reorthogonalized

Reorthogonalized refers to procedures that restore or preserve mutual orthogonality among a set of vectors after they have become non-orthogonal due to finite-precision arithmetic. In exact arithmetic, methods like Gram-Schmidt yield an orthogonal (or orthonormal) basis, but floating-point computations introduce rounding errors that cause loss of orthogonality, which can degrade numerical methods that rely on an orthogonal basis.

Reorthogonalization is commonly employed in Gram-Schmidt procedures to counteract this loss. Full reorthogonalization applies an additional Gram-Schmidt pass of the new vector against all previously computed vectors, further reducing components along them. Selective reorthogonalization targets only those vectors whose inner products with the new vector exceed a prescribed threshold, reducing cost. Periodic or partial reorthogonalization strategies balance accuracy and efficiency by limiting how often or against how many vectors orthogonality is enforced.

Key contexts include the Gram-Schmidt process itself and Krylov subspace methods such as the Arnoldi and Lanczos algorithms. In these methods, loss of orthogonality among basis or Lanczos vectors can lead to degraded convergence properties or spurious eigenvalues; reorthogonalization helps maintain numerical stability and accuracy, especially in high-precision requirements or large-scale problems.

Trade-offs involve increased computational cost and memory usage, since maintaining orthogonality requires extra vector operations and storage of the basis. Practical implementations often use selective or periodic reorthogonalization to achieve a balance between numerical reliability and performance. Reorthogonalization is a standard technique in numerical linear algebra libraries when robust orthogonality is essential.