Single-Precision Calculation of Iterative Refinement of Eigenpairs of a Real Symmetric-Definite Generalized Eigenproblem by Using a Filter Composed of a Single Resolvent

By using a filter, we calculate approximate eigenpairs of a real symmetric-definite generalized eigenproblem Av = λBv whose eigenvalues are in a specified interval. In our experiments in this paper, the IEEE-754 single-precision floating-point (binary 32bit) number system is used for calculations. In general, a filter is constructed by using some resolvents with different shifts ρ. For a given vector x, an action of a resolvent is given by solving a system of linear equations C(ρ)y = Bx for y, here the coefficient C(ρ) = A − ρB is symmetric. We assume to solve this system of linear equations by matrix factorization of C(ρ), for example by the modified Cholesky method (LDLT decomposition method). When both matrices A and B are banded, C(ρ) is also banded and the modified Cholesky method for banded system can be used to solve the system of linear equations. The filter we used is either a polynomial of a resolvent with a real shift, or a polynomial of an imaginary part of a resolvent with an imaginary shift. We use only a single resolvent to construct the filter in order to reduce both amounts of calculation to factor matrices and especially storage to hold factors of matrices. The most disadvantage when we use only a single resolvent rather than many is, such a filter have poor properties especially when compuation is made in single-precision. Therefore, approximate eigenpairs required are not obtained in good accuracy if they are extracted from the set of vectors made by an application of a combination of B-orthonormalization and filtering to a set of initial random vectors. However, experiments show approximate eigenpairs required are refined well if they are extracted from the set of vectors obtained by a few applications of a combination of B-orthonormalization and filtering to a set of initial random vectors.