018, title={Convergence analysis of an SVD-based algorithm for the best rank-1 tensor approximation}, volume={555}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85048228789&partnerID=MN8TOARS}, DOI={10.1016/j.laa.2018.06.006}, abstractNote={This paper revisits the classical problem of finding the best rank-1 approximation to a generic tensor. The main focus is on providing a mathematical proof for the convergence of the iterates of an SVD-based algorithm. In contrast to the conventional approach by the so called alternating least squares (ALS) method that works to adjust one factor a time, the SVD-based algorithms improve two factors simultaneously. The ALS method is easy to implement, but suffers from slow convergence and easy stagnation at a local solution. It has been suggested recently that the SVD-algorithm might have a better limiting behavior leading to better approximations, yet a theory of convergence has been elusive in the literature. This note proposes a simple tactic to partially close that gap.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, publisher={Elsevier BV}, author={Guan, Yu and Chu, Moody T. and Chu, Delin}, year={2018}, month={Oct}, pages={53–69} }
 @book{chu_jiang_2018, title={Decoupling of Lattice Vibration}, author={Chu, Moody T. and Jiang, N.}, year={2018} }
 @book{chu_jiang_dong_2018, title={Global rank-1 approximation for order-3 tensors}, url={https://mtchu.math.ncsu.edu/Research/Papers/Rank1_Order3_Global.pdf}, author={Chu, Moody T. and Jiang, N. and Dong, B.}, year={2018} }
 @article{guan_chu_chu_2018, title={SVD-BASED ALGORITHMS FOR THE BEST RANK-1 APPROXIMATION OF A SYMMETRIC TENSOR}, volume={39}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85048210526&partnerID=MN8TOARS}, DOI={10.1137/17M1136699}, abstractNote={This paper revisits the problem of finding the best rank-1 approximation to a symmetric tensor and makes three contributions. First, in contrast to the many long and lingering arguments in the lite...}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Guan, Yu and Chu, Moody T. and Chu, Delin}, year={2018}, pages={1095–1115} }
 @article{wu_chu_2017, title={Markov chains with memory, tensor formulation, and the dynamics of power iteration}, volume={303}, ISSN={["1873-5649"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85012267381&partnerID=MN8TOARS}, DOI={10.1016/j.amc.2017.01.030}, abstractNote={A Markov chain with memory is no different from the conventional Markov chain on the product state space. Such a Markovianization, however, increases the dimensionality exponentially. Instead, Markov chain with memory can naturally be represented as a tensor, whence the transitions of the state distribution and the memory distribution can be characterized by specially defined tensor products. In this context, the progression of a Markov chain can be interpreted as variants of power-like iterations moving toward the limiting probability distributions. What is not clear is the makeup of the “second dominant eigenvalue” that affects the convergence rate of the iteration, if the method converges at all. Casting the power method as a fixed-point iteration, this paper examines the local behavior of the nonlinear map and identifies the cause of convergence or divergence. As an application, it is found that there exists an open set of irreducible and aperiodic transition probability tensors where the Z-eigenvector type power iteration fails to converge.}, journal={APPLIED MATHEMATICS AND COMPUTATION}, publisher={Elsevier BV}, author={Wu, Sheng-Jhih and Chu, Moody T.}, year={2017}, month={Jun}, pages={226–239} }
 @article{wu_chu_2017, title={Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements}, volume={33}, ISSN={["1361-6420"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85026767399&partnerID=MN8TOARS}, DOI={10.1088/1361-6420/aa76c4}, abstractNote={An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing–Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.}, number={8}, journal={INVERSE PROBLEMS}, author={Wu, Sheng-Jhih and Chu, Moody T.}, year={2017}, month={Aug} }
 @book{chu_2016, title={A Comment on the best rank-1 approximation of a symmetric tensor}, author={Chu, Moody}, year={2016} }
 @book{chu_liao_2016, title={Numerical Methods for Gradient Dynamics}, author={Chu, Moody T. and Liao, L.Z.}, year={2016} }
 @book{chu_2016, title={On the adjoint of tensors}, author={Chu, Moody}, year={2016} }
 @book{chu_2016, title={On the dynamics of maximin flows}, author={Chu, Moody}, year={2016} }
 @article{chu_2016, title={On the first degree Fejer-Riesz factorization and its applications to X plus A*X(-1)A = Q}, volume={489}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84945151931&partnerID=MN8TOARS}, DOI={10.1016/j.laa.2015.09.051}, abstractNote={Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejér–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X+A⁎X−1A=Q. In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, publisher={Elsevier BV}, author={Chu, Moody T.}, year={2016}, month={Jan}, pages={123–143} }
 @article{wang_chu_bo_2015, title={A computational framework of gradient flows for general linear matrix equations}, volume={68}, ISSN={["1572-9265"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85027952357&partnerID=MN8TOARS}, DOI={10.1007/s11075-014-9885-1}, number={1}, journal={NUMERICAL ALGORITHMS}, author={Wang, Liqi and Chu, Moody T. and Bo, Yu}, year={2015}, month={Jan}, pages={121–141} }
 @article{wu_chu_2015, title={Constructing optimal transition matrix for Markov chain Monte Carlo}, volume={487}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84941950319&partnerID=MN8TOARS}, DOI={10.1016/j.laa.2015.09.016}, abstractNote={The notion of asymptotic variance has been used as a means for gauging the performance of Markov chain Monte Carlo (MCMC) methods. For an effective MCMC simulation, it is imperative to first construct a Markov model with minimal asymptotic variance. The construction of such a stochastic matrix with prescribed stationary distribution as well as optimal asymptotic variance amounts to an interesting variationally constrained inverse eigenvector problem. Cast against a specially defined oblique coordinate system, the worst-case analysis of the asymptotic variance can be formulated as a problem of minimizing the logarithmic 2-norm of a restricted resolvent matrix over a convex and compact monoid. Based on this framework, this paper proposes employing global optimization techniques as a general instrument for numerical construction of optimal transition matrices. Numerical experiments manifest the complexity of the underlying problem. First, new global solutions different from the conventional structure characterized in the literature are found across the board for reversible problems. Second, local solutions with high frequencies of occurrence appear widespread for nonreversible problems. In all, the approach via the global optimization techniques is a feasible and practical means for numerical construction of the optimal Markov chain.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Wu, Sheng-Jhih and Chu, Moody T.}, year={2015}, month={Dec}, pages={184–202} }
 @article{wang_chu_yu_2015, title={ORTHOGONAL LOW RANK TENSOR APPROXIMATION: ALTERNATING LEAST SQUARES METHOD AND ITS GLOBAL CONVERGENCE}, volume={36}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84925305684&partnerID=MN8TOARS}, DOI={10.1137/130943133}, abstractNote={With the notable exceptions of two cases---that tensors of order 2, namely, matrices, always have best approximations of arbitrary low ranks and that tensors of any order always have the best rank-1 approximation, it is known that high-order tensors may fail to have best low rank approximations. When the condition of orthogonality is imposed, even under the modest assumption of semiorthogonality where only one set of components in the decomposed rank-1 tensors is required to be mutually perpendicular, the situation is changed completely---orthogonal low rank approximations always exist. The purpose of this paper is to discuss the best low rank approximation subject to semiorthogonality. The conventional high-order power method is modified to address the desirable orthogonality via the polar decomposition. Algebraic geometry technique is employed to show that for almost all tensors the orthogonal alternating least squares method converges globally.}, number={1}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Wang, Liqi and Chu, Moody T. and Yu, Bo}, year={2015}, pages={1–19} }
 @article{chu_lin_2015, title={On the finite rank and finite-dimensional representation of bounded semi-infinite Hankel operators}, volume={35}, ISSN={["1464-3642"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84943275892&partnerID=MN8TOARS}, DOI={10.1093/imanum/dru001}, abstractNote={Bounded, semi-infinite Hankel matrices of finite rank over the space 2 of square-summable sequences occur frequently in classical analysis and engineering applications. The notion of finite rank often appears under different contexts and the literature is diverse. The first part of this paper reviews some elegant, classical criteria and establishes connections among the various characterizations of finite rank in terms of rational functions, recursion, matrix factorizations and sinusoidal signals. All criteria require 2d parameters, though with different meanings, for a matrix of rank d . The Vandermonde factorization, in particular, permits immediately a singular-value preserving, finite-dimensional representation of the original semi-infinite Hankel matrix and, hence, makes it possible to retrieve the nonzero singular values of the semi-infinite Hankel matrix. The second part of this paper proposes using the LDL∗ decomposition of a specially constructed sample matrix to find the unitarily equivalent finite-dimensional representation. This approach enjoys several advantages, including the ease of computation by avoiding infinitedimensional vectors, the ability to reveal rank deficiency and the established pivoting strategy for stability. No error analysis is given, but several computational issues are discussed.}, number={3}, journal={IMA JOURNAL OF NUMERICAL ANALYSIS}, author={Chu, Moody T. and Lin, Matthew M.}, year={2015}, month={Jul}, pages={1256–1276} }
 @article{chu_lin_wang_2014, title={A study of singular spectrum analysis with global optimization techniques}, volume={60}, ISSN={["1573-2916"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84920878763&partnerID=MN8TOARS}, DOI={10.1007/s10898-013-0117-3}, number={3}, journal={JOURNAL OF GLOBAL OPTIMIZATION}, publisher={Springer Science and Business Media LLC}, author={Chu, Moody T. and Lin, Matthew M. and Wang, Liqi}, year={2014}, month={Nov}, pages={551–574} }
 @article{wu_hwang_chu_2014, title={Attaining the Optimal Gaussian Diffusion Acceleration}, volume={155}, ISSN={["1572-9613"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84898545692&partnerID=MN8TOARS}, DOI={10.1007/s10955-014-0963-5}, number={3}, journal={JOURNAL OF STATISTICAL PHYSICS}, author={Wu, Sheng-Jhih and Hwang, Chii-Ruey and Chu, Moody T.}, year={2014}, month={May}, pages={571–590} }
 @article{dong_lin_chu_2014, title={Nonnegative rank factorization-a heuristic approach via rank reduction}, volume={65}, ISSN={["1572-9265"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84875296090&partnerID=MN8TOARS}, DOI={10.1007/s11075-013-9704-0}, number={2}, journal={NUMERICAL ALGORITHMS}, author={Dong, Bo and Lin, Matthew M. and Chu, Moody T.}, year={2014}, month={Feb}, pages={251–274} }
 @article{wang_chu_2014, title={ON THE GLOBAL CONVERGENCE OF THE ALTERNATING LEAST SQUARES METHOD FOR RANK-ONE APPROXIMATION TO GENERIC TENSORS}, volume={35}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84907810454&partnerID=MN8TOARS}, DOI={10.1137/130938207}, abstractNote={Tensor decomposition has important applications in various disciplines, but it remains an extremely challenging task even to this date. A slightly more manageable endeavor has been to find a low rank approximation in place of the decomposition. Even for this less stringent undertaking, it is an established fact that tensors beyond matrices can fail to have best low rank approximations, with the notable exception that the best rank-one approximation always exists for tensors of any order. Toward the latter, the most popular approach is the notion of alternating least squares whose specific numerical scheme appears in the form as a variant of the power method. Though the limiting behavior of the objective values is well understood, a proof of global convergence for the iterates themselves has been elusive. This paper partially addresses the missing piece by showing that for almost all tensors, the iterates generated by the alternating least squares method for the rank-one approximation converge globally. Th...}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Wang, Liqi and Chu, Moody T.}, year={2014}, pages={1058–1072} }
