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The algebraic eigenvalue problem

Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of…
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Rounding errors in algebraic processes

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Error Analysis of Direct Methods of Matrix Inversion

TLDR
This paper determines error bounds for a number of the most effective direct methods of inverting a matrix by analyzing the effect of the rounding errors made in the solution of the equations.
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Inverse Iteration, Ill-Conditioned Equations and Newton’s Method

Inverse iteration is one of the most widely used algorithms in practical linear algebra but an understanding of its main numerical properties has developed piecemeal over the last thirty years: a…
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Note on matrices with a very ill-conditioned eigenproblem

SummaryGives a bound for the distance of a matrix having an ill-conditioned eigenvalue problem from a matrix having a multiple eigenvalue which is generally sharper than that which has been published…
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The Least Squares Problem and Pseudo-Inverses

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AN ESTIMATE FOR THE CONDITION NUMBER OF A MATRIX

It is important in practice when solving linear systems to have an economical method for estimating the condition number $\kappa (A)$ of the matrix of coefficients. An algorithm involving $O(n^2 )$…
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Ill-conditioned eigensystems and the computation of the Jordan canonical form

TLDR
Several of the more stable methods for computing the Jordan canonical form are discussed together with the alternative approach of computing well-defined bases (usually orthogonal) of the relevant invariant subspaces.
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Note on the quadratic convergence of the cyclic Jacobi process

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Error analysis of floating-point computation

This paper consists of two main sections. In the first the bounds are derived for the rounding errors made in the fundamental floating-point arithmetic operations. In the second, these results are…
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