Displacement operator based decompositions of matrices using circulants or other group matrices

Inverting linear combinations of identity and generalized Catalan matrices

Recent research presents a technique to enumerate all valid assignments of “twiddle factors” for power-of-two fast Fourier transform (FFT) flow graphs. Brute-force search employing state-of-the-art Boolean satisfiability (SAT) solvers can then be used to find FFT algorithms within this large solution space which have desirable characteristics. Surprisingly, this approach has discovered FFT algorithms requiring fewer operations than the split-radix algorithm even when all twiddle factors are nth roots of unity. This paper reviews and then extends this prior research to examine fast discrete convolution algorithms when implemented via FFT and inverse FFT (IFFT) algorithms. In particular, we find that the combination of FFT and IFFT algorithms in fast convolution permits greater freedom when selecting valid twiddle factor assignments. We exploit this freedom and use SAT solvers to find new fast convolution algorithms with the lowest operation counts known.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {L}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the collection of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> × </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lower triangular Toeplitz matrices and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the collection of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> × </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lower triangular Toeplitz contractions. We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact and strictly convex, in the spectral norm, with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper L Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {L}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is compact, convex and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript German upper L Sub Subscript upper N Baseline German upper T Subscript upper N subset-of-or-equal-to e x t German upper T Subscript upper N"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> <mml:mo> ⊆ </mml:mo> <mml:mi>ext</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">T</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\partial _{\mathfrak {L}_{N}} \mathfrak {T}_{N} \subseteq \operatorname {ext}\mathfrak {T}_{N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript German upper L Sub Subscript upper N Baseline left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub>

The expression of a bound of the uniform norm of infinite lower triangular Toeplitz matrices with nonnegative entries is found. All the results are obtained by studying the behavior of the resolvent kernel and of the fundamental matrix of the recurrence relation, which generates the sequence of the entries of the considered matrix.

Based on fast convolution algorithm, this paper proposes improved parallel FIR filter structures for linear-phase FIR filters where the number of taps is a multiple of parallelism. The proposed parallel FIR structures not only use fast convolution algorithm to reduce the number of subfilter, but also exploit the symmetric (or antisymmetric) coefficients of linear-phase FIR filter to reduce half the number of multiplications in subfilter section at the expense of additional adders in preprocessing and postprocessing blocks. The proposed parallel FIR structures save a large amount of hardware cost for symmetric (or antisymmetric) coefficients from the reported FFA parallel FIR filter structures, especially when the length of the filter is large, e.g., the proposed 4-parallel FIR filter structure has eight subfilter blocks in total and four subfilter blocks contain symmetric coefficients, whereas the improved FFA structure has nine subfilter blocks in total and four contain symmetric coefficients. Specifically, for a 4-parallel 576-tap filter, the proposed design saves 144 multipliers (14.3%), 135 adders (10.2%) and 143 delay elements (11.1%).

A fast algorithm for chirp Z-transforms is improved form chirp Z-transform, which is developed by using two fast Fourier transforms and an analytical Gaussian kernel. Its computational complexity is less than a fast convolution algorithm. However, there are some problems when the algorithm is implemented, such as the discarding of the data, the smallness of the response domain, the bigness of the computational complexity and so on. To avoid the problems mentioned above, we make a change on the implementing of the algorithm in this paper. Then we compare the numerical results of some chirp systems with the analytical ones. The accuracy of Fourier transforms of Gaussian function is higher than the 10-15 order for most cases, and the accuracy of Fourier transforms of rectangle function is about the 10-3 order, which is essentially limited by the accuracy of the fast Fourier transform. Finially this algorithm is used to calculate some typical systems of scalar diffraction and fractional-order Fourier transforms, and the results are in good agreement with other published results in the literatures.

We give necessary and sufficient conditions for cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part.

In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on R, as well as some more general singular integrals. After mapping R into a finite interval, we discretize the integral operator using a modified midpoint rule. The result of this procedure can be cast as a discrete convolution, which can be evaluated efficiently using the Fast-Fourier Transform (FFT). The method provides an efficient, second-order accurate approximation to the fractional Laplacian, without the need to truncate the domain. We first prove that the method gives a second-order approximation for the fractional Laplacian and other related singular integrals; then, we detail the implementation of the method using a fast convolution algorithm, and give numerical examples that support its efficacy and efficiency; finally, as an example of its applicability to an evolution problem, we employ the method for the discretization of the nonlocal part of the one-dimensional cubic fractional Schrödinger equation in the focusing case.