 @article{chu_kuo_lin_2013, title={Tensor Spline Approximation in Economic Dynamics with Uncertainties}, volume={42}, ISSN={["1572-9974"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84879954290&partnerID=MN8TOARS}, DOI={10.1007/s10614-012-9331-1}, number={2}, journal={COMPUTATIONAL ECONOMICS}, publisher={Springer Science and Business Media LLC}, author={Chu, Moody T. and Kuo, Chun-Hung and Lin, Matthew M.}, year={2013}, month={Aug}, pages={175–198} }
 @article{zhang_chu_2012, title={Computing absolute maximum correlation}, volume={32}, ISSN={["0272-4979"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84863079144&partnerID=MN8TOARS}, DOI={10.1093/imanum/drq029}, abstractNote={Optimizing correlations between sets of variables is an important task in many areas of applications. There are plenty of algorithms for computing the maximum correlation. Most disappointingly, however, is that these methods typically cannot guarantee attaining the absolute maximum correlation, which would have significant impact on practical applications. This paper makes two contributions. Firstly, some distinctive traits of the absolute maximum correlation are characterized. By exploiting these attributes it is possible to propose an effective starting-point strategy that significantly increases the likelihood of attaining the absolute maximum correlation. Numerical testing of the classical Horst algorithm with the starting-point strategy seems to evince the potency of this approach. Secondly, the Horst algorithm is but one aggregated Jacobi-type power method. Following the innate iterative structure, a generalization to the Gauss–Seidel formulation is proposed as a natural improvement on the power method. Monotone convergence of the Gauss–Seidel algorithm is proved. When combined with the starting-point strategy, the newer Gauss–Seidel approach leads to faster computation of the absolute maximum correlation.}, number={1}, journal={IMA JOURNAL OF NUMERICAL ANALYSIS}, author={Zhang, Lei-Hong and Chu, Moody T.}, year={2012}, month={Jan}, pages={163–184} }
 @article{hughes-oliver_brooks_welch_khaledi_hawkins_young_patil_howell_ng_chu_2011, title={ChemModLab: A web-based cheminformatics modeling laboratory}, volume={11}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84859536904&partnerID=MN8TOARS}, DOI={10.3233/CI-2008-0016}, abstractNote={ChemModLab, written by the ECCR @ NCSU consortium under NIH support, is a toolbox for fitting and assessing quantitative structure-activity relationships (QSARs). Its elements are: a cheminformatic front end used to supply molecular descriptors for use in modeling; a set of methods for fitting models; and methods for validating the resulting model. Compounds may be input as structures from which standard descriptors will be calculated using the freely available cheminformatic front end PowerMV; PowerMV also supports compound visualization. In addition, the user can directly input their own choices of descriptors, so the capability for comparing descriptors is effectively unlimited. The statistical methodologies comprise a comprehensive collection of approaches whose validity and utility have been accepted by experts in the fields. As far as possible, these tools are implemented in open-source software linked into the flexible R platform, giving the user the capability of applying many different QSAR modeling methods in a seamless way. As promising new QSAR methodologies emerge from the statistical and data-mining communities, they will be incorporated in the laboratory. The web site also incorporates links to public-domain data sets that can be used as test cases for proposed new modeling methods. The capabilities of ChemModLab are illustrated using a variety of biological responses, with different modeling methodologies being applied to each. These show clear differences in quality of the fitted QSAR model, and in computational requirements. The laboratory is web-based, and use is free. Researchers with new assay data, a new descriptor set, or a new modeling method may readily build QSAR models and benchmark their results against other findings. Users may also examine the diversity of the molecules identified by a QSAR model. Moreover, users have the choice of placing their data sets in a public area to facilitate communication with other researchers; or can keep them hidden to preserve confidentiality.}, number={1-2}, journal={In Silico Biology}, author={Hughes-Oliver, J.M. and Brooks, A.D. and Welch, W.J. and Khaledi, M.G. and Hawkins, D. and Young, S.S. and Patil, K. and Howell, G.W. and Ng, R.T. and Chu, M.T.}, year={2011}, pages={61–81} }
 @article{chu_lin_2011, title={Dynamical System Characterization of the Central Path and Its Variants-A Revisit}, volume={10}, ISSN={["1536-0040"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-80055015129&partnerID=MN8TOARS}, DOI={10.1137/100802955}, abstractNote={The notion of central path plays a fundamental role in the development of interior point methods which, in turn, have become important tools for solving various optimization problems. The central path equation is algebraic in nature and is derived from the KKT optimality conditions of a certain logarithmic barrier problem; meanwhile, the primal variable portion of the very same central path can also be cast precisely as the integral curve, known as the affine scaling trajectory, of a certain gradient-type dynamical system. The justification is easy to establish in the context of linear programming. Though expected, the generalization of such a concept to semidefinite programming is not quite as obvious due to the difficulty of addressing noncommutativity in matrix multiplication. This paper revisits the dynamical system characterization of these flows and addresses the needed details for extension to semidefinite programming by means of a simple notion of operators and a specially defined inner product. F...}, number={3}, journal={SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS}, author={Chu, Moody T. and Lin, Matthew M.}, year={2011}, pages={887–905} }
 @article{lin_dong_chu_2010, title={Inverse mode problems for real and symmetric quadratic models}, volume={26}, ISSN={["1361-6420"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77952566534&partnerID=MN8TOARS}, DOI={10.1088/0266-5611/26/6/065003}, abstractNote={Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that given merely the desirable structure of the spectrum, namely given the cardinalities of real or complex eigenvalues but not of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed.}, number={6}, journal={INVERSE PROBLEMS}, author={Lin, Matthew M. and Dong, Bo and Chu, Moody T.}, year={2010}, month={Jun} }
 @article{lin_chu_2010, title={On the nonnegative rank of Euclidean distance matrices}, volume={433}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77953135527&partnerID=MN8TOARS}, DOI={10.1016/j.laa.2010.03.038}, abstractNote={The Euclidean distance matrix for n distinct points in Rr is generically of rank r + 2. It is shown in this paper via a geometric argument that its nonnegative rank for the case r = 1 is generically n.}, number={3}, journal={Linear Algebra and Its Applications}, author={Lin, M.M. and Chu, Moody}, year={2010}, pages={681–689} }
 @book{chu_2010, title={On the nonnegative rank of Euclidean distance matrices, II}, author={Chu, Moody}, year={2010} }
 @article{lin_dong_chu_2010, title={Semi-definite programming techniques for structured quadratic inverse eigenvalue problems}, volume={53}, ISSN={["1572-9265"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77950457631&partnerID=MN8TOARS}, DOI={10.1007/s11075-009-9309-9}, abstractNote={In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.}, number={4}, journal={NUMERICAL ALGORITHMS}, author={Lin, Matthew M. and Dong, Bo and Chu, Moody T.}, year={2010}, month={Apr}, pages={419–437} }
 @book{chu_lin_dong_2009, title={Integer matrix factorization and its applications}, author={Chu, Moody T. and Lin, M. and Dong, B.}, year={2009} }
 @article{dong_lin_chu_2009, title={Parameter reconstruction of vibration systems from partial eigeninformation}, volume={327}, ISSN={["1095-8568"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-69049084391&partnerID=MN8TOARS}, DOI={10.1016/j.jsv.2009.06.026}, abstractNote={Quadratic matrix polynomials are fundamental to vibration analysis. Because of the predetermined interconnectivity among the constituent elements and the mandatory nonnegativity of the physical parameters, most given vibration systems will impose some inherent structure on the coefficients of the corresponding quadratic matrix polynomials. In the inverse problem of reconstructing a vibration system from its observed or desirable dynamical behavior, respecting the intrinsic structure becomes important and challenging both theoretically and practically. The issue of whether a structured inverse eigenvalue problem is solvable is problem dependent and has to be addressed structure by structure. In an earlier work, physical systems that can be modeled under the paradigm of a serially linked mass–spring system have been considered via specifically formulated inequality systems. In this paper, the framework is generalized to arbitrary generally linked systems. In particular, given any configuration of interconnectivity in a mass–spring system, this paper presents a mechanism that systematically and automatically generates a corresponding inequality system. A numerical approach is proposed to determine whether the inverse problem is solvable and, if it is so, computes the coefficient matrices while providing an estimate of the residual error. The most important feature of this approach is that it is problem independent, that is, the approach is general and robust for any kind of physical configuration. The ideas discussed in this paper have been implemented into a software package by which some numerical experiments are reported.}, number={3-5}, journal={JOURNAL OF SOUND AND VIBRATION}, author={Dong, Bo and Lin, Matthew M. and Chu, Moody T.}, year={2009}, month={Nov}, pages={391–401} }
 @article{chu_chu_lin_2009, title={QUADRATIC MODEL UPDATING WITH SYMMETRY, POSITIVE DEFINITENESS, AND NO SPILL-OVER}, volume={31}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-72449157410&partnerID=MN8TOARS}, DOI={10.1137/080726136}, abstractNote={Updating a system modeled as a real symmetric quadratic eigenvalue problem to match observed spectral information has been an important task for practitioners in different disciplines. It is often desirable in the process to match only the newly measured data without tampering with the other unmeasured and often unknown eigenstructure inherent in the original model. Such an updating, known as no spill-over, has been critical yet challenging in practice. Only recently, a mathematical theory on updating with no spill-over has begun to be understood. However, other imperative issues such as maintaining positive definiteness in the coefficient matrices remain to be addressed. This paper highlights several theoretical aspects about updating that preserves both no spill-over and positive definiteness of the mass and the stiffness matrices. In particular, some necessary and sufficient conditions for the solvability conditions are established in this investigation.}, number={2}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, Delin and Chu, Moody and Lin, Wen-Wei}, year={2009}, pages={546–564} }
 @article{chu_xu_2009, title={SPECTRAL DECOMPOSITION OF REAL SYMMETRIC QUADRATIC lambda-MATRICES AND ITS APPLICATIONS}, volume={78}, ISSN={["1088-6842"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-58149231237&partnerID=MN8TOARS}, DOI={10.1090/S0025-5718-08-02128-5}, abstractNote={Spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. Spectral decomposition is of fundamental importance in many applications. The well-known GLR theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees, but its development is based on complex numbers. This paper modifies the GLR theory for the special application to real symmetric quadratic matrix polynomials, Q( A) = MX2 + CX + K, M nonsingular, subject to the specific restriction that all matrices in the representation be realvalued. It is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which, in turn, can be constructed from the spectral data. Applications to a variety of challenging inverse problems are discussed.}, number={265}, journal={MATHEMATICS OF COMPUTATION}, author={Chu, Moody T. and Xu, Shu-Fang}, year={2009}, pages={293–313} }
