An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I~QR is the most reliable algorithm when A is ill-conditioned.

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LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices

In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...

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Nineteen Dubious Ways to Compute the Exponential of a Matrix

Factorization methods for discrete sequential estimation

We analyze perturbation experiments performed on real and idealized ecological com- munities. A community may be considered as a black box in the sense that the individual species grow and interact in complicated ways that are difficult to discern. Yet, by observing the response (output) of the system to natural or human-induced disturbances (inputs), information can be gained regarding the character and strengths of species interactions. We define a perturbation as selective alteration of the density of one or more members of the community, and we distinguish two quite different kinds of perturbations. A PULSE perturbation is a relatively instantaneous alteration of species numbers, after which the system is studied as it "relaxes" back to its previous equilibrium state. A PRESS perturbation is a sustained alteration of species densities (often a complete elimination of particular species); it is maintained until the unperturbed species reach a new equilibrium. The measure of interest in PRESS perturbation is the net change in densities of the unperturbed species. There is a very important difference between these two approaches: PULSE experiments yield in- formation only on direct interactions (e.g., terms in the interaction matrix), while PRESS experiments yield information on direct interactions mixed together with the indirect effects mediated through other species in the community. We develop mathematical techniques that yield measures of ecolog- ical interaction between species from both types of experimental designs. Particular caution must be exercised in interpreting results from PRESS experiments, particularly when some species are lumped into functional categories and others are neglected altogether in the experimental design. We also suggest mathematical methods to deal with temporal and random variation during experiments. Fi- nally, we critically review techniques that rely on natural variation in numbers to estimate species interaction coefficients. The problems with such studies are formidable.

Perturbation Experiments in Community Ecology: Theory and Practice

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

The theory of irrationalities of the third degree

Computational Methods of Linear Algebra.

Numerische Methoden der linearen Algebra

Preliminary information about the embedding problem The compatibility condition the embedding problem with Abelian kernel The embedding problem for local fields the embedding problem with non-Abelian kernel for algebraic number fields Appendix Bibliography Subject index.

D. K. Faddeev

Papers

Computational methods of linear algebra

The theory of irrationalities of the third degree

Computational Methods of Linear Algebra.

Numerische Methoden der linearen Algebra

The Embedding Problem in Galois Theory