bijnasinguliere

Bijnasinguliere is a Dutch term used in mathematics and numerical analysis to describe objects that are almost but not exactly singular. In common usage it refers especially to matrices or linear systems that are close to losing full rank, or to transformations that are nearly degenerate. The term highlights the practical instability that arises when an object is near the threshold between well-posed and ill-posed behavior.

For matrices, a near-singular (bijna-singuliere) matrix has a determinant close to zero and, more robustly, a very small smallest singular value. The consequence is a large condition number, defined as the ratio of the largest to the smallest singular value. Such matrices are highly sensitive to perturbations: small changes in the input or in data can produce large changes in the solution of A x = b. In numerical computations this leads to unstable results and amplified round-off errors.

Near-singularity also appears in the context of Jacobians and differential maps. If the Jacobian of a function at a point has rank deficiency or a tiny determinant, the local behavior becomes ill-conditioned, complicating tasks such as solving nonlinear systems, performing optimization, or applying Newton-type methods. Regularization, reparameterization, or using a pseudo-inverse are common remedies when facing bijna-singuliere structures.

Typical examples include small perturbations of a full-rank matrix that reduce its determinant or a Jacobian that is nearly singular at a solution, as well as ill-conditioned data-fitting problems. In practice, detecting near-singularity involves examining singular values, condition numbers, or the rank of matrices, and addressing it through numerical techniques or model adjustments.

See also: singular value decomposition, ill-conditioning, Moore-Penrose pseudoinverse, regularization, Jacobian.