 @inproceedings{chu_2008, title={Data mining and applied linear algebra}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-50149102774&partnerID=MN8TOARS}, DOI={10.1109/ICKS.2008.39}, abstractNote={In this era of hyper-technological innovation, massive amounts of data are being generated at almost every level of applications in almost every area of disciplines. Extracting interesting knowledge from raw data, or data mining in a broader sense, has become an indispensable task. Nevertheless, data collected from complex phenomena represent often the integrated result of several interrelated variables, whereas these variables are less precisely defined. The basic principle of data mining is to distinguish which variable is related to which and how the variables are related. In many situations, the digitized information is gathered and stored as a data matrix. It is often the case, or so assumed, that the exogenous variables depend on the endogenous variables in a linear relationship. Retrieving "useful" information therefore can often be characterized as finding "suitable" matrix factorization. This paper offers a synopsis from this prospect on how linear algebra techniques can help to carry out the task of data mining. Examples from factor analysis, cluster analysis, latent semantic indexing and link analysis are used to demonstrate how matrix factorization helps to uncover hidden connection and do things fast. Low rank matrix approximation plays a fundamental role in cleaning the data and compressing the data. Other types of constraints, such as nonnegativity, will also be briefly discussed.}, booktitle={Proceedings - International Conference on Informatics Education and Research for Knowledge-Circulating Society, ICKS 2008}, author={Chu, Moody}, year={2008}, pages={20–25} }
 @article{chu_2008, title={Linear algebra algorithms as dynamical systems}, volume={17}, ISSN={0962-4929 1474-0508}, url={http://dx.doi.org/10.1017/S0962492906340019}, DOI={10.1017/S0962492906340019}, abstractNote={Any logical procedure that is used to reason or to infer either deductively or inductively, so as to draw conclusions or make decisions, can be called, in a broad sense, a realization process. A realization process usually assumes the recursive form that one state develops into another state by following a certain specific rule. Such an action is generally formalized as a dynamical system. In mathematics, especially for existence questions, a realization process often appears in the form of an iterative procedure or a differential equation. For years researchers have taken great effort to describe, analyse, and modify realization processes for various applications.}, journal={Acta Numerica}, publisher={Cambridge University Press (CUP)}, author={Chu, Moody T.}, year={2008}, month={Apr}, pages={1–86} }
 @article{chu_lin_2008, title={Low-dimensional polytope approximation and its applications to nonnegative matrix factorization}, volume={30}, ISSN={["1064-8275"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-55549091744&partnerID=MN8TOARS}, DOI={10.1137/070680436}, abstractNote={In this study, nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets. Working on the probability simplex has the advantage that data are limited to a compact set with a known boundary, making it easier to trace the approximation procedure. In particular, the supporting hyperplane that separates a point from a disjoint polytope, a fact asserted by the Hahn-Banach theorem, can be calculated in finitely many steps. This approach leads to a convenient way of computing the proximity map which, in contrast to most existing algorithms where only an approximate map is used, finds the unique and global minimum per iteration. This paper sets up a theoretical framework, outlines a numerical algorithm, and suggests an effective implementation. Testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods.}, number={3}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Chu, Moody T. and Lin, Matthew M.}, year={2008}, pages={1131–1155} }
 @article{kelley_liao_qi_chu_reese_winton_2008, title={PROJECTED PSEUDOTRANSIENT CONTINUATION}, volume={46}, ISSN={["1095-7170"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-55349118529&partnerID=MN8TOARS}, DOI={10.1137/07069866X}, abstractNote={We propose and analyze a pseudotransient continuation algorithm for dynamics on subsets of $R^N$. Examples include certain flows on manifolds and the dynamic formulation of bound-constrained optimization problems. The method gets its global convergence properties from the dynamics and inherits its local convergence properties from any fast locally convergent iteration.}, number={6}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Kelley, C. T. and Liao, Li-Zhi and Qi, Liqun and Chu, Moody T. and Reese, J. P. and Winton, C.}, year={2008}, pages={3071–3083} }
 @inbook{chu_2008, title={Quadratic Inverse Eigenvalue Problem and Its Applications to Model Updating — An Overview}, ISBN={9783540788409 9783540788416}, ISSN={1612-3956}, url={http://dx.doi.org/10.1007/978-3-540-78841-6_15}, DOI={10.1007/978-3-540-78841-6_15}, abstractNote={Modeling is one of the most fundamental tools that we use to simulate the complex world. The goal of modeling is to come up with a representation that is simple enough for mathematical manipulation yet powerful enough for describing, inducing, and reasoning complicated phenomena. When modeling physical systems, the resulting mathematical models are sometimes of a very high order too expensive for simulation. One remedy is the notion of model reduction that assists in approximating very high order mathematical models with lower order models. As is evidenced in this collection, model reduction has been under extensive study and rapid development over the past few years with many physical and engineering applications. On the other hand, precise mathematical models of physical systems are hardly available in practice. Many factors, including inevitable disturbances to the measurement and imperfect characterization of the model, attribute to the inexactitude. Since the model reduction process begets only a partial effect of the original model, it is reasonable to expect that the reduced model might not be consonant with realistic data either. For various reasons, it often becomes necessary to update a primitive model to attain consistency with empirical results. This procedure of updating or revising an existing model is another essential ingredient for establishing an effective model. The emphasis of the following discussion is on the model updating of quadratic pencils.}, booktitle={Mathematics in Industry}, publisher={Springer Berlin Heidelberg}, author={Chu, Moody T.}, year={2008}, pages={323–340} }
 @inbook{quadratic inverse eigenvalue problem and its applications to model updating - an overview_2008, url={https://link.springer.com/book/10.1007/978-3-540-78841-6}, DOI={10.1007/978-3-540-78841-6}, abstractNote={The idea for this book originated during the workshop “Model order reduction, coupled problems and optimization” held at the Lorentz Center in Leiden from S- tember 19–23, 2005. During one of the disc}, booktitle={Model Order Reduction: Theory, Research Aspects and Applications}, year={2008} }
 @article{chu_datta_lin_xu_2008, title={Spillover phenomenon in quadratic model updating}, volume={46}, ISSN={["1533-385X"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-43649088765&partnerID=MN8TOARS}, DOI={10.2514/1.31320}, abstractNote={Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. This paper studies the spillover phenomenon in the updating of quadratic pencils. In particular, it is shown that an updating with no spillover is always possible for undamped quadratic pencils, whereas spillover for damped quadratic pencils is generally unpreventable.}, number={2}, journal={AIAA JOURNAL}, author={Chu, Moody T. and Datta, Biswa and Lin, Wen-Wei and Xu, Shufang}, year={2008}, month={Feb}, pages={420–428} }
 @article{chu_del buono_2008, title={Total decoupling of general quadratic pencils, Part I: Theory}, volume={309}, ISSN={["0022-460X"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-35448964347&partnerID=MN8TOARS}, DOI={10.1016/j.jsv.2007.05.058}, abstractNote={The notion of quadratic pencils, λ2M+λC+K, where M, C, and K are n×n real matrices with or without some additional properties such as symmetry or positive definiteness, plays critical roles in many important applications. It has been long desirable, yet with very limited success, to reduce a complicated high-degree-of-freedom system to some simpler low-degree-of-freedom subsystems. Recently, Garvey, Friswell and Prells proposed a promising approach by which, under some mild assumptions, a general quadratic pencils can be converted by real-valued isospectral transformations into a totally decoupled system. This approach, if numerically feasible, would reduce the original n-degree-of-freedom second-order system to n totally independent single-degree-of-freedom second-order subsystems. Such a claim would be a striking breakthrough in the common knowledge that generally no three matrices M, C, and K can be diagonalized simultaneously. This paper intends to serve three purposes: to clarify some of the ambiguities in the original proposition, to simplify some of the computational details and, most importantly, to complete the theory of existence by matrix polynomial factorization tactics.}, number={1-2}, journal={JOURNAL OF SOUND AND VIBRATION}, author={Chu, Moody T. and Del Buono, Nicoletta}, year={2008}, month={Jan}, pages={96–111} }
 @article{chu_del buono_2008, title={Total decoupling of general quadratic pencils, Part II: Structure preserving isospectral flows}, volume={309}, ISSN={["1095-8568"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-35448978753&partnerID=MN8TOARS}, DOI={10.1016/j.jsv.2007.05.052}, abstractNote={Abstract Quadratic pencils, λ 2 M + λ C + K , where M, C, and K are n × n real matrices with or without some additional properties such as symmetry, connectivity, bandedness, or positive definiteness, arise in many important applications. Recently an existence theory has been established, showing that almost all n-degree-of-freedom second-order systems can be reduced to n totally independent single-degree-of-freedom second-order subsystems by real-valued isospectral transformations. In contrast to the common knowledge that generally no three matrices can be diagonalized simultaneously by equivalence transformations, these isospectral transformations endeavor to maintain a special linearization form called the Lancaster structure and do break down M, C and K into diagonal matrices simultaneously. However, these transformations depend on the matrices in a rather complicated way and, hence, are difficult to construct directly. In this paper, a second part of a continuing study, a closed-loop control system that preserves both the Lancaster structure and the isospectrality is proposed as a means to achieve the diagonal reduction. Consequently, these transformations are acquired.}, number={1-2}, journal={JOURNAL OF SOUND AND VIBRATION}, author={Chu, Moody T. and Del Buono, Nicoletta}, year={2008}, month={Jan}, pages={112–128} }
 @book{chu_golub_2007, title={Inverse Eigenvalue Problems: Theory, Algorithms, and Applications}, volume={9780198566649}, ISBN={0198566646}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84919596855&partnerID=MN8TOARS}, DOI={10.1093/acprof:oso/9780198566649.001.0001}, abstractNote={The basic goal of an inverse eigenvalue problem is to reconstruct the physical parameters of a certain system from the knowledge or desire of its dynamical behavior. Depending on the application, inverse eigenvalue problems appear in many different forms. This book discusses the fundamental questions, some known results, many applications, mathematical properties, a variety of numerical techniques, as well as several open problems.}, journal={Inverse Eigenvalue Problems: Theory, Algorithms, and Applications}, publisher={Oxford: Oxford University Press}, author={Chu, Moody and Golub, G.}, year={2007}, pages={1–406} }
 @article{ragni_chu_diele_marangi_2007, title={On the estimation of the consumption matrix from inexact data in the leontief model}, volume={2}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84947050612&partnerID=MN8TOARS}, number={3-4}, journal={Journal of Numerical Analysis, Industrial and Applied Mathematics}, author={Ragni, S. and Chu, M.T. and Diele, F. and Marangi, C.}, year={2007}, pages={139–156} }
 @article{chu_del buono_yu_2007, title={Structured quadratic inverse eigenvalue problem, I. Serially linked systems}, volume={29}, ISSN={["1095-7197"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-41849138387&partnerID=MN8TOARS}, DOI={10.1137/060672510}, abstractNote={Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a mass-spring system, are serially linked. The discussion recasts both undamped and damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described.}, number={6}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Chu, Moody T. and Del Buono, Nicoletta and Yu, Bo}, year={2007}, pages={2668–2685} }
