High-level linear algebra

This chapter describes all of the linear algebra operations which are not basic enough to fall within the BLAS (Basic Linear Algebra Subprograms) framework. In particular, algorithms which would traditionally have fallen into the domain of LAPACK (Linear Algebra PACKage), such as factorizations and matrix decompositions, are placed here.

• Reduction to condensed form
  • Hermitian to tridiagonal
  • Square to Hessenberg
  • General to bidiagonal
• Matrix decompositions
  • Hermitian eigensolver
  • Skew-Hermitian eigensolver
  • Hermitian generalized-definite eigensolvers
  • Unitary eigensolver
  • Normal eigensolver
  • Schur decomposition
  • Hermitian SVD
  • Polar decomposition
  • SVD
• Factorizations
  • Cholesky factorization
  • LDL factorization
  • LU factorization
  • LQ factorization
  • QR factorization
  • Generalized QR factorization
  • RQ factorization
  • Generalized RQ factorization
  • Interpolative Decomposition (ID)
  • Skeleton decomposition
• Matrix functions
  • Direct inversion
  • Hermitian functions
  • Pseudoinverse
  • Square root
  • Sign
• Permutations
  • Permutation vectors
  • Pivot sequences
  • Converting pivot sequences to permutations
• Matrix properties
  • Condition number
  • Determinant
  • Matrix norms
  • Pseudospectra
  • Trace
  • Inertia
• Linear solvers
  • HPD solve
  • Symmetric solve
  • Hermitian solve
  • Gaussian elimination
  • Least Squares
  • General (Gauss-Markov) Linear Model (GLM)
  • Equality-constrained Least Squares (LSE)
  • Multi-shift Hessenberg solves
• Utilities
  • Householder reflectors
  • Sorting
• Tuning parameters
  • Hermitian to tridiagonal