Some large-scale matrix computation problems
Zhaojun Bai, Gark Fahey, Gene Howard Golub · Journal of Computational and Applied Mathematics · 1996
There are numerous applications in physics, statistics and electrical circuit simulation where it is required to bound entries and the trace of the inverse and the determinant of a large sparse matrix. All these computational tasks are related to the central mathematical problem studied in this paper, namely, bounding the bilinear form uTf(A)v for a given matrix A and vectors u and v, where f is a given smooth function and is defined on the spectrum of A. We will study a practical numerical algorithm for bounding the bilinear form, where the matrix A is only referenced through matrix-vector multiplications. A Monte Carlo method is also presented to efficiently estimate the trace of the inverse and the determinant of a large sparse matrix.