 @article{chu_lin_xu_2007, title={Updating quadratic models with no spillover effect on unmeasured spectral data}, volume={23}, ISSN={["1361-6420"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33947697703&partnerID=MN8TOARS}, DOI={10.1088/0266-5611/23/1/013}, abstractNote={Model updating concerns the modification of an existing but inaccurate model with measured data. For models characterized by quadratic pencils, the measured data usually involve incomplete knowledge of natural frequencies, mode shapes, or other spectral information. In conducting the updating, it is often desirable to match only the part of observed data without tampering with the other part of unmeasured or unknown eigenstructure inherent in the original model. Such an updating, if possible, is said to have no spillover. Model updating with no spillover has been a very challenging task in applications. This paper provides a complete theory on when such an updating with no spillover is possible.}, number={1}, journal={INVERSE PROBLEMS}, author={Chu, Moody T. and Lin, Wen-Wei and Xu, Shu-Fang}, year={2007}, month={Feb}, pages={243–256} }
 @article{chu_chu_2006, title={LOW RANK UPDATE OF SINGULAR VALUES}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.6877}, author={Chu, Delin and Chu, Moody}, year={2006} }
 @article{chu_chu_2006, title={Low rank update of singular values}, volume={75}, ISSN={0025-5718}, url={http://dx.doi.org/10.1090/s0025-5718-06-01825-4}, DOI={10.1090/S0025-5718-06-01825-4}, abstractNote={The notion of a low rank update arises in many important applications. This paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values. The setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated. Precise and easy-to-check necessary and sufficient conditions under which the problem is solvable are completely characterized, generalizing some traditional Weyl inequalities for singular values. The constructive proof makes it possible to compute such a solution numerically. A pseudo algorithm is outlined.}, number={255}, journal={Mathematics of Computation}, publisher={American Mathematical Society (AMS)}, author={Chu, Delin and Chu, Moody}, year={2006}, month={Feb}, pages={1351–1367} }
 @article{chu_chu_2006, title={Reachable matrices by a QR step with shift}, volume={5}, ISSN={["1536-0040"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33644931671&partnerID=MN8TOARS}, DOI={10.1137/040617844}, abstractNote={One of the most interesting dynamical systems used in numerical analysis is the $QR$ algorithm. An added maneuver to improve the convergence behavior is the $QR$ iteration with shift which is of fundamental importance in eigenvalue computation. This paper is a theoretical study of the set of all isospectral matrices "reachable" by the dynamics of the $QR$ algorithm with shift. A matrix B is said to be reachable by A if $B = RQ + \mu I$, where $A - \mu I = QR$ is the $QR$ decomposition for some $\mu \in \mathbb{R}$. It is proved that in general the $QR$ algorithm with shift is neither reflexive nor symmetric. Examples are given to demonstrate that this relation is neither transitive nor antisymmetric. It is further discovered that the reachable set from a given $n \times n$ matrix A forms $2^{n-1}$ disjoint open loops if n is even and $2^{n-2}$ disjoint components each of which is no longer a loop when n is odd.}, number={1}, journal={SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS}, author={Chu, DL and Chu, M}, year={2006}, pages={91–107} }
 @article{chu_plemmons_2005, title={Nonnegative Matrix Factorization And Applications}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.7507}, author={Chu, Moody and Plemmons, Robert}, year={2005} }
 @article{nonnegative matrix factorization and applications_2005, url={https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=73a543613d5eee22fd0172f5c1397ac23d8c09c0}, journal={IMAGE}, year={2005} }
 @article{chu_plemmons_2005, title={Nonnegative matrix factorization and applications}, volume={34}, journal={IMAGE}, author={Chu, Moody T. and Plemmons, R.}, year={2005}, pages={1–5} }
 @article{chu_xu_2005, title={On computing minimal realizable spectral radii of non-negative matrices}, volume={12}, ISSN={["1099-1506"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-20744446749&partnerID=MN8TOARS}, DOI={10.1002/nla.395}, abstractNote={For decades considerable efforts have been exerted to resolve the inverse eigenvalue problem for non‐negative matrices. Yet fundamental issues such as the theory of existence and the practice of computation remain open. Recently, it has been proved that, given an arbitrary (n–1)‐tuple ℒ︁ = (λ2,…,λn) ∈ ℂn–1 whose components are closed under complex conjugation, there exists a unique positive real number ℛ(ℒ︁), called the minimal realizable spectral radius of ℒ︁, such that the set {λ1,…,λn} is precisely the spectrum of a certain n × n non‐negative matrix with λ1 as its spectral radius if and only if λ1 ⩾ ℛ(ℒ︁). Employing any existing necessary conditions as a mode of checking criteria, this paper proposes a simple bisection procedure to approximate the location of ℛ(ℒ︁). As an immediate application, it offers a quick numerical way to check whether a given n‐tuple could be the spectrum of a certain non‐negative matrix. Copyright © 2004 John Wiley & Sons, Ltd.}, number={1}, journal={NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS}, author={Chu, MT and Xu, SF}, year={2005}, month={Feb}, pages={77–86} }
 @article{chu_diele_ragni_2005, title={On the inverse problem of constructing symmetric pentadiagonal Toeplitz matrices from their three largest eigenvalues}, volume={21}, ISSN={["1361-6420"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-28544445478&partnerID=MN8TOARS}, DOI={10.1088/0266-5611/21/6/005}, abstractNote={The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem—can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? Given three real numbers ν ⩽ µ ⩽ λ, this paper finds that the ratio , including infinity if µ = ν, determines whether there is a symmetric pentadiagonal Toeplitz matrix with ν, µ and λ as its three largest eigenvalues. It is shown that such a matrix of size n × n does not exist if n is even and α is too large or if n is odd and α is too close to 1. When such a matrix does exist, a numerical method is proposed for the construction.}, number={6}, journal={INVERSE PROBLEMS}, author={Chu, MT and Diele, F and Ragni, S}, year={2005}, month={Dec}, pages={1879–1894} }
 @article{chu_del buono_lopez_politi_2005, title={On the low-rank approximation of data on the unit sphere}, volume={27}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33144482741&partnerID=MN8TOARS}, DOI={10.1137/S0895479803433295}, abstractNote={In various applications, data in multidimensional space are normalized to unit length. This paper considers the problem of best fitting given points on the m-dimensional unit sphere Sm-1 by k-dimensional great circles with k much less than m. The task is cast as an algebraically constrained low-rank matrix approximation problem. Using the fidelity of the low-rank approximation to the original data as the cost function, this paper offers an analytic expression of the projected gradient which, on one hand, furnishes the first order optimality condition and, on the other hand, can be used as a numerical means for solving this problem.}, number={1}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, M and Del Buono, N and Lopez, L and Politi, T}, year={2005}, pages={46–60} }
 @book{chu_diele_plemmons_ragni_2005, title={Optimality, Computation and Interpretation of nonnegative matrix factorizations}, author={Chu, Moody T. and Diele, F. and Plemmons, R. and Ragni, S.}, year={2005} }
 @article{chu_diele_sgura_2004, title={Gradient flow methods for matrix completion with prescribed eigenvalues}, volume={379}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0742268367&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(03)00393-8}, abstractNote={Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. The challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable. An intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix. Constructive proofs exist to a certain point (and those proofs, such as the classical Schur–Horn theorem, are amazingly elegant enough in their own right) beyond which very few theories or numerical algorithms are available. In this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries. The gradient flow is proposed as a numerical means to tackle the construction. This approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities.}, number={1-3 SPEC. ISS}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT and Diele, F and Sgura, I}, year={2004}, month={Mar}, pages={85–112} }
 @book{chu_2004, title={Group theory, linear transformations, and flows: Dynamical systems on manifolds}, author={Chu, Moody}, year={2004} }
 @book{chu_chu_brown_2004, title={On the least squares Euclidean distance matrix approximation and completion}, author={Chu, Moody T. and Chu, D.I. and Brown, H.}, year={2004} }
 @article{chu_del buono_diele_politi_ragni_2004, title={On the semigroup of standard symplectic matrices and its applications}, volume={389}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-3943079249&partnerID=MN8TOARS}, DOI={10.1016/j.laa.2004.03.017}, abstractNote={A matrix Z∈R2n×2n is said to be in the standard symplectic form if Z enjoys a block LU-decomposition in the sense of A0−HIZ=IG0AT, where A is nonsingular and both G and H are symmetric and positive definite in Rn×n. Such a structure arises naturally in the discrete algebraic Riccati equations. This note contains two results: First, by means of a parameter representation it is shown that the set of all 2n×2n standard symplectic matrices is closed under multiplication and, thus, forms a semigroup. Secondly, block LU-decompositions of powers of Z can be derived in closed form which, in turn, can be employed recursively to induce an effective structure-preserving algorithm for solving the Riccati equations. The computational cost of doubling and tripling of the powers is investigated. It is concluded that doubling is the better strategy.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, M and Del Buono, N and Diele, F and Politi, T and Ragni, S}, year={2004}, month={Sep}, pages={215–225} }
 @article{chu_chu_2004, title={SINGULAR VALUE REASSIGNMENT WITH LOW RANK MATRICES}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.71.8290}, author={Chu, Delin and Chu, Moody}, year={2004} }
 @article{chu_chu_2004, title={SINGULAR VALUE REASSIGNMENT WITH LOW RANK MATRICES (DRAFT: June 11, 2004)}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.6384}, author={Chu, Delin and Chu, Moody}, year={2004} }
 @book{chu_orlowski_schlorff_blevins_canas_funderlic_2004, title={The effect of ties on convergence in the k-modes variants for clustering categorical data}, author={Chu, Moody T. and Orlowski, N. and Schlorff, D. and Blevins, J. and Canas, D. and Funderlic, R.}, year={2004} }
 @article{chu_kuo_lin_2003, title={On inverse quadratic eigenvalue problems with partially prescribed eigenstructure}, volume={25}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-8344269351&partnerID=MN8TOARS}, DOI={10.1137/S0895479803404484}, abstractNote={The inverse eigenvalue problem of constructing real and symmetric square matrices M, C, and K of size $n \times n$ for the quadratic pencil $Q(\lambda) = \lambda^2 M + \lambda C + K$ so that $Q(\lambda)$ has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but different problems.The first part deals with the inverse problem where M and K are required to be positive definite and semidefinite, respectively. It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$. The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil. The eigenstructure of the resulting $Q(\lambda)$ is completely analyzed.The second part deals with the inverse problem where M is a fixed positive definite matrix (and hence may be assumed to be t...}, number={4}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Kuo, YC and Lin, WW}, year={2003}, pages={995–1020} }
 @article{chu_diele_sgura_2003, title={On robust matrix completion with prescribed eigenvalues}, volume={19}, ISSN={["0167-739X"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0042692965&partnerID=MN8TOARS}, DOI={10.1016/s0167-739x(03)00040-2}, abstractNote={Matrix completion with prescribed eigenvalues is a special kind of inverse eigenvalue problems. Thus far, only a handful of specific cases concerning its existence and construction have been studied in the literature. The general problem where the prescribed entries are at arbitrary locations with arbitrary cardinalities proves to be challenging both theoretically and computationally. This paper investigates some continuation techniques by recasting the completion problem as an optimization of the distance between the isospectral matrices with the prescribed eigenvalues and the affine matrices with the prescribed entries. The approach not only offers an avenue to solving the completion problem in its most general setting but also makes it possible to seek a robust solution that is least sensitive to perturbation.}, number={7}, journal={FUTURE GENERATION COMPUTER SYSTEMS}, author={Chu, MT and Diele, F and Sgura, I}, year={2003}, month={Oct}, pages={1139–1153} }
