The origins and development of Zuwīla, Libyan Sahara: an archaeological and historical overview of an ancient oasis town and caravan centre

Zuwīla in southwestern Libya (Fazzān) was one of the most important early Islamic centres in the Central Sahara, but the archaeological correlates of the written sources for it have been little explored. This paper brings together for the first time a detailed consideration of the relevant historical and archaeological data, together with new AMS radiocarbon dates from several key monuments. The origins of the settlement at Zuwīla were pre-Islamic, but the town gained greater prominence in the early centuries of Arab rule of the Maghrib, culminating with the establishment of an Ibāḍī state ruled by the dynasty of the Banū Khaṭṭāb, with Zuwīla its capital. The historical sources and the accounts of early European travellers are discussed and archaeological work at Zuwīla is described (including the new radiocarbon dates). A short gazetteer of archaeological monuments is provided as an appendix. Comparisons and contrasts are also drawn between Zuwīla and other oases of the ash-Sharqiyāt region of Fazzān. The final section of the paper presents a series of models based on the available evidence, tracing the evolution and decline of this remarkable site

An Implementation Of The Qmr Method Based On Coupled Two-Term Recurrences

. Recently, the authors have proposed a new Krylov subspace iteration, the quasi-minimal residual algorithm (QMR), for solving non-Hermitian linear systems. In the original implementation of the QMR method, the Lanczos process with look-ahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of three-term recurrences. It has been observed that, in finite precision arithmetic, vector iterations based on three-term recursions are usually less robust than mathematically equivalent coupled two-term vector recurrences. This paper presents a look-ahead algorithm that constructs the Lanczos basis vectors by means of coupled two-term recursions. Implementation details are given, and the look-ahead strategy is described. A new implementation of the QMR method, based on this coupled two-term algorithm, is proposed. A simplified version of the QMR algorithm without look-ahead is also presented, and the specia..

An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices Part II

this paper, we have presented an implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices. Here, we show how the look-ahead Lanczos process --- combined with a quasi-minimal residual (QMR) approach --- can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. In particular, BCG iterates can be recovered stably from the QMR process. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problems and linear systems. Also, program listings of FORTRAN implementations of the look-ahead Lanczos algorithm and the QMR method are included. Categories and Subject Descriptors: G.1.3 [Numerical Analysis]: Numerical Linear Algebra--eigenvalues; linear systems (direct and iterative methods); sparse and very large systems; G.4 [Mathematics of Computing]: Mathematical Software General Terms: Large sparse linear systems and eigenvalue problems, iterative methods Additional Key Words and Phrases: Non-Hermitian linear systems, quasi-minimal residual property, Lanczos method, biconjugate gradients The work of R. W. Freund and N. M. Nachtigal was supported in part by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA). Authors' addresses: R. W. Freund, RIACS, Mail Stop Ellis Street, NASA Ames Research Center, Moffett Field, CA 94035, and Institut fur Angewandte Mathematik, Universit at Wurzburg, D--W8700 Wurzburg, Federal Republic of Germany; N. ..

Combining The Qmr Method With First Principles Electronic Structure Codes

. First principles methods are used to aid the metallurgist in the investigation and design of new materials. However, these methods suffer from an O(N 3 ) scaling which restricts the problem sizes that can be addressed. By incorporating the QMR method into a first principles algorithm we are able to attain O(N) scaling for the problem sizes of interest. This advancement will provide researchers with the necessary tools to treat large systems. Key words. electronic structure calculations, local density approximation, quasi-minimal residual method AMS subject classifications. 65C20, 65F10 1. Introduction. The design of new technologically advanced materials is of extreme importance to the industrial and economical competitiveness of the U.S. For example, the design of a high temperature ductile intermetallic alloy could save the power generation and aerospace industries billions of dollars. The modern design of new materials makes use of localdensity approximation (LDA) based, firs..

An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices

. The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. We present an implementation of a look-ahead version of the Lanczos algorithm that---except for the very special situation of an incurable breakdown--- overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead. Key words. Lanczos method, orthogonal polynomials, look-ahead steps, eigenvalue problems, iterative methods, non-Hermitian matrices, sparse linear systems AMS(MOS) subject classifications. 65F15, 65F10 1. Introduction. In 1950, Lanczos [20] proposed a me..