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Computing the singular values of a bidiagonal matrix is the fin al phase of the standard algow rithm for the singular value decomposition of a general matrix. We present a new algorithm hich computes all the singular values of a bidiagonal matrix to high relative accuracy independent of their magnitudes. In contrast, the standard algorithm for bidiagonal matrices may compute small singular values with no relative accuracy at all. Numerical experiments show that the new algorithm is comparable in speed to the standard algorithm , and frequently faster.

High dimensionality of text can be a deterrent in applying complex learners such as Support Vector Machines to the task of text classification. Feature clustering is a powerful alternative to feature selection for reducing the dimensionality of text data. In this paper we propose a new informationtheoretic divisive algorithm for feature/word clustering and apply it to text classification. Existing techniques for such “distributional clustering ” of words are agglomerative in nature and result in (i) sub-optimal word clusters and (ii) high computational cost. In order to explicitly capture the optimality of word clusters in an information theoretic framework, we first derive a global criterion for feature clustering. We then present a fast, divisive algorithm that monotonically decreases this objective function value. We show that our algorithm minimizes the “within-cluster Jensen-Shannon divergence ” while simultaneously maximizing the “between-cluster Jensen-Shannon divergence”. In comparison to the previously proposed agglomerative strategies our divisive algorithm is much faster and achieves comparable or higher classification accuracies. We further show that feature clustering is an effective technique for building smaller class models in hierarchical classification. We present detailed experimental results using Naive Bayes and Support Vector Machines on the 20Newsgroups data set and a 3-level hierarchy of HTML documents collected from the Open Directory project (www.dmoz.org).

Vector Microprocessors by Krste Asanovic Doctor of Philosophy in Computer Science University of California, Berkeley Professor John Wawrzynek, Chair Most previous research into vector architectures has concentrated on supercomputing applications and small enhancements to existing vector supercomputer implementations. This thesis expands the body of vector research by examining designs appropriate for single-chip full-custom vector microprocessor implementations targeting a much broader range of applications. I present the design, implementation, and evaluation of T0 (Torrent-0): the first single-chip vector microprocessor. T0 is a compact but highly parallel processor that can sustain over 24 operations per cycle while issuing only a single 32-bit instruction per cycle. T0 demonstrates that vector architectures are well suited to full-custom VLSI implementation and that they perform well on many multimedia and human-machine interface tasks. The remainder of the thesis contains ...

Floating-point division is generally regarded as a low frequency, high latency operation in typical floating-point applications. However, the increasing emphasis on high performance graphics and the industry-wide usage of performance benchmarks forces processor designers to pay close attention to all aspects of floating-point computation. Many algorithms are suitable for implementing division in hardware. This paper presents four major classes of algorithms in a unified framework, namely digit recurrence, functional iteration, very high radix, and variable latency. Digit recurrence algorithms, the most common of which is SRT, use subtraction as the fundamental operator, and they converge to a quotient linearly. Division by functional iteration converges to a quotient quadratically using multiplication. Very high radix division algorithms are similar to digit recurrence algorithms, but they incorporate multiplication to reduce the latency. Variable latency division algorithms reduce the...

We consider computing the singular value decomposition of a bidiagonal matrixB. This problem arises in the singular value decomposition of a general matrix, and in the eigenproblem for a symmetric positive de nite tridiagonal matrix. We show that if the entries of B are known with high relative accuracy, the singular values and singular vectors ofB will be determined to much higher accuracy than the standard perturbation theory suggests. We also show that the algorithm in [Demmel and Kahan] computes the singular vectors as well as the singular values to this accuracy. We also give a Hamiltonian interpretation of the algorithm and use di erential equation methods to prove many of the basic facts. The Hamiltonian approach suggests a way to use ows to predict the accumulation of error in other eigenvalue algorithms as well.

We present the design and testing of an algorithm for iterative refinement of the solution of linear equations, where the residual is computed with extra precision. This algorithm was originally proposed in the 1960s [6, 22] as a means to compute very accurate solutions to all but the most ill-conditioned linear systems of equations. However two obstacles have until now prevented its adoption in standard subroutine libraries like LAPACK: (1) There was no standard way to access the higher precision arithmetic needed to compute residuals, and (2) it was unclear how to compute a reliable error bound for the computed solution. The completion of the new BLAS Technical Forum Standard [5] has recently removed the first obstacle. To overcome the second obstacle, we show how a single application of iterative refinement can be used to compute an error bound in any norm at small cost, and use this to compute both an error bound in the usual infinity norm, and a componentwise relative error bound. We report extensive test results on over 6.2 million matrices of dimension 5, 10, 100, and 1000. As long as a normwise (resp. componentwise) condition number computed by the algorithm is less than 1/max{10, √ n}εw, the computed normwise (resp. componentwise) error bound is at most

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We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The first is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new 2 \Theta 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm. We present proofs of stability and high accuracy of the 2 \Theta 2 GSVD algorithm, and demonstrate it using examples on which all previous algorithms fail. 1 Introduction The purpose of this paper is to describe a variation of Paige's algorithm [28] for computing the following generalized singular value decomposition (GSVD) introduced by Van Loan [33], and Paige and Saunders [25]. This is also called the quotient singular value decomposition (QSVD) in [8]. Theorem 1.1 Let A 2 IR m\Thetan and B 2 IR p\Thetan have rank(A T ; B T ) = n. 1 Then there are orthogonal matrices U , V and Q su...

Abstract. The IA-64 architecture defers floating point and integer division to software. To ensure correctness and maximum efficiency, Intel provides a number of recommended algorithms which can be called as subroutines or inlined by compilers and assembly language programmers. All these algorithms have been subjected to formal verification using the HOL Light theorem prover. As well as improving our level of confidence in the algorithms, the formal verification process has led to a better understanding of the underlying theory, allowing some significant efficiency improvements. 1