... In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.

In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue problems, electronic structures computations, and signal processing. In addition to the new algorithms, we show how the geometrical framework gives penetrating new insights allowing us to create, understand, and compare algorithms. The theory proposed here provides a taxonomy for numerical linear algebra algorithms that provide a top level mathematical view of previously unrelated algorithms. It is our hope that developers of new algorithms and perturbation theories will benefit from the theory, methods, and examples in this paper.

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. Categories and Subject Descriptors: G.1.2 [Numerical Analysis]: ApprorJmation--least squares approximation; G.1.3 [Numerical Analysis]: Numerical Linear Algebra--linear systems (direct and

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Abstract-In this paper we present a new placement method for cellbased layout styles. It is composed of alternating and interacting global optimization and partitioning steps that are followed by an optimization of the area utilizaiton. Methods using the divide-and-conquer paradigm usually lose the global view by generating smaller and smaller subproblems. In contrast, GORDIAN maintains the simultaneous treatment of all cells over all global optimization steps, thereby considering constraints that reflect the current dissection of the circuit. The global optimizations are performed by solving quadratic programming problems that possess unique global minima. Improved partitioning schemes for the stepwise refinement of the placement are introduced. The area utilization is optimized by an exhaustive slicing procedure. The placement method has been applied to real world problems and excellent results in terms of both placement quality and computation time have been obtained. I.

This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient–type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell–Boeing collection and from Tim Davis’s collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.

Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to non-Hermitian matrices. We also review GMRES and related methods, as well as CG-like algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczos-based iterative methods for general non-Hermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using look-ahead techniques. Moreover, we describe the quasi-minimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the look-ahead Lanczos process to obtain quasi-optimal approximate solutions. Next, a survey of transposefree Lanczos-based methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameter-dependent algorithms that, unlike CG-type schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameter-dependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal

Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q - 1, for which u = g k . The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...

We theoretically and experimentally analyse the efficiency with which a wide range of important scientific computations can be performed on bulk synchronous parallel architectures.