Scientific Computing · Cluster
Numerical Linear Algebra
Factorizations and sparse methods.
- 01 LU Factorization with Partial Pivoting and Element Growth How the pivot search bounds every multiplier by one, why that bound still does not bound the factors, and what the growth factor costs in practice.
- 02 Cholesky Factorization as a Positive-Definiteness Test Why the symmetric factorization needs no pivot search, where the factor-of-two saving comes from, and how a failed square root becomes a definiteness verdict.
- 03 Householder QR and Orthogonality Loss in Gram-Schmidt Why the reflector-based factorization keeps its computed factors orthogonal when classical Gram-Schmidt does not, and what that costs in flops.
- 04 Condition Number, Residual, and Backward Error Why a small residual does not imply a small error, how the condition number splits the blame between problem and algorithm, and how to estimate it without forming an inverse.
- 05 Compressed Sparse Row Storage and the Cost of Insertion How three flat arrays encode a sparse matrix, why the matrix-vector product is limited by memory rather than arithmetic, and why assembly happens in a different format.
- 06 Fill-In, Elimination Ordering, and Symbolic Factorization Why eliminating a sparse matrix creates new nonzeros, how ordering heuristics bound the damage, and how the pattern of the factor is predicted before any arithmetic runs.
- 07 Krylov Subspace Iteration and Why GMRES Restarts How a solver builds a subspace from repeated matrix-vector products, why the symmetric case needs three vectors of memory and the general case needs many, and when restarting destroys convergence.
- 08 Incomplete LU Preconditioning and Dropped Fill How a factorization restricted to the original sparsity pattern still clusters the spectrum, what the level-of-fill and threshold variants trade, and why the triangular solves limit parallelism.
- 09 Multifrontal Factorization and the Dense Front Matrix How a sparse factorization becomes a tree of small dense problems, why that conversion is what makes it fast, and how pivoting survives in a static data structure.
- 10 Lanczos Iteration and Shift-Invert for Extreme Eigenvalues How a three-term recurrence reduces a large symmetric matrix to a tridiagonal one, which eigenvalues that reduction can resolve, and why shift-invert changes the question being asked.