Christopher C. Paige

Bio: Christopher C. Paige is an academic researcher from McGill University. The author has contributed to research in topics: Matrix (mathematics) & Least squares. The author has an hindex of 34, co-authored 81 publications receiving 11241 citations. Previous affiliations of Christopher C. Paige include University of London & Kent State University.

LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

Abstract: An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I~QR is the most reliable algorithm when A is ill-conditioned.

Solution of Sparse Indefinite Systems of Linear Equations

Abstract: The method of conjugate gradients for solving systems of linear equations with a symmetric positive definite matrix A is given as a logical development of the Lanczos algorithm for tridiagonalizing...

Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems

Abstract: Received 4 June 1980; revised 23 September 1981, accepted 28 February 1982 This work was supported by Natural Sciences and Engineering Research Council of Canada Grant A8652, by the New Zealand Department of Scientific and Industrial Research; and by U S. National Science Foundation Grants MCS-7926009 and ECS-8012974, the Department of Energy under Contract AM03-76SF00326, PA No. DE-AT03-76ER72018, the Office of Naval Research under Contract N00014-75-C-0267, and the Army Research Office under Contract DAA29-79-C-0U0, Authors' addresses: C. C. Paige, School of Computer Science, McGill University, Montreal, Quebec, Canada H3A 2K6; M. A Saundem, Department of Operations Research, Stanford University, Stanford, CA 94305. Permmsion to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notme is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1982 ACM 0098-3500/82/0600-0[95 $00 75

Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms

Abstract: An algorithm is presented in this paper for computing state-space balancing transformations directly from a state-space realization. The algorithm requires no "squaring up" or unnecessary matrix products. Various algorithmic aspects are discussed in detail. A key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product. Other contragredient transformation applications are also described. It is further shown that a similar approach may be taken, involving the generalized singular value decomposition, to the classical simultaneous diagonalization problem. These SVD-based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems.

Towards a Generalized Singular Value Decomposition

Abstract: We suggest a form for, and give a constructive derivation of, the generalized singular value decomposition of any two matrices having the same number of columns We outline its desirable characteristics and compare it to an earlier suggestion by Van Loan [SIAM J Numer Anal, 13 (1976), pp 76–83] The present form largely follows from the work of Van Loan, but is slightly more general and computationally more amenable than that in the paper cited We also prove a useful extension of a theorem of Stewart [SIAM Rev 19 (1977), pp 634–662] on unitary decompositions of submatrices of a unitary matrix

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Numerical Optimization

Abstract: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Iterative Methods for Sparse Linear Systems

Abstract: Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems

Abstract: We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

Atomic Decomposition by Basis Pursuit

Abstract: The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.