DETDEF

DETDEF, short for Determinant Deflation and Efficient Factorization, is a framework in numerical linear algebra designed to improve the computation of determinants and the solution of linear systems for large, sparse matrices. The method combines determinant deflation—isolating influential spectral components—with efficient matrix factorization techniques to reduce problem size and improve numerical stability. DETDEF is applicable to symmetric, positive definite, and general matrices and is used in contexts where log-determinants or determinant evaluations arise, such as in statistical physics, lattice computations, and large-scale simulations.

The core idea of DETDEF is to separate the troublesome part of the spectrum from a matrix and solve the deflated system on a reduced space. By projecting onto a complementary subspace that captures the dominant or near-singular behavior, the method produces a smaller, better-conditioned problem whose factorization is cheaper to obtain. The determinant and solutions of the original system are then reconstructed from the factorization of the deflated matrix and contributions from the deflated subspace, often via a Schur-like decomposition or related assembly scheme. This separation can improve stability and reduce the number of expensive operations required for repeated solves.

Implementation considerations include how the deflation subspace is chosen, such as approximate eigenvectors or singular vectors, and how the deflated matrix is formed. The approach is compatible with common factorizations (LU or LDL^T) and can incorporate preconditioning and iterative refinement. DETDEF is most advantageous when the spectrum has identifiable challenging components and when multiple linear solves or determinant evaluations are needed.

See also: numerical linear algebra, determinant computation, deflation, matrix factorization, Schur complement.