Verified bounds for the determinant of real or complex point or interval matrices (Q2297164)

In verified computation one wants to compute a result with bounds for the numerical error. When computing a determinant of a large or ill-conditioned matrix, it is known that interval Gaussian elimination with pivoting is slow because it requires scalar computations and is prone to premature singularities. In this paper several steps to compute the determinant via LU factorization are analysed to sharpen the bounds. If for a permutation matrix \(P\) we get \(PA=LU-F\), then \(\det(A)=\det(P)\det(U)/\det(I+A^{-1}F)\), so that the analysis involves elements such as scaling the matrix, the determinant of a triangular matrix or a slightly perturbed identity matrix, and preconditioning. To achieve this, interval matrices are considered. A midpoint-radius notation is used: \((M,R)=\{A: M-R\le A\le M+R\}\). For small radius, the previous techniques can be applied, while for large radius it is not advised to use preconditioning since an Hadamard bound is better. The paper is using MATLAB notation and gives short snippets of code that implement the proposed techniques.