 @article{chu_plemmons_2003, title={Real-valued, low rank, circulant approximation}, volume={24}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0041663838&partnerID=MN8TOARS}, DOI={10.1137/S0895479801383166}, abstractNote={Partially due to the fact that the empirical data collected by devices with finite bandwidth often neither preserves the specified structure nor induces a certain desired rank, retrieving the nearest structured low rank approximation from a given data matrix becomes an imperative task in many applications. This paper investigates the case of approximating a given target matrix by a real-valued circulant matrix of a specified, fixed, and low rank. A fast Fourier transform (FFT)-based numerical procedure is proposed to speed up the computation. However, since a conjugate-even set of eigenvalues must be maintained to guarantee a real-valued matrix, it is shown by numerical examples that the nearest real-valued, low rank, and circulant approximation is sometimes surprisingly counterintuitive.}, number={3}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Plemmons, RJ}, year={2003}, pages={645–659} }
 @article{chu_funderlic_plemmons_2003, title={Structured low rank approximation}, volume={366}, ISSN={["0024-3795"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0037409774&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(02)00505-0}, abstractNote={This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system. The task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix. This paper addresses some of the theoretical and numerical issues involved in the problem. Two procedures for constructing the nearest structured low rank matrix are proposed. The procedures are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness. The techniques can also be easily implemented by utilizing available optimization packages. The special case of symmetric Toeplitz structure using the Frobenius matrix norm is used to exemplify the ideas throughout the discussion. The concept, rather than the implementation details, is the main emphasis of the paper.}, number={SPEC. ISS.}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT and Funderlic, RE and Plemmons, RJ}, year={2003}, month={Jun}, pages={157–172} }
 @article{chu_golub_2002, title={Structured inverse eigenvalue problems}, volume={11}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85095826310&partnerID=MN8TOARS}, DOI={10.1017/S0962492902000016}, abstractNote={An inverse eigenvalue problem concerns the reconstruction of a structured 
matrix from prescribed spectral data. Such an inverse problem arises in many 
applications where parameters of a certain physical system are to be determined 
from the knowledge or expectation of its dynamical behaviour. Spectral 
information is entailed because the dynamical behaviour is often governed by 
the underlying natural frequencies and normal modes. Structural stipulation 
is designated because the physical system is often subject to some feasibility 
constraints. The spectral data involved may consist of complete or only partial 
information on eigenvalues or eigenvectors. The structure embodied by the 
matrices can take many forms. The objective of an inverse eigenvalue problem 
is to construct a matrix that maintains both the specific structure as well as 
the given spectral property. In this expository paper the emphasis is to provide 
an overview of the vast scope of this intriguing problem, treating some of its 
many applications, its mathematical properties, and a variety of numerical 
techniques.}, journal={Acta Numerica}, author={Chu, M.T. and Golub, G.H.}, year={2002}, pages={1–71} }
 @article{chu_fundelic_2002, title={The centroid decomposition: Relationships between discrete variational decompositions and SVDs}, volume={23}, ISSN={["1095-7162"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0036401834&partnerID=MN8TOARS}, DOI={10.1137/S0895479800382555}, abstractNote={The centroid decomposition, an approximation for the singular value decomposition (SVD), has a long history among the statistics/psychometrics community for factor analysis research. We revisit the centroid method in its original context of factor analysis and then adapt it to other than a covariance matrix. The centroid method can be cast as an ${\cal O}(n)$-step ascent method on a hypercube. It is shown empirically that the centroid decomposition provides a measurement of second order statistical information of the original data in the direction of the corresponding left centroid vectors. One major purpose of this work is to show fundamental relationships between the singular value, centroid, and semidiscrete decompositions. This unifies an entire class of truncated SVD approximations. Applications include semantic indexing in information retrieval.}, number={4}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Fundelic, RE}, year={2002}, month={May}, pages={1025–1044} }
 @book{chu_2001, title={On the statistical meaning of truncated singular value decomposition}, author={Chu, Moody}, year={2001} }
 @article{chu_trendafilov_2001, title={The orthogonally constrained regression revisited}, volume={10}, ISSN={["1061-8600"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0035565472&partnerID=MN8TOARS}, DOI={10.1198/106186001317243430}, abstractNote={The Penrose regression problem, including the orthonormal Procrustes problem and rotation problem to a partially specified target, is an important class of data matching problems arising frequently in multivariate analysis, yet its optimality conditions have never been clearly understood. This work offers a way to calculate the projected gradient and the projected Hessian explicitly. One consequence of this calculation is the complete characterization of the first order and the second order necessary and sufficient optimality conditions for this problem. Another application is the natural formulation of a continuous steepest descent ow that can serve as a globally convergent numerical method. Applications to the orthonormal Procrustes problem and Penrose regression problem with partially specified target are demonstrated in this article. Finally, some numerical results are reported and commented.}, number={4}, journal={JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS}, author={Chu, MT and Trendafilov, NT}, year={2001}, month={Dec}, pages={746–771} }
 @article{chu_2000, title={A fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values}, volume={37}, ISSN={["0036-1429"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0012087393&partnerID=MN8TOARS}, DOI={10.1137/S0036142998339301}, abstractNote={The Weyl--Horn theorem characterizes a relationship between the eigenvalues and the singular values of an arbitrary matrix. Based on that characterization, a fast recursive algorithm is developed to construct numerically a matrix with prescribed eigenvalues and singular values. Besides being of theoretical interest, the technique could be employed to create test matrices with desired spectral features. Numerical experiment shows this algorithm to be quite efficient and robust.}, number={3}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Chu, MT}, year={2000}, month={Mar}, pages={1004–1020} }
 @article{chu_pauca_plemmons_sun_2000, title={A mathematical framework for the linear reconstructor problem in adaptive optics}, volume={316}, ISSN={["0024-3795"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0034392470&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(00)00019-7}, abstractNote={The wave front field aberrations induced by atmospheric turbulence can severely degrade the performance of an optical imaging system. Adaptive optics refers to the process of removing unwanted wave front distortions in real time, i.e., before the image is formed, with the use of a phase corrector. The basic idea in adaptive optics is to control the position of the surface of a deformable mirror in such a way as to approximately cancel the atmospheric turbulence effects on the phase of the incoming light wave front. A phase computation system, referred to as a reconstructor, transforms the output of a wave front sensor into a set of drive signals that control the shape of a deformable mirror. The control of a deformable mirror is often based on a linear wave front reconstruction algorithm that is equivalent to a matrix–vector multiply. The matrix associated with the reconstruction algorithm is called the reconstructor matrix. Since the entire process, from the acquisition of wave front measurements to the positioning of the surface of the deformable mirror, must be performed at speeds commensurate with the atmospheric changes, the adaptive optics control imposes several challenging computational problems. The goal of this paper is twofold: (i) to describe a simplified yet feasible mathematical framework that accounts for the interactions among main components involved in an adaptive optics imaging system, and (ii) to present several ways to estimate the reconstructor matrix based on this framework. The performances of these various reconstruction techniques are illustrated using some simple computer simulations.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT and Pauca, VP and Plemmons, RJ and Sun, XB}, year={2000}, month={Sep}, pages={113–135} }
 @article{trendafilov_chu_1999, title={A Continuous-Time Approach to the Oblique Procrustes Problem}, volume={26}, ISSN={0385-7417 1349-6964}, url={http://dx.doi.org/10.2333/bhmk.26.167}, DOI={10.2333/bhmk.26.167}, abstractNote={In the paper proposed we will make use of the gradient flow approach to consider a generalization of the well-known oblique Procrustes rotation problem, involving oblique simple structure rotation of both the core and component matrices resulting from three-mode factor analysis. The standard oblique Procrustes rotations to specified factor-structure and factor-pattern follw as special cases. The approach adopted leads to globally convergent algorithm and includes solving of initial value problem for certain matrix ordinary differential equation. Necessary conditions are established for the solution of the problem. The same approach is extended easily to the weighted oblique Procrustes rotation. Finally, some simulated numerical results are given and commented.}, number={2}, journal={Behaviormetrika}, publisher={Springer Science and Business Media LLC}, author={Trendafilov, Nickolay T. and Chu, Moody T.}, year={1999}, month={Jul}, pages={167–181} }
 @book{chu_1999, title={On an adaptive control algorithm for the adaptive optics problems}, author={Chu, Moody}, year={1999} }
 @article{chu_1999, title={On constructing matrices with prescribed singular values and diagonal elements}, volume={288}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0039035737&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(98)10124-6}, abstractNote={Similar to the well known Schur-Horn theorem that characterizes the relationship between the diagonal entries and the eigenvalues of a Hermitian matrix, the Sing-Thompson theorem characterizes the relationship between the diagonal entries and the singular values of an arbitrary matrix. It is noted in this paper that, based on the induction principle, such a matrix can be constructed numerically by a fast recursive algorithm, provided that the given singular values and diagonal elements satisfy the Sing-Thompson conditions.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT}, year={1999}, month={Feb}, pages={11–22} }
 @article{chu_guo_1998, title={A numerical method for the inverse stochastic spectrum problem}, volume={19}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0032339054&partnerID=MN8TOARS}, DOI={10.1137/S0895479896292418}, abstractNote={The inverse stochastic spectrum problem involves the construction of a stochastic matrix with a prescribed spectrum. The problem could be solved by first constructing a nonnegative matrix with the same prescribed spectrum. A differential equation aimed to bring forth the steepest descent flow in reducing the distance between isospectral matrices and nonnegative matrices, represented in terms of some general coordinates, is described. The flow is further characterized by an analytic singular value decomposition to maintain the numerical stability and to monitor the proximity to singularity. This flow approach can be used to design Markov chains with specified structure. Applications are demonstrated by numerical examples.}, number={4}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Guo, QL}, year={1998}, month={Oct}, pages={1027–1039} }
