[Search][Subject Index][MathMap][Tour][Help!]

	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 65F: Numerical linear algebra

Introduction

Numerical linear algebra topics include: solutions of linear systems AX = B, eigenvalues and eigenvectors, matrix factorizations.

History

Applications and related fields

This image slightly hand-edited for clarity.

Subfields

• 65F05: Direct methods for linear systems and matrix inversion
• 65F10: Iterative methods for linear systems [See also 65N22]
• 65F15: Eigenvalues, eigenvectors
• 65F18: Inverse eigenvalue problems
• 65F20: Overdetermined systems, pseudoinverses
• 65F22: Ill-posedness, regularization
• 65F25: Orthogonalization
• 65F30: Other matrix algorithms
• 65F35: Matrix norms, conditioning, scaling [See also 15A12, 15A60]
• 65F40: Determinants
• 65F50: Sparse matrices
• 65F99: None of the above, but in this section

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

There is a substantial pool of information on numerical issues in linear algebra available on the internet -- see for example the newsgroup sci.math.num-analysis.

Software and tables

Note that mathematical algorithms constitute one of the most web-accessible bodies of human knowledge available. A search at one of these sites can easily save considerable programming headaches! Highly recommended are NETLIB and GAMS. See also TOMS.

Other web sites with this focus

• SIAM's Linear Algebra and Matrix Theory page.
• UTK archives page
• GAMS : Guide to Available Mathematical Software
• Netlib Repository at UTK/ORNL (Software)

Selected topics at this site

• Advice for numerical work on large (1000 x 1000) matrices
• Origin and scope of the instability associated with the Hilbert matrix
• Computing the determinant of the Hilbert matrix.
• Using rank-revealing factorizations to determine (approximate) linear dependence cheaply and stably.
• Hotelling's method of inverting matrices (Newton's method).
• Using the Arnoldi method to find the largest (or smallest) eigenvalues of a matrix.
• Arnoldi (Lánczos) algorithms to determine a few largest eigenvalues of a large sparse matrix.
• How do sparse matrices "usually" arise?
• Regularization, error propagation, and confidence intervals
• Regularization and (pseudo-) inverses of ill-conditioned matrices
• Basic code for Cholesky decomposition
• Stability of Cholesky decomposition for a large matrix
• Comparing Jacobi-Davidson with Arnoldi/Lanczos methods for finding largest eigenvalues
• Power method: successive Rayleigh quotients converge to dominant eigenvalue of a matrix
• Solving a tridiagonal n x n linear system
• Ox -- free package for matrix computation
• Computational complexity of matrix inversion
• Efficient computation of exponential of a matrix
• Pointers to evaluations of basic functions (exp, ln) on matrices
• Computational complexity of eigenvalue-related problems
• Updating the solution to a linear system after a small change in the coefficient matrix (Sherman-Morrison-Woodbury Formula)
• Solving Toeplitz linear systems
• Deflating a matrix after some eigenvectors are found, to find orthogonal ones
• Gerschgorin circles method of bounding eigenvalues
• Factorization into rectangular matrices of size equal to rank
• Basic topics in computing some matrix factorizations
• How to find Q in the QR factorization?
• QZ algorithm for simultaneous factorization of two matrices