Numerical Linear Algebra

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8 domain-specific abstractions whose origin domain is Numerical Linear Algebra.

• Block LU decomposition — Factorization of a partitioned matrix into lower and upper block-triangular factors, governed by Schur complements.
• Crout matrix decomposition — An LU factorization convention placing arbitrary diagonal entries in the lower triangular factor and unit diagonal entries in the upper factor, with pivoting when needed.
• Gauss–Seidel Method — A stationary linear-system iteration that sweeps coordinates in order and immediately reuses each newly computed component within the same sweep.
• Jacobi Method — Solve a linear system by isolating its diagonal and synchronously recomputing every component from the same previous iterate, with the resulting iteration matrix governing convergence.
• Linear least squares — Approximation of an overdetermined or rank-deficient linear system by choosing parameters that minimize a quadratic residual norm.
• Locally Optimal Block Preconditioned Conjugate Gradient — Compute a few extreme eigenpairs of a large Hermitian-definite problem by repeatedly Rayleigh–Ritz optimizing a block over current Ritz vectors, preconditioned eigen-residuals, and compressed prior directions.
• QR Algorithm — An eigenvalue iteration that repeatedly QR-factorizes a shifted matrix and reverses the factors, preserving similarity while driving it toward real or complex Schur form for deflation.
• Schur complement method — A nonoverlapping domain-decomposition method eliminating subdomain interiors and solving the remaining interface Schur-complement system.