 @inproceedings{chu_funderlic_plemmons_1998, title={Approximation by structured lower rank matrices}, volume={3461}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0032224051&partnerID=MN8TOARS}, DOI={10.1117/12.325687}, abstractNote={We consider two general procedures for constructing the nearest approximation of a given matrix by one with any lower rank and any linear structure. Nearness can be measured in any matrix norm. Structured low rank matrices arise in various applications, including image enhancement and model reduction. In practice, the empirical data collected in the matrix often either do not maintain the specified structure or do not induce the desirable rank.It is an important task to search for the nearest structured lower rank approximation of a given matrix. The techniques developed in this paper can easily be implemented for numerical computation. In particular, it is shown that the computations can be approached using efficient optimization packages. The special case of Toeplitz structure using the Frobenius matrix norm is discussed in detail to illustrate the ideas, and numerical test are reported. The procedures developed herein can be generalized to consider a much broader range of approximation problems.}, booktitle={Proceedings of SPIE - The International Society for Optical Engineering}, author={Chu, M.T. and Funderlic, R.E. and Plemmons, R.J.}, year={1998}, pages={268–279} }
 @book{davis_chu_mcconnell_dolan_norris_ortiz_plemmon_ridgeway_scaife_stewart_et al._1998, title={Cornelius Lanczos: Collected published papers with commentaries}, ISBN={0929493003}, publisher={Raleigh, NC: College of Physical and Mathematical Sciences, North Carolina State University}, author={Davis, W. R. and Chu, M. T. and McConnell, J. R. and Dolan, P. and Norris, L. K. and Ortiz, E. and Plemmon, R. J. and Ridgeway, D. and Scaife, B.K.P. and Stewart, W. J. and et al.}, year={1998} }
 @misc{chu_1998, title={Inverse eigenvalue problems}, volume={40}, ISSN={["1095-7200"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0032012860&partnerID=MN8TOARS}, DOI={10.1137/S0036144596303984}, abstractNote={A collection of inverse eigenvalue problems are identified and classified according to their characteristics. Current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper. This exposition also reveals many open questions that deserve further study. An extensive bibliography of pertinent literature is attached.}, number={1}, journal={SIAM REVIEW}, author={Chu, MT}, year={1998}, month={Mar}, pages={1–39} }
 @article{chu_trendafilov_1998, title={On a differential equation approach to the weighted or orthogonal Procrustes problem}, volume={8}, ISSN={["0960-3174"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0042725448&partnerID=MN8TOARS}, DOI={10.1023/A:1008934100736}, number={2}, journal={STATISTICS AND COMPUTING}, author={Chu, MT and Trendafilov, NT}, year={1998}, month={Jun}, pages={125–133} }
 @article{chu_quanlin_1998, title={On the least squares approximation of symmetric-definite pencils subject to generalized spectral constraints}, volume={19}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0040581657&partnerID=MN8TOARS}, DOI={10.1137/s0895479895285135}, abstractNote={A general framework for the least squares approximation of symmetric-definite pencils subject to generalized eigenvalues constraints is developed in this paper. This approach can be adapted to different applications, including the inverse eigenvalue problem. The idea is based on the observation that a natural parameterization for the set of symmetric-definite pencils with the same generalized eigenvalues is readily available. In terms of these parameters, descent flows on the isospectral surface aimed at reducing the distance to matrices of the desired structure can be derived. These flows can be designed to carry certain other interesting properties and may be integrated numerically.}, number={1}, journal={SIAM Journal on Matrix Analysis and Applications}, author={Chu, Moody and Quanlin, G.}, year={1998}, pages={1–20} }
 @article{chu_1998, title={On the optimal consistent approximation to pairwise comparison matrices}, volume={272}, ISSN={["1873-1856"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0041856720&partnerID=MN8TOARS}, DOI={10.1016/S0024-3795(97)00329-7}, abstractNote={Consistency retrieval from a biased relative preference table is an imperative task in decision theory. This paper considers the least squares approximation of a pairwise comparison matrix by consistent matrices. It is observed that the highly nonlinear manifold of consistent matrices can be changed into a linear subspace by the componentwise logarithmic transformation. A first order optimality condition therefore can be described in terms of coordinates in the linear subspace. This approach facilitates the otherwise much more complicated optimality condition if working with the variables in the original manifold. Fast nonlinear equation solvers can be employed to solve the problem efficiently.}, number={1-3}, journal={LINEAR ALGEBRA AND ITS APPLICATIONS}, author={Chu, MT}, year={1998}, month={Mar}, pages={155–168} }
 @article{chu_trendafilov_1998, title={Orthomax Rotation Problem. A Differential Equation Approach}, volume={25}, ISSN={0385-7417 1349-6964}, url={http://dx.doi.org/10.2333/bhmk.25.13}, DOI={10.2333/bhmk.25.13}, abstractNote={In the present paper the ORTHOMAX rotation problem is reconsidered. It is shown that its solution can be presented as a steepest ascent flow on the manifold of orthogonal matrices. A matrix formulation of the ORTHOMAX problem is given as an initial value problem for matrix differential equation of first order. The solution can be found by any available ODE numerical integrator. Thus the paper proposes a convergent method for direct matrix solution of the ORTHOMAX problem. The well-known first order necessary condition for the VARIMAX maximizer is reestablished for the ORTHOMAX case without using Lagrange multipliers. Additionally new second order optimality conditions are derived and as a consequence an explicit second order necessary condition for further classification of the ORTHOMAX maximizer is obtained.}, number={1}, journal={Behaviormetrika}, publisher={Springer Science and Business Media LLC}, author={Chu, Moody T. and Trendafilov, Nickolay T.}, year={1998}, month={Jan}, pages={13–23} }
 @article{chu_funderlic_golub_1998, title={Rank modifications of semidefinite matrices associated with a secant update formula}, volume={20}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0032216893&partnerID=MN8TOARS}, DOI={10.1137/S0895479896306021}, abstractNote={This paper analyzes rank modification of symmetric positive definite matrices H of the form H −M + P , where H −M denotes a step of reducing H to a lower-rank, symmetric and positive semidefinite matrix and (H −M) + P denotes a step of restoring H −M to a symmetric positive definite matrix. These steps and their generalizations for rectangular matrices are fully characterized. The well-known BFGS and DFP updates used in Hessian and inverse Hessian approximations provided the motivation for the generalizations and are special cases with H and P having rank one.}, number={2}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Funderlic, RE and Golub, GH}, year={1998}, month={Oct}, pages={428–436} }
 @article{chu_funderlic_golub_1997, title={On a variational formulation of the generalized singular value decomposition}, volume={18}, ISSN={["0895-4798"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0031520186&partnerID=MN8TOARS}, DOI={10.1137/S0895479895287079}, abstractNote={A variational formulation for the generalized singular value decomposition (GSVD) of a pair of matrices $A \in R^{m \times n}$ and $B \in R^{p \times n}$ is presented. In particular, a duality theory analogous to that of the SVD provides new understanding of left and right generalized singular vectors. It is shown that the intersection of row spaces of $A$ and $B$ plays a key role in the GSVD duality theory. The main result that characterizes left GSVD vectors involves a generalized singular value deflation process.}, number={4}, journal={SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS}, author={Chu, MT and Funderlic, RE and Golub, GH}, year={1997}, month={Oct}, pages={1082–1092} }
 @article{chen_chu_1996, title={On the least squares solution of inverse eigenvalue problems}, volume={33}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0009284930&partnerID=MN8TOARS}, DOI={10.1137/S0036142994264742}, abstractNote={An inverse eigenvalue problem, where a matrix is to be constructed from some or all of its eigenvalues, may not have a real-valued solution at all. An approximate solution in the sense of least squares is sometimes desirable. Two types of least squares problems are formulated and explored in this paper. In spite of their different appearance, the two problems are shown to be equivalent. Thus one new numerical method, modified from the conventional alternating projection method, is proposed. The method converges linearly and globally and can be used to generate good starting values for other faster but more expensive and locally convergent methods. The idea can be applied to multiplicative inverse eigenvalue problems for the purpose of preconditioning. Numerical examples are presented.}, number={6}, journal={SIAM Journal on Numerical Analysis}, author={Chen, X. and Chu, M.T.}, year={1996}, pages={2417–2430} }
 @article{chu_1995, title={A list of matrix flows with applications}, url={http://dx.doi.org/10.1090/fic/003/07}, DOI={10.1090/fic/003/07}, abstractNote={Many mathematical problems such as existence questions are studied by using an appropriate realization process either iteratively or continu ously This article is a collection of di erential equations that have been proposed as special continuous realization processes In some cases there are remarkable connections between smooth ows and discrete numerical algorithms In other cases the ow approach seems advantageous in tack ling very di cult problems The ow approach has potential applications ranging from new development of numerical algorithms to the theoretical solution of open problems Various aspects of the recent development and applications of the ow approach are reviewed in this article}, journal={Hamiltonian and Gradient Flows, Algorithms and Control}, publisher={American Mathematical Society}, author={Chu, Moody}, year={1995}, month={May}, pages={87–97} }
 @article{chu_1995, title={Constructing a Hermitian Matrix from Its Diagonal Entries and Eigenvalues}, volume={16}, url={http://dx.doi.org/10.1137/s0895479893243177}, DOI={10.1137/s0895479893243177}, abstractNote={Given two vectors $a,\lambda \in R^{n}$, the Schur--Horn theorem states that $a$ majorizes $\lambda$ if and only if there exists a Hermitian matrix $H$ with eigenvalues $\lambda$ and diagonal entries $a$. While the theory is regarded as classical by now, the known proof is not constructive. To construct a Hermitian matrix from its diagonal entries and eigenvalues therefore becomes an interesting and challenging inverse eigenvalue problem. Two algorithms for determining the matrix numerically are proposed in this paper. The lift and projection method is an iterative method that involves an interesting application of the Wielandt--Hoffman theorem. The projected gradient method is a continuous method that, besides its easy implementation, offers a new proof of existence because of its global convergence property.}, number={1}, journal={SIAM Journal on Matrix Analysis and Applications}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T.}, year={1995}, month={Jan}, pages={207–217} }
 @book{chu_plemmons_1995, title={Numerical methods for adaptive-optics systems}, author={Chu, Moody T. and Plemmons, R.}, year={1995} }
 @book{chu_funderlic_golub_1995, title={On a new geometric meaning of the BFGS update}, author={Chu, Moody T. and Funderlic, R.E. and Golub, G.H.}, year={1995} }
 @book{chu_1995, title={On the refinement of a Newton method for the inverse Toeplitz eigenvalue Problem}, author={Chu, Moody}, year={1995} }
 @article{chu_funderlic_golub_1995, title={Rank-one reduction formula and its applications to matrix factorizations}, volume={37}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0029543087&partnerID=MN8TOARS}, DOI={10.1137/1037124}, abstractNote={Let $A \in R^{m \times n} $ denote an arbitrary matrix. If $x \in R^n $ and $y \in R^m $ are vectors such that $\omega = y^T Ax \ne 0$, then the matrix $B: = A - \omega ^{ - 1} Axy^T A$ A has rank exactly one less than the rank of A. This Wedderburn rank-one reduction formula is easy to prove, yet the idea is so powerful that perhaps all matrix factorizations can be derived from it. The formula also appears in places such as the positive definite secant updates BFGS and DFP as well as the ABS methods. By repeatedly applying the formula to reduce ranks, a biconjugation process analogous to the Gram–Schmidt process with oblique projections can be developed. This process provides a mechanism for constructing factorizations such as ${\text{LDM}}^T $, QR, and SVD under a common framework of a general biconjugate decomposition $V^T AU = \Omega $ that is diagonal and nonsingular. Two characterizations of biconjugation provide new insight into the Lanczos method and its breakdown. One characterization shows that ...}, number={4}, journal={SIAM Review}, author={Chu, Moody T. and Funderlic, Robert E. and Golub, Gene H.}, year={1995}, pages={512–530} }
