Abstract
The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form w T f(A)v with \(A\in {\mathbb {R}}^{n\times n}\) a large matrix, f a function, and \(v,w\in {\mathbb {R}}^{n}\) arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v, and derivatives of f do not change sign in the convex hull of the spectrum of A, a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating v T f(A)v by a pair of Gauss and Gauss-Radau quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with A T. The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with A T than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.
Similar content being viewed by others
References
Baglama, J., Fenu, C., Reichel, L., Rodriguez, G.: Analysis of directed networks via partial singular value decomposition and Gauss quadrature. Linear Algebra Appl. 456, 93–121 (2014)
Bai, Z., Day, D., Ye, Q.: ABLE: An Adaptive block Lanczos method for non-Hermitian eigenvalue problems. SIAM J. Matrix Anal. Appl. 20, 1060–1082 (1999)
Bai, Z., Fahey, M., Golub, G.H.: Some large scale matrix computation problems. J. Comput. Appl. Math. 74, 71–89 (1996)
Batagelj, V., Mrvar, A.: Pajek data sets. http://vlado.fmf.uni-lj.si/pub/networks/data/ (2006)
Bellalij, M., Reichel, L., Rodriguez, G., Sadok, H.: Bounding matrix functionals via partial global block Lanczos decomposition. Appl. Numer. Math. 94, 127–139 (2015)
Benzi, M., Boito, P.: Quadrature rule-based bounds for functions of adjacency matrices. Linear Algebra Appl. 433, 637–652 (2010)
Benzi, M., Estrada, E., Klymko, C.: Ranking hubs and authorities using matrix functions. Linear Algebra Appl. 438, 2447–2474 (2013)
Benzi, M., Klymko, C.: Total communicability as a centrality measure. J. Complex Networks 1, 1–26 (2013)
Biological Networks Data Sets of Newcastle University. Available at http://www.biological-networks.org/
Brezinski, C., Fika, P., Mitrouli, M.: Moments of a linear operator on a Hilbert space, with applications to the trace of the inverse of matrices and the solution of equations. Numer. Linear Algebra Appl. 19, 937–953 (2012)
Brezinski, C., Fika, P., Mitrouli, M.: Estimations of the trace of powers of positive self-adjoint operators by extrapolation of the moments. Electron. Trans. Numer. Anal. 39, 144–155 (2012)
Calvetti, D., Golub, G.H., Reichel, L.: An adaptive Chebyshev iterative method for nonsymmetric linear systems of equations based on modified moments. Numer. Math. 67, 21–40 (1997)
Calvetti, D., Hansen, P.C., Reichel, L.: L-curve curvature bounds via Lanczos bidiagonalization. Electron. Trans. Numer. Anal. 14, 20–35 (2002)
Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Computable error bounds and estimates for the conjugate gradient method. Numer. Algorithms 25, 79–88 (2000)
Calvetti, D., Reichel, L.: Application of a block modified Chebyshev algorithm to iterative solution of symmetric linear systems with multiple right hand side vectors. Numer. Math. 68, 3–16 (1994)
Calvetti, D., Reichel, L., Sgallari, F.: Application of anti-Gauss quadrature rules in linear algebra. In: Gautschi, W., Golub, G.H., Opfer, G. (eds.) Applications and Computation of Orthogonal Polynomials, pp 41–56. Birkhäuser, Basel (1999)
Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)
Estrada, E., Higham, D.J.: Network properties revealed through matrix functions. SIAM Rev. 52, 696–714 (2010)
Fenu, C., Martin, D., Reichel, L., Rodriguez, G.: Network analysis via partial spectral factorization and Gauss quadrature. SIAM J. Sci. Comput. 35, A2046–A2068 (2013)
Fenu, C., Martin, D., Reichel, L., Rodriguez, G.: Block Gauss and anti-Gauss quadrature with application to networks. SIAM J. Matrix Anal. Appl. 34, 1655–1684 (2013)
Fika, P., Mitrouli, M., Roupa, P.: Estimates for the bilinear form x T a −1 y with applications to linear algebra problems. Electron. Trans. Numer. Anal. 43, 70–89 (2014)
Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput. 13, 1236–1264 (1992)
Gautschi, W.: Orthogonal Polynomials: Approximation and computation. Oxford University Press, Oxford (2004)
Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Griffiths, D.F., Watson, G.A. (eds.) Numerical Analysis 1993, pp. 105–156. Longman, Essex, England (1994)
Golub, G.H., Meurant, G.: Matrices, moments and quadrature with applications. Princeton University Press, Princeton (2010)
Gragg, W.B.: Matrix interpretation and applications of the continued fraction algorithm Rocky Mountain. J. Math 4, 213–225 (1974)
Higham, N.J.: Functions of matrices: Theory and computation. SIAM, Philadelphia (2008)
Jeong, H., Mason, S., Barabási, A.-L., Oltvai, Z.N.: Lethality and centrality of protein networks. Nature 411, 41–42 (2001)
Lambers, J.V.: Enhancement of Krylov subspace spectral methods by block Lanczos iteration. Electron. Trans. Numer. Anal. 31, 86–109 (2008)
Laurie, D.P.: Anti-Gaussian quadrature formulas. Math. Comp. 65, 735–747 (1996)
Marcelino, J., Kaiser, M.: Critical Paths in a Metapopulation Model of H1N1: Efficiently delaying influenza spreading through flight cancellation. PLoS Currents Influenza (2012)
Morigi, S., Reichel, L., Sgallari, F.: An iterative Lavrentiev regularization method. BIT 46, 589–606 (2006)
Notaris, S.: Gauss-kronrod quadrature formulae - a survey of fifty years of research. Electron. Trans. Numer. Anal. 45, 371–404 (2016)
Pozza, S., Pranić, M.S., Strakoš, Z.: Gauss quadrature for quasi-definite linear functionals, IMA J. Numer. Anal., in press
Sun, S., Ling, L., Zhang, N., Li, G., Chen, R.: Topological structure analysis of the protein-protein interaction network in budding yeast. Nucleic Acids Res. 31, 2443–2450 (2003)
SNAP Network Data Sets. Available at http://snap.stanford.edu/data/index.html
Acknowledgements
The authors would like to thank Gérard Meurant for comments that lead to improvements of the presentation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Dirk Laurie on the occasion of his 70th birthday.
Rights and permissions
About this article
Cite this article
Alqahtani, H., Reichel, L. Simplified anti-Gauss quadrature rules with applications in linear algebra. Numer Algor 77, 577–602 (2018). https://doi.org/10.1007/s11075-017-0329-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-017-0329-6