nearsingular

Nearsingular is an informal term used in mathematics and numerical analysis to describe a matrix, linear operator, or system that is close to being singular (non-invertible). In practice, this means the determinant is small, the spectrum contains eigenvalues near zero, and the smallest singular value is small. Such objects are almost but not quite invertible, and their behavior under perturbations is highly sensitive. The concept is closely tied to the condition number of a matrix, with near-singular cases typically exhibiting a large condition number, and to the distance to singularity, which in many norms is related to the smallest singular value.

In numerical computations, near-singularity leads to ill-conditioning, meaning small round-off errors or data perturbations can produce large changes in the solution. This can manifest when solving Ax = b with a nearly singular A or when computing inverses or decompositions that rely on stability near zero singular values. Indicators include a large cond(A), very small σ_min(A), and a determinant close to zero.

Handling near-singular cases involves strategies to improve stability. Regularization, such as Tikhonov methods, introduces a controlled bias to stabilize solutions. Using the pseudoinverse via singular value decomposition, applying preconditioning, and proper scaling can mitigate ill-conditioning. Reformulating as a least-squares problem or increasing precision are common approaches. In practice, awareness of near-singularity guides model design and numerical method choices to balance fidelity and stability.

Related concepts include ill-conditioning, degenerate systems, and proximity to singularity in eigenvalue problems and optimization.