This paper is concerned with the solution of systems of equations whose associated matrix is block Toeplitz. First a recursive in nature algorithm is developed for the general problem of block system solution, which i...
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This paper is concerned with the solution of systems of equations whose associated matrix is block Toeplitz. First a recursive in nature algorithm is developed for the general problem of block system solution, which is then specialized to block Toeplitz structures. Subsequently cases, where the right hand side member is not arbitrary but rather relates to the Toeplitz matrix are analyzed and a new fast algorithm is deduced, with applications in smoothing and prediction. Finally, within the above context, block banded Toeplitz systems are discussed and two new efficient methods are presented.
The partial fraction decomposition of a proper rational function whose denominator has degree n and is given in general factored form can be done in O(nlog<span class="mn" id="MathJax-Span-10" s...
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We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX ('GIGAQBX') and target-specific expansions. GI...
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We develop an algorithm for the asymptotically fast evaluation of layer potentials close to and on the source geometry, combining Geometric Global Accelerated QBX ('GIGAQBX') and target-specific expansions. GIGAQBX is a fast high-order scheme for evaluation of layer potentials based on Quadrature by Expansion ('QBX') using local expansions formed via the fast Multipole Method (FMM). Target-specific expansions serve to lower the cost of the formation and evaluation of QBX local expansions, reducing the associated computational effort from 0 ((p + 1)(2)) to 0 (p + 1) in three dimensions, without any accuracy loss compared with conventional expansions, but with the loss of source/target separation in the expansion coefficients. GIGAQBX is a 'global' QBX scheme, meaning that the potential is mediated entirely through expansions for points close to or on the boundary. In our scheme, this single global expansion is decomposed into two parts that are evaluated separately: one part incorporating near-field contributions using target-specific expansions, and one part using conventional spherical harmonic expansions of far-field contributions, noting that convergence guarantees only exist for the sum of the two sub-expansions. By contrast, target-specific expansions were originally introduced as an acceleration mechanism for 'local' QBX schemes, in which the far-field does not contribute to the QBX expansion. Compared with the unmodified GIGAQBX algorithm, we show through a reproducible, time-calibrated cost model that the combined scheme yields a considerable cost reduction for the near-field evaluation part of the computation. We support the effectiveness of our scheme through numerical results demonstrating performance improvements for Laplace and Helmholtz kernels. (C) 2019 Elsevier Inc. All rights reserved.
We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clust...
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We consider the problem of clustering with the longest-leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the Laplacian eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the labeling accuracy of the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient, easy to implement approximation algorithm for the LLPD based on a multiscale analysis of adjacency graphs, which allows for the runtime of LLPD spectral clustering to be quasilinear in the number of data points.
The continuous wavelet transform (CWT) is a powerful technique for signal analysis. Direct CWT computation by FFT requires O(N log(2) N) operations per scale, where N is the data length, The a trous algorithm and the ...
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The continuous wavelet transform (CWT) is a powerful technique for signal analysis. Direct CWT computation by FFT requires O(N log(2) N) operations per scale, where N is the data length, The a trous algorithm and the Shensa algorithm are two fast methods to compute CWT recursively that require only O(N) operations per scale. Both of them can be described by the multiresolution analysis (MRA) structure but with different MRA filters. This paper proposes methods to design the MRA filters of the two algorithms to improve their accuracy on CWT computation. We begin with the formulation of the CWT computation error using the MRA structure, The MRA filters of the two algorithms are then designed to minimize the error, In either algorithm, both the lowpass and bandpass MRA filters can be optimized. The a trous algorithm has closed-form solutions for the two filters. The Shensa algorithm, on the other hand, has an analytic solution for the bandpass filter only. Finding the optimum lowpass filter requires a multidimensional numerical search. Simulation studies show that by using the proposed optimum filters, the Shensa algorithm, in general, outperforms the ri trous algorithm.
We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexi...
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We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse operation. (C) 2009 Elsevier Inc. All rights reserved.
fast decoding algorithms are described for a number of established coded aperture systems. The fast decoding algorithms for all these systems offer significant reductions in the number of calculations required when re...
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fast decoding algorithms are described for a number of established coded aperture systems. The fast decoding algorithms for all these systems offer significant reductions in the number of calculations required when reconstructing images formed by a coded aperture system and hence require less computation time to produce the images. The algorithms may therefore be of use in applications that require fast image reconstruction, such as near real-time nuclear medicine and location of hazardous radioactive spillage. Experimental tests confirm the efficacy of the fast decoding techniques. (C) 2014 Elsevier B.V. All rights reserved.
We construct new fast evaluation algorithms for elementary algebraic and inverse functions based on application of two methods: A.A. Karatsuba's method of 1960 and the author's FEE method of 1990. The computat...
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We construct new fast evaluation algorithms for elementary algebraic and inverse functions based on application of two methods: A.A. Karatsuba's method of 1960 and the author's FEE method of 1990. The computational complexity is close to the optimal. The algorithms admit partial parallelization.
We present new modular algorithms for the squarefree factorization of a primitive polynomial in Z[x] and for computing the rational part of the integral of a rational function in Q(x). We analyze both algorithms with ...
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We present new modular algorithms for the squarefree factorization of a primitive polynomial in Z[x] and for computing the rational part of the integral of a rational function in Q(x). We analyze both algorithms with respect to classical and fast arithmetic and argue that the latter variants are - up to logarithmic factors - asymptotically optimal. Even for classical arithmetic, the integration algorithm is faster than previously known methods.
fast decoding algorithms are described for the class of coded aperture designs known as geometric coded apertures which were introduced by Gourlay and Stephen. When compared to the direct decoding method, the algorith...
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fast decoding algorithms are described for the class of coded aperture designs known as geometric coded apertures which were introduced by Gourlay and Stephen. When compared to the direct decoding method, the algorithms significantly reduce the number of calculations required when performing the decoding for these apertures and hence speed up the decoding process. Experimental tests confirm the efficacy of these fast algorithms, demonstrating a speed up of approximately two to three orders of magnitude over direct decoding. (C) 2015 Elsevier B.V. All rights reserved.
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