论文信息 - Approximation of the Determinant of Large Sparse Symmetric Positive Definite Matrices

Approximation of the Determinant of Large Sparse Symmetric Positive Definite Matrices

This paper is concerned with the problem of approximating det(A)1/n for a large sparse symmetric positive definite matrix A of order n. It is shown that an efficient solution of this problem is obtained by using a sparse approximate inverse of A. The method is explained and theoretical properties are discussed. The method is ideal for implementation on a parallel computer. Numerical experiments are described that illustrate the performance of this new method and provide a comparison with Monte Carlo--type methods from the literature.

Arnold Reusken | A. Reusken

[1] J. Lamperti. ON CONVERGENCE OF STOCHASTIC PROCESSES , 1962 .

[2] Philip Rabinowitz,et al. Methods of Numerical Integration , 1985 .

[3] L. Kolotilina,et al. On a family of two-level preconditionings of the incomplete block factorization type , 1986 .

[4] M. Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines , 1989 .

[5] L. Kolotilina,et al. Factorized Sparse Approximate Inverse Preconditionings I. Theory , 1993, SIAM J. Matrix Anal. Appl..

[6] I. Montvay,et al. Quantum Fields on a Lattice: Introduction , 1994 .

[7] P. Forcrand. Progress on lattice QCD algorithms , 1995, hep-lat/9509082.

[8] G. Golub,et al. Some large-scale matrix computation problems , 1996 .

[9] J. Sexton,et al. Error estimate for the valence approximation and for a systematic expansion of full QCD , 1997 .

[10] A. Frommer,et al. Exploiting Structure in Krylov Subspace Methods for the Wilson Fermion Matrix , 1997 .

[11] Marcus J. Grote,et al. Parallel Preconditioning with Sparse Approximate Inverses , 1997, SIAM J. Sci. Comput..

[12] M. Hasenbusch. Speeding up finite step-size updating of full QCD on the lattice , 1998, hep-lat/9807031.

[13] B. Medeke. On Algebraic Multilevel Preconditioners in Lattice Gauge Theory , 2000 .