A fast algorithm for matrix balancing

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the matrix can be balanced, that is we can find a diagonal scaling of A that is doubly stochastic. A number of algorithms have been proposed to achieve the balancing, the most well known of these being Sinkhorn-Knopp. In this paper we derive new algorithms based on inner-outer iteration schemes. We show that Sinkhorn-Knopp belongs to this family, but other members can converge much more quickly. In particular, we show that while stationary iterative methods offer little or no improvement in many cases, a scheme using a preconditioned conjugate gradient method as the inner iteration can give quadratic convergence at low cost

New Year\u27s Come

Why chime the bells so merrilyWhy seem ye all so gay?Is it because the New Year\u27s comeAnd the old has pass\u27d away?Oh! Can ye look upon the pastAnd feel no sorrow nowThat thus ye sing so joyouslyAnd smiles light ev\u27ry brow?Oh! If ye can be blithe and gayThe song, troul gaily onAnd the burden be the New Year\u27s come and the Old Year\u27s gone. And the burden be the New Year\u27s come and the Old Year\u27s gone. The old man gazes on your mirthHe smiles not like the restHe sits in silence by the hearthAnd seems with grief oppress\u27dHe sees not in the merry throngThe child who was his prideHe listens for her joyous songShe is not by his side!But scarce a twelvemonth she was thereAnd now he is aloneYet still ye sing the New Year\u27s come and the Old Year\u27s gone. Yet still ye sing the New Year\u27s come and the Old Year\u27s gone. Dance on! dance on! be blithe and gayNor pause to think the whileThat ere this year has pass\u27d awayYe to may cease to smileFor time in his resistless flightBrings changes sad and drearThe sunny hopes of youth to blightWith ev\u27ry coming yearBut still be happy while ye mayAnd let the dance go onStill gaily sing the New Year\u27s come and the Old Year\u27s gone!Still gaily sing the New Year\u27s come and the Old Year\u27s gone

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Eta De Cicco, Mike Farmer and James Hargrave, Using the Internet in Secondary Schools, London: Kogan Page (2nd edn), 2001. ISBN: 0–7494–3459–7. Softback, x + 192 pages, £16.99

The Sinkhorn-Knopp algorithm : convergence and applications

As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear, and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices. We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that, with an appropriate modi. cation, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets

Fast rectangular matrix multiplication and QR decomposition

AbstractIn the last twenty-five years there has been much research into “fast” matrix multiplication methods: ones that have an asymptotically smaller operation count than conventional multiplication. Most fast methods are derived for square matrices, but they can be applied to rectangular matrices by a blocking technique. We obtain an expression for the order of the operation count for this blocked multiplication of rectangular matrices. We derive an exact operation count for Strassen's method with rectangular matrices and determine the recursion threshold that minimizes the operation count. We also show that when Strassen's method is used to multiply rectangular matrices it is more efficient to use the method on the whole product than to apply the method to square submatrices. Fast multiplication methods can be exploited in calculating a QR decomposition of an m × n matrix. We show that the operation count can be reduced from O(mn2) to O(mn1+(1(4-α))) by using a fast multiplication method with exponent α in conjunction with Bischof and Van Loan's WY representation of a product of Householder transformations

Philosophical pragmatism and religious belief: interpreting Christian non-realism through john Dewey and Richard Rorty

In this thesis I consider the account of religious non-realism in the work of Don Cupitt and in other prominent writers belonging to the 'Sea of Faith' network. I argue that the appropriate context of the non-realists understanding of religious belief is provided by philosophical pragmatism as this is presented in the work of John Dewey and Richard Rorty. This context outlines important aspects of the 'Sea of Faith' religious non-realists' self-understanding; and provides them with an argumentative resource which they can employ against alternative critical-realist approaches to religious belief I show that John Dewey's understanding of religious faith coheres with many of the ideas expressed by religious non-realists and that Rorty's pragmatism provides religious non-realism with a contemporary philosophical articulation of its theology. In order to defend this assertion I argue that Rorty's pragmatism does not necessarily lead to radical subjectivism nor to a dangerous political ideology as some interpreters have suggested. Further, I argue that his ideas are open to theological appropriation and that his rejection of religious belief is tempered by a tolerance toward those who still find a use for it. Rorty, I claim, has such a use. He employs the term 'God' as a backdrop against which he can present his own account of a pragmatic culture. I show that his work contains positive references to the influence that religious belief has had on the development of such a culture and argue that this pragmatic culture fits well with a non-realist understanding of religious belie

Network models and biproportional rounding for fair seat allocations in the UK elections

Systems for allocating seats in an election offer a number of socially and mathematically interesting problems. We discuss how to model the allocation process as a network flow problem, and propose a wide choice of objective functions and allocation schemes. Biproportional rounding, which is an instance of the network flow problem, is used in some European countries with multi-seat constituencies. We discuss its application to single seat constituencies and the inevitable consequence that seats are allocated to candidates with little local support. However, we show that variants can be selected, such as regional apportionment, to mitigate this problem. In particular, we introduce a parameter based family of methods, which we call Balanced Majority Voting, that can be tuned to meet the public's demand for local and global ``fairness''. Using data from the 2010 and 2015 UK General Elections, we study a variety of network models and implementations of biproportional rounding, and address conditions of existence and uniqueness

Space Shuttle 2 Advanced Space Transportation System. Volume 1: Executive Summary

An investigation into the feasibility of establishing a second generation space transportation system is summarized. Incorporating successful systems from the Space Shuttle and technological advances made since its conception, the second generation shuttle was designed to be a lower-cost, reliable system which would guarantee access to space well into the next century. A fully reusable, all-liquid propellant booster/orbiter combination using parallel burn was selected as the base configuration. Vehicle characteristics were determined from NASA ground rules and optimization evaluations. The launch profile was constructed from particulars of the vehicle design and known orbital requirements. A stability and control analysis was performed for the landing phase of the orbiter's flight. Finally, a preliminary safety analysis was performed to indicate possible failure modes and consequences

Space Shuttle 2 advanced space transportation system, volume 2

To determine the best configuration from all candidate configurations, it was necessary first to calculate minimum system weights and performance. To optimize the design, it is necessary to vary configuration-specific variables such as total system weight, thrust-to-weight ratios, burn durations, total thrust available, and mass fraction for the system. Optimizing each of these variables at the same time is technically unfeasible and not necessarily mathematically possible. However, discrete sets of data can be generated which will eliminate many candidate configurations. From the most promising remaining designs, a final configuration can be selected. Included are the three most important designs considered: one which closely approximates the design criteria set forth in a Marshall Space Flight Center study of the Shuttle 2; the configuration used in the initial proposal; and the final configuration. A listing by cell of the formulas used to generate the aforementioned data is included for reference