Dixon's factorization method

Abstract: The factorization method is an operational procedure which enables us to answer, in a direct manner, questions about eigenvalue problems which are of importance to physicists. The underlying idea is to consider a pair of first-order differential-difference equations which are equivalent to a given second-order differential equation with boundary conditions. For a large class of such differential equations the method enables us to find immediately the eigenvalues and a manufacturing process for the normalized eigenfunctions. These results are obtained merely by consulting a table of the six possible factorization types.The manufacturing process is also used for the calculation of transition probabilities.The method is generalized so that it will handle perturbation problems.

Abstract: I . E . Borel, Lefons sur lea series divergentes (Paris, 1901), 164 et seq. 2. P. Dienes, The Taylor Series (Oxford, 1931). 3 . Y. Okada, \" Uber die Annaherung analytischer Funktionen\", Math. Zeitschrift, 23 (1925), 62-71. 4. L. L. Silverman, \" On the definition of the sum of a divergent series,\" University of Missouri Studies, Math. Ser. 1 (1913), 1-96. 5. O. Toeplitz, \" tJber allgemeine lineare Mittelbildungen,\" Prace mat. fix., 22 (1911), 113— 120.

Abstract: In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(O) factorization without using the concept of level of fill-in. There are two traditional ways of developing incomplete factorization preconditioners. The first uses a symbolic factorization approach in which a level of fill is attributed to each fill-in element using only the graph of the matrix. Then each fill-in that is introduced is dropped whenever its level of fill exceeds a certain threshold. The second class of methods consists of techniques derived from modifications of a given direct solver by including a dropoff rule, based on the numerical size of the fill-ins introduced, traditionally referred to as threshold preconditioners. The first type of approach may not be reliable for indefinite problems, since it does not consider numerical values. The second is often far more expensive than the standard ILU(O). The strategy we propose is a compromise between these two extremes.

Abstract: The Factorization of Rational Matrix Functions.- Decomposing Algebras of Matrix Functions.- Canonical Factorizations of Continuous Matrix Functions.- Factorization of Triangular Matrix Functions.- Factorization of Continuous Self-Adjoint Matrix Functions on the Unit Circle.- Miscellaneous Results on Factorization Relative to a Contour.- Generalized Factorization.- Further Results Concerning Generalized Factorization.- Local Principles in the Theory of Factorization.- Perturbations and Stability.