zbMATH — the first resource for mathematics

Some large-scale matrix computation problems. (English) Zbl 0870.65035
A numerical method is outlined to bound the bilinear form \(u^T f(A)v\) for a given matrix \(A\) and vectors \(u\) and \(v\). A simulation method is also given to estimate the trace of the inverse and the determinant of sparse matrices. Computational results are presented.

65F30 Other matrix algorithms (MSC2010)
65F05 Direct numerical methods for linear systems and matrix inversion
65F40 Numerical computation of determinants
65F50 Computational methods for sparse matrices
Full Text: DOI
[1] Davis, P.; Rabinowitz, P., Methods of numerical integration, (1984), Academic Press New York
[2] Dong, S.; Liu, K., Stochastic estimation with z2 noise, Phys. lett. B, 328, 130-136, (1994)
[3] Gautschi, W., A survey of Gauss-Christoffel quadrature formulae, (), 73-157
[4] Golub, G., Some modified matrix eigenvalue problems, SIAM rev., 15, 318-334, (1973) · Zbl 0254.65027
[5] Golub, G.; Gutknecht, M., Modified moments for indefinite weight functions, Numer. math., 57, 607-624, (1989) · Zbl 0679.30029
[6] Golub, G.; Meurant, G., Matrics, moments and quadrature, () · Zbl 0888.65050
[7] Golub, G.; Strakos, Z., Estimates in quadratic formulas, () · Zbl 0822.65022
[8] Golub, G.; Van Loan, C., Matrix computations, (1989), Johns Hopkins University Press Baltimore, MD · Zbl 0733.65016
[9] Higham, N.J., The test matrix toolbox for Matlab, () · Zbl 0681.65029
[10] Hockney, G.M., Comparison of inversion algorithms for Wilson fermions, (), 301-304
[11] Hutchinson, M., A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines, Commun. statist. simula., 18, 1059-1076, (1989) · Zbl 0695.62113
[12] Jayanthi, C.S.; Wu, S.Y.; Cocks, J., The nature of vibrational modes of a self-similar fractal, ()
[13] Kittel, C., Introduction to solid state physics, (1972), Wiley New York
[14] Lanczos, C., An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. res. natl. bur. stand., 45, 225-280, (1950)
[15] Mitchell, A.R.; Griffiths, D.F., The finite difference method in partial differential equations, (1980), Wiley New York · Zbl 0417.65048
[16] Montvay, I.; Münster, G., Quantum fields on a lattice, (1994), Cambridge Univ. Press Cambridge
[17] Pollard, D., Convergence of stochastic processes, (1984), Springer New York · Zbl 0544.60045
[18] Robinson, P.; Wathen, J., Variational bounds on the entries of the inverse of a matrix, IMAJ.. numer. anal., 12, 463-486, (1992) · Zbl 0759.15016
[19] Sapoval, B.; Gobron, Th.; Margolina, A., Vibrations of fractal drums, Phy. rev. lett., 67, 2974-2977, (1991)
[20] Sexton, J.C.; Weingarten, D.H., The numerical estimation of the error induced by the valence approximation, (), 361-363
[21] Struble, G., Orthogonal polynomials: variable-signed weight functions, Numer. math., 5, 88-94, (1963) · Zbl 0107.05502
[22] Wu, S.Y.; Cocks, J.A.; Jayanthi, C.S., An accelerated inversion algorithm using the resolvent matrix method, Comput. phys. commun., 71, 125-133, (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.