 @article{chu_1995, title={Scaled Toda-like flows}, volume={215}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0040928421&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(93)00091-D}, abstractNote={This paper discusses the class of isospectral flows Ẋ = [X, A ∘ X], where ∘ denotes the Hadamard product and [., .] is the Lie bracket. The presence of A allows arbitrary and independent scaling for each element in the matrix X. The time-1 mapping of the scaled Toda-like flow still enjoys a QR-like iteration. The scaled structure includes the classical Toda flow, Brockett's double bracket flow, and other interesting flows as special cases. Convergence proof is thus unified and simplified. The effect of scaling on a variety of applications is demonstrated by examples.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.T.}, year={1995}, pages={261–273} }
 @article{chu_wright_1995, title={The educational testing problem revisited}, volume={15}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-21844482940&partnerID=MN8TOARS}, DOI={10.1093/imanum/15.1.141}, abstractNote={The educational testing problem is a convex non-smooth optimization problem. We recast the problem so that classical non-smooth optimization techniques such as the ellipsoid method can be readily applicable. Attention is paid to Dikin's method where a special barrier function and interior ellipsoids for the feasible domain are explicitly formulated. The implementation is much easier than that by Fletcher. The convergence property is numerically demonstrated}, number={1}, journal={IMA Journal of Numerical Analysis}, author={Chu, M.T. and Wright, J.W.}, year={1995}, pages={141–160} }
 @inbook{a list of matrix flows with applications_1994, url={https://bookstore.ams.org/fic-3}, booktitle={Hamiltonian and Gradient Flows, Algorithms and Control}, year={1994} }
 @book{chu_brown_ellison_plemmons_1994, place={Philadelphia, PA}, title={Proceedings of the Cornelius Lanczos International Centenary Conference}, ISBN={9780898713398}, publisher={SIAM}, year={1994} }
 @article{chu_erbrecht_1994, title={Symmetric Toeplitz Matrices with Two Prescribed Eigenpairs}, volume={15}, url={http://dx.doi.org/10.1137/s0895479891221757}, DOI={10.1137/s0895479891221757}, abstractNote={The inverse problem of constructing a real symmetric Toeplitz matrix based on two prescribed eigenpairs is considered. Two new results are obtained. First, it is shown that the dimension of the subspace of Toeplitz matrices with two generically prescribed eigenvectors is independent of the size of the problem, and, in fact, is either two, three, or four, depending upon whether the eigenvectors are symmetric or skew-symmetric and whether $n$ is even or odd. This result is quite notable in that when only one eigenvector is prescribed the dimension is known to be at least $\[(n + 1)/2\]$. Taking into account the prescribed eigenvalues, the authors then show how each unit vector in the null subspace of a certain matrix uniquely determines a Toeplitz matrix that satisfies the prescribed eigenpairs constraint. The cases where two prescribed eigenpairs uniquely determine a Toeplitz matrix are explicitly characterized.}, number={2}, journal={SIAM Journal on Matrix Analysis and Applications}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Erbrecht, Melissa A.}, year={1994}, month={Apr}, pages={623–635} }
 @article{chu_watterson_1993, title={On a Multivariate Eigenvalue Problem, Part I: Algebraic Theory and a Power Method}, volume={14}, url={http://dx.doi.org/10.1137/0914066}, DOI={10.1137/0914066}, abstractNote={Multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated. By using the method of Lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem. For over 30 years an iterative method proposed by Horst [Psychometrika, 26 (1961), pp. 129–149J has been used for solving the multivariate eigenvalue problem. Yet the theory of convergence has never been complete. The number of solutions to the multivariate eigenvalue problem also remains unknown. This paper contains two new results. By using the degree theory, a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved. A convergence property of Horst’s method by forming it as a generalization of the so-called power method is then proved. The discussion leads to new formulations of nume...}, number={5}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Watterson, J. Loren}, year={1993}, month={Sep}, pages={1089–1106} }
 @book{chu_1993, title={On the differential equation dX dt = [X; k(X)] where k is a Toeplitz annihilator}, author={Chu, Moody}, year={1993} }
 @article{chu_1993, title={The stability group of symmetric Toeplitz matrices}, volume={185}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-43949170372&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(93)90208-6}, abstractNote={In regard to the linear subspace T(n) of n × n symmetric Toeplitz matrices over the real field, the collection L(n) of all real and orthogonal matrices Q such that QTQT ϵ L(n) whenever T ϵ T(n) forms a group, called the stability group of T(n). This paper shows that L(n) is finite. In fact, L(n) has exactly eight elements regardless of the dimension n. Group elements in L(n) are completely characterized.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.T.}, year={1993}, pages={119–123} }
 @article{chu_1992, title={Matrix differential equations: a continuous realization process for linear algebra problems}, volume={18}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0346928644&partnerID=MN8TOARS}, DOI={10.1016/0362-546X(92)90157-A}, number={12}, journal={Nonlinear Analysis}, author={Chu, M.T.}, year={1992}, pages={1125–1146} }
 @article{chu_1992, title={Numerical methods for inverse singular value problems}, volume={29}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0026883934&partnerID=MN8TOARS}, DOI={10.1137/0729054}, abstractNote={Two numerical methods—one continuous and the other discrete—are proposed for solving inverse singular value problems. The first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function. The second method generalizes an iterative process proposed originally by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving inverse eigenvalue problems. With the geometry understood from the first method, it is shown that the second method (also, the method proposed by Friedland, Nocedal, and Overton for inverse eigenvalue problems) is a variation of the Newton method. While the continuous method is expected to converge globally at a slower rate (in finding a stationary point of the objective function), the discrete method is proved to converge locally at a quadratic rate (if there is a solution). Some numerical examples are presented.}, number={3}, journal={SIAM Journal on Numerical Analysis}, author={Chu, Moody T.}, year={1992}, pages={885–903} }
 @book{chu_1992, title={On the inverse eigenvalue problem for real circulant matrices}, author={Chu, Moody}, year={1992} }
 @article{chu_1991, title={A continuous Jacobi-like approach to the simultaneous reduction of real matrices}, volume={147}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0039546401&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(91)90230-T}, abstractNote={The problem of simultaneous reduction of real matrices by either orthogonal similarity or orthogonal equivalence transformations is considered. Based on the Jacobi idea of minimizing the sum of squares of the complementary part of the desired form to which matrices are reduced, the projected-gradient method is used in this paper. It is shown that the projected gradient of the objective function can be formulated explicitly. This gives rise to a system of ordinary differential equations that can be readily solved by numerical software. The advantages of this approach are that the desired form to which matrices are reduced can be almost arbitrary, and that if a desired form is not attainable, then the limit point of the corresponding differential equation gives a way of measuring the distance from the best reduced matrices to the nearest matrices that have the desired form. The general procedure for deriving these differential equations is discussed. The framework can be generalized to the complex-valued case. Some applications are given.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.T.}, year={1991}, pages={75–96} }
 @article{chu_driessel_1991, title={Constructing Symmetric Nonnegative Matrices with Prescribed Eigenvalues by Differential Equations}, volume={22}, url={http://dx.doi.org/10.1137/0522088}, DOI={10.1137/0522088}, abstractNote={The inverse eigenvalue problem is solved for symmetric nonnegative matrices by means of a differential equation. If the given spectrum is feasible, then a symmetric nonnegative matrix can be constructed simply by following the solution curve of the differential system. The choice of the vector field is based on the idea of minimizing the distance between the cone of symmetric nonnegative matrices and the isospectral surface determined by the given spectrum. The projected gradient of the objective function is explicitly described. Using center manifold theory, it is also shown that the $\omega$-limit set of any solution curve is a single point. Some numerical examples are presented.}, number={5}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Driessel, Kenneth R.}, year={1991}, month={Sep}, pages={1372–1387} }
 @article{chu_1991, title={Least Squares Approximation by Real Normal Matrices with Specified Spectrum}, volume={12}, url={http://dx.doi.org/10.1137/0612009}, DOI={10.1137/0612009}, abstractNote={The problem of best approximating a given real matrix in the Frobenius norm by real, normal matrices subject to a prescribed spectrum is considered. The approach is based on using the projected gradient method. The projected gradient of the objective function on the manifold of constraints can be formulated explicitly. This gives rise to a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality condition from which some interesting properties of the stationary points are related to the well-known Wielandt–Hoffman theorem.}, number={1}, journal={SIAM Journal on Matrix Analysis and Applications}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T.}, year={1991}, month={Jan}, pages={115–127} }
 @article{chu_driessel_1990, title={Projected gradient method for least squares matrix approximations with spectral constraints}, volume={27}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0025474861&partnerID=MN8TOARS}, DOI={10.1137/0727062}, abstractNote={The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly. This gives rise to the construction of a descent flow that can be followed numerically. The explicit form also facilitates the computation of the second-order optimality conditions. Examples of applications are discussed. With slight modifications, the procedure can be extended to solve least squares problems for general matrices subject to singular-value constraints.}, number={4}, journal={SIAM Journal on Numerical Analysis}, author={Chu, Moody T. and Driessel, Kenneth R.}, year={1990}, pages={1050–1060} }
 @article{chu_1990, title={Solving additive inverse eigenvalue problems for symmetric matrices by the homotopy method}, volume={10}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0039206135&partnerID=MN8TOARS}, DOI={10.1093/imanum/10.3.331}, number={3}, journal={IMA Journal of Numerical Analysis}, author={Chu, M.T.}, year={1990}, pages={331–342} }
 @book{chu_driessel_1990, title={Some numerical experiments with isospectral flows}, number={90-01}, institution={Idaho State University}, author={Chu, Moody T. and Driessel, K.R.}, year={1990} }
 @article{chu_guiguis_1989, title={A numerical method for solving interface problems arising in two-point boundary value problems}, volume={74}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0024733864&partnerID=MN8TOARS}, DOI={10.1016/0045-7825(89)90089-3}, abstractNote={Abstract A numerical approach iterating on the position of interface points is suggested for solving interface problems arising in two-point boundary value problems. Given an interface problem, it is decomposed into several standard local boundary value problems which are coupled at the interface points. Meanwhile, a nonlinear equation involving the interface points are formulated from the interface conditions. The advantages of this approach are that the boundary conditions for each local problem can easily be selected by considering the natural physical requirements, and that each of the local problems can be solved independently by standard numerical BVP techniques.}, number={1}, journal={Computer Methods in Applied Mechanics and Engineering}, author={Chu, M.T. and Guiguis, G.H.}, year={1989}, pages={99–113} }
 @book{chu_driessel_1989, title={Can real symmetric Toeplitz matrices have arbitrary real spectra?}, author={Chu, Moody T. and Driessel, K.R.}, year={1989} }
 @book{chu_1988, title={A derivative-free iterative method for locating the hand position of a robot manipulator}, number={MCSP48-0189}, institution={Argonne National Laboratory}, author={Chu, Moody T.}, year={1988}, month={Dec} }
