The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Computational methods of linear algebra

Methods for solving symmetric indefinite systems are surveyed including a new one which is stable and almost as fast as the Cholesky method.

Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations

Several decompositions ofsymmetric matrices for calculating inertia and solving systems of linear equations are discussed. New partial pivoting strategies for decomposing symmetric matrices are introduced and analyzed.

/pdf/some-stable-methods-for-calculating-inertia-and-solving-23xs62gdhl.pdf

Some stable methods for calculating inertia and solving symmetric linear systems

This paper surveys the state of the art in sparse matrix research in January 1976. Much of the survey deals with the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms. In the symmetric case, relevant terms from graph theory are defined. Band systems and matrices arising from the discretization of partial differential equations are treated as separate cases. Preordering techniques are surveyed with particular emphasis on partitioning (to block triangular form) and tearing (to bordered block triangular form). Methods for solving the least squares problem and for sparse linear programming are also reviewed. The sparse eigenproblem is discussed with particular reference to some fairly recent iterative methods. There is a short discussion of general iterative techniques, and reference is made to good standard texts in this field. Design considerations when implementing sparse matrix algorithms are examined and finally comments are made concerning the availability of codes in this area.

A survey of sparse matrix research

A real symmetric matrix H is called “well-behaved” if the small relative changes in its elements cause small relative changes in its eigenvalues. We show that an algorithm which implements decomposition followed by Jacobi-type iteration computes the eigenvalues with nearly the same accuracy. The positive definite case was treated in [2], our results cover general indefinite matrices.

/pdf/accurate-symmetric-eigenreduction-by-a-jacobi-method-wudxklz6uc.pdf

Accurate Symmetric Eigenreduction by a Jacobi Method

A backwards error analysis of the diagonal pivoting method for solving symmetric (indefinite) systems of linear equations shows that the elements of the associated error matrix can be bounded in te...

Analysis of the Diagonal Pivoting Method

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Analysis of the Diagonal Pivoting Method