 @article{chu_1988, title={A note on the homotopy method for linear algebraic eigenvalue problems}, volume={105}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-38249028600&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(88)90015-8}, abstractNote={Recently the homotopy method has been applied to solve linear algebraic eigenvalue problems. On the basis of theoretical advantages and practical experiments, the method has been suggested as a serious alternative to EISPACK for finding all isolated eigenpairs of large, sparse linear algebraic eigenvalue problems on SIMD machines. This note offers a simpler proof than Li and Sauer's of the existence of homotopy curves for eigenvalue problems of general matrices.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.T.}, year={1988}, pages={225–236} }
 @article{chu_li_sauer_1988, title={Homotopy Method for General $\lambda $-Matrix Problems}, volume={9}, url={http://dx.doi.org/10.1137/0609043}, DOI={10.1137/0609043}, abstractNote={This paper describes a homotopy method used to solve the kth-degree $\lambda $-matrix problem $( A_k \lambda ^k + A_{k - 1} \lambda ^{k - 1} + \cdots + A_1 \lambda + A_0 )x = 0$. A special homotopy equation is constructed for the case where all coefficients are general $n\times n$ complex matrices. Smooth curves connecting trivial solutions to desired eigenpairs are shown to exist. The homotopy equations maintain the nonzero structure of the underlying matrices (if there is any) and the curves correspond only to different initial values of the same ordinary differential equation. Therefore, the method might be used to find all isolated eigenpairs for large-scale $\lambda $-matrix problems on single-instruction multiple data (SIMD) machines.}, number={4}, journal={SIAM Journal on Matrix Analysis and Applications}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Li, T. Y. and Sauer, Tim}, year={1988}, month={Oct}, pages={528–536} }
 @article{chu_norris_1988, title={Isospectral Flows and Abstract Matrix Factorizations}, volume={25}, url={http://dx.doi.org/10.1137/0725080}, DOI={10.1137/0725080}, abstractNote={A general framework for constructing isospectral flows in the space $\operatorname{gl}(n)$ of n by n matrices is proposed. Depending upon how $\operatorname{gl}(n)$ is split, this framework gives rise to different types of abstract matrix factorizations. When sampled at integer times, these flows naturally define special iterative processes, and each flow is associated with the sequence generated by the corresponding abstract factorization. The proposed theory unifies as special cases the well-known matrix decomposition techniques used in numerical linear algebra and is likely to offer a broader approach to the general matrix factorization problem.}, number={6}, journal={SIAM Journal on Numerical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Norris, Larry K.}, year={1988}, month={Dec}, pages={1383–1391} }
 @article{chu_1988, title={On the continuous realization of iterative processes}, volume={30}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0024073497&partnerID=MN8TOARS}, DOI={10.1137/1030090}, abstractNote={Many important mathematical problems are solved by iterative methods. Many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems. This paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods.}, number={3}, journal={SIAM Review}, author={Chu, Moody T.}, year={1988}, pages={375–387} }
 @book{chu_1987, title={On a differential equation approach to the additive inverse eigenvalue problems}, author={Chu, Moody}, year={1987} }
 @article{chu_hamilton_1987, title={Parallel Solution of ODE’<scp>s</scp> by Multiblock Methods}, volume={8}, url={http://dx.doi.org/10.1137/0908039}, DOI={10.1137/0908039}, abstractNote={The notion of linear multi-step methods for solving ordinary differential equations is generalized to a class of multi-block methods. In a multi-block method step values are all obtained together in a single block advance which is accomplished by allocating the parallel tasks on separate processors. The expected benefit of multi-block methods is the speedup in the computation of solutions. The basic formulation is described. Examples are given to demonstrate the existence of such schemes. The predictor-corrector type combination is formed and the resulting stability problem is considered. Test results of one of these multi-block methods on the Denelcor HEP machine are reported.}, number={3}, journal={SIAM Journal on Scientific and Statistical Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T. and Hamilton, Hans}, year={1987}, month={May}, pages={342–353} }
 @book{chu_hamilton_1987, title={Some remarks on the zero-stability of multiblock methods}, author={Chu, Moody T. and Hamilton, H.}, year={1987} }
 @article{chu_1986, title={A continuous approximation to the generalized Schur decomposition}, volume={78}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0038953719&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(86)90019-4}, abstractNote={We consider the problem of approximating the generalized Schur decomposition of a matrix pencil A − λB by a family of differentiable orthogonal transformations. It is shown that when B is nonsingular this approach is feasible and can be fully expressed as an autonomous differential system. When B is singular, we show that the location of zero diagonal entries of B affects the feasibility of such an approach, and hence we conclude that at least the QZ algorithm cannot be extended continuously.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.}, year={1986}, pages={119–132} }
 @article{chu_1986, title={A differential equation approach to the singular value decomposition of bidiagonal matrices}, volume={80}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0002063624&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(86)90278-8}, abstractNote={We consider the problem of approximating the singular value decomposition of a bidiagonal matrix by a one-parameter family of differentiable matrix flows. It is shown that this approach can be fully expressed as an autonomous, homogeneous, and cubic dynamical system. Asymptotic behavior is justified by the established theory of the Toda lattice.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.T.}, year={1986}, pages={71–79} }
 @book{chu_1986, title={Continuous power method}, author={Chu, Moody}, year={1986} }
 @article{chu_1986, title={Curves on $S^{n - 1} $ That Lead to Eigenvalues or Their Means of a Matrix}, volume={7}, url={http://dx.doi.org/10.1137/0607048}, DOI={10.1137/0607048}, abstractNote={This paper discusses two dynamical systems on the unit sphere $S^{n - 1} $ in $\mathbb{R}^n $ space, each defined in terms of a real square matrix M. The solutions of these systems are found to converge to points which provide essential information about eigenvalues of the matrix M. It is shown, in particular, how the dynamics of the second flow is analogous to that of the Rayleigh quotient iterations.}, number={3}, journal={SIAM Journal on Algebraic Discrete Methods}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T.}, year={1986}, month={Jul}, pages={425–432} }
 @article{chu_1985, title={Asymptotic analysis of Toda lattice on diagonalizable matrices}, volume={9}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0022011580&partnerID=MN8TOARS}, DOI={10.1016/0362-546X(85)90072-0}, abstractNote={Soit X˙=[X, Π 0 (G(X))], X(0)=X 0 , X est une matrice a valeur complexe et G(X) une integrale de contour a valeur matricielle. On etudie la dynamique de X(t) sous les hypotheses suivantes: 1) X(0)=X 0 ∈R n×n est diagonalisable; 2) G(Z)=Z; 3) X 0 est une matrice de Hessenberg superieure irreductible}, number={2}, journal={Nonlinear Analysis}, author={Chu, M.T.}, year={1985}, pages={193–201} }
 @article{chu_1985, title={Symbolic calculation of the trace of the power of a tridiagonal matrix}, volume={35}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0022240153&partnerID=MN8TOARS}, DOI={10.1007/BF02240193}, number={3-4}, journal={Computing}, author={Chu, M.T.}, year={1985}, pages={257–268} }
 @article{chu_1984, title={A simple application of the homotopy method to symmetric eigenvalue problems}, volume={59}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0001806232&partnerID=MN8TOARS}, DOI={10.1016/0024-3795(84)90160-5}, abstractNote={The homotopy method is used to find all eigenpairs of symmetric matrices. A special homotopy is constructed for Jacobi matrices. It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. These curves are solutions of a certain ordinary differential equation with different initial values. Hence, they can be followed numerically. Incorporated with sparse matrix techniques, this method might be used to solve eigenvalue problems for large scale matrices.}, number={C}, journal={Linear Algebra and Its Applications}, author={Chu, M.T.}, year={1984}, pages={85–90} }
 @article{chu_1984, title={On the Global Convergence of the Toda Lattice for Real Normal Matrices and Its Applications to the Eigenvalue Problem}, volume={15}, ISSN={0036-1410 1095-7154}, url={http://dx.doi.org/10.1137/0515004}, DOI={10.1137/0515004}, abstractNote={The asymptotic behavior of the Toda lattice, when acting on real normal matrices, is studied. It is shown that the solution flow eventually converges to a diagonal block form where for a real eigenvalue the associated block is of size $1 \times 1$ with that eigenvalue as its element and for complex-conjugate pairs of eigenvalues the associated block is of size $2 \times 2$ with the real part as its diagonal elements and the (negative) imaginary part as its off-diagonal elements. This result generalizes the well-known asymptotic behavior of Jacobi matrices and is consistent with that from the $QR$-algorithm.}, number={1}, journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T.}, year={1984}, month={Jan}, pages={98–104} }
 @article{chu_1984, title={The Generalized Toda Flow, the QR Algorithm and the Center Manifold Theory}, volume={5}, url={http://dx.doi.org/10.1137/0605020}, DOI={10.1137/0605020}, abstractNote={A continuous version of the classical QR algorithm, known as the Toda flow, is generalized to complex-valued, full and nonsymmetric matrices. It is shown that this generalized Toda flow, when sampled at integer times, gives the same sequence of matrices as the OR algorithm applied to the matrix exp $(G(X_0 ))$. When $G(X) = X$, global convergence is deduced for the case of distinct real eigenvalues. This convergence property can also be understood locally by the center manifold theory. It is shown that the manifold of upper triangular matrices with decreasing main diagonal entries is the stable center manifold for the Toda flow. One interesting example is given to demonstrate geometrically the dynamical behavior of this flow.}, number={2}, journal={SIAM Journal on Algebraic Discrete Methods}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Chu, Moody T.}, year={1984}, month={Jun}, pages={187–201} }
 @article{chu_1983, title={An automatic multistep method for solving stiff initial value problems}, volume={9}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0020822212&partnerID=MN8TOARS}, DOI={10.1016/0377-0427(83)90016-X}, abstractNote={A multistep method with matricial coefficients is developed. It can be used to solve stiff initial value problems of the form y′ = Ay + g(x, y). This method bears the nature of the classical Adams—Bashforth—Moulton PC formula and can be shown to be consistent, convergent and A-stable. A careful reformulation of this method legitimatizes the implementation of this algorithm in a variable-step variable-order fashion. Numerical test results from a PECE mode of this method show its possible advantages.}, number={3}, journal={Journal of Computational and Applied Mathematics}, author={Chu, M.T.}, year={1983}, pages={229–238} }
 @article{chu_1983, title={On a numerical treatment for the curve-tracing of the homotopy method}, volume={42}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0006797929&partnerID=MN8TOARS}, DOI={10.1007/BF01389577}, number={3}, journal={Numerische Mathematik}, author={Chu, M.T.}, year={1983}, pages={323–329} }
 @article{chu, title={Data Mining and Applied Linear Algebra}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.149.6302}, author={Chu, Moody} }
 @article{plemmonsy, title={Low Rank Circulant Approximation}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.6788}, author={Plemmonsy, Moodyt. Chuandrobertj.} }
 @article{chu_chu_lin, title={REAL SYMMETRIC QUADRATIC MODEL UPDATING THAT PRESERVES POSITIVE DEFINITENESS AND NO SPILL-OVER}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.149.5746}, author={Chu, Delin and Chu, Moody and Lin, Wen-wei} }
 @article{chu_funderlic_gene_golub, title={Rank Modifications Of Semi-Definite Matrices With Applications To Secant Updates}, url={http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.6137}, author={Chu, Moody and Funderlic, R. E. and Gene and Golub, H.} }