The numerical studies of magnetization dynamics in micromagnetic systems with spatially distributed magnetization crucially rely on the computation of magnetostatic fields generated by the ferromagnet. The solution of the magnetostatic problem can be written as convolution of an appropriate Green’s function with the magnetization vector field. Since the Green’s function is long-range, after discretization of the problem in N cells, the direct numerical computation of the convolution integral has a computational cost scaling as N2.In this work, we use, as it is often done in micromagnetics, the Fast Fourier Transform (FFT) method to reduce the computational cost scaling to N logN. We consider micromagnetic systems with thin-film geometry where the magnetization can be considered constant across the thickness of the magnet. In these situations, only the field averaged across the thickness is relevant to the magnetostatic interactions and both the magnetization and the averaged field inside the magnet depend only on two spatial variables. The usual approach to deal with this type of problem is to use FFT to compute the discretized version of the convolution integral relating the field and the magnetization [1]. The dual approach to the usual one, is based on the discretization of the integral relation between the field (averaged across the thickness) and the magnetization in the reciprocal space (k-space). The connection in the k-space is given by the Fourier transform of Green function of the magnetostatic problem which, in case of infinite thin-films, can be written in a special form taking into account the thin-film geometry [2]. The analysis in the k-space has been extensively used in connection with linear spin wave dynamics in thin films working as magnonic waveguide, where all the relation of practical use (e.g dispersion relations), are easily obtained by algebraic manipulations. When the dimensions of the thin film are finite, the reconstruction of the magnetostatic field with the infinite film Green's function, is altered by the presence of replicas of the thin film in the ordinary space due to the sampling operation in the k-space. This effect is usually smoothed by using zero-padding, which increases the distance between two interacting replicas weakening their coupling.In this work, a fast convolution algorithm in the k-space able to remove the influence of replicas in the reconstruction of the magnetostatic field in thin films of finite size, is proposed and applied to study linear spin wave dynamics. It is based on the use of the Fourier transform of truncated Green function, which has been recently used to compute volume potentials defined on free-space in a finite (truncated) domain [3]. Such method, guarantees a regularization of the Green function by cutting off the interaction in physical space beyond the domain of interest and this in turn produces spectral accuracy [5] for the Fourier transform evaluation by FFT algorithm. The accuracy of the proposed method is studied by comparing the results with analytical calculations and with those obtained by fast convolution with the infinite thin film Green's function as a function of mesh size. For example, in the figure, the computation of the demagnetizing factor along the x direction for a thin film of dimensions Lx = 300nm, Ly = 100nm and Lz = 3nm as a function of the mesh dimension (kxmax = kymax = kmax) and discretization step-size (Δkx = Δky = Δk) in the k-space is considered. With solid lines, the curves in the infinite film approximation (infin.) are indicated, while curves with symbols results from the proposed approach (trunc.). The horizontal dotted line, represents the analytical value of the demagnetizing factor evaluated with Newell’s formula [6]. Computation with Δk&lt; π/300 nm-1, for the duality property of the Fourier transform, are equivalent to use zero padding in the ordinary space. This produces a substantial improving of estimation accuracy with respect to the analytical value, in case of infinite film approximation. As a benchmark of the proposed approach, it results that the curves computed with the truncated Green function, show a weak dependence from Δk for Δk&gt;π/R and collapse into the same one for Δk&lt;π/R, where R is the radius of a disk region which defines the compact support of the truncated Green function. This, as a confirmation of the fact, that when the region that exceeds the support of the truncated Green's function is zero padded, there is no effect on the computation of the magnetostatic field. Moreover, for sufficient value of kmax the asymptotic value tends towards analytical one, since when Δk&lt;π/R the influence of replicas on the magnetostatic field computation is removed. **

The paper presents a method for the computation of the radar cross section of microstrip patches. The approach is based on a combination of ideas derived from the mixed potential integral equation formulation for microstrip antennas, fast convolution algorithms and the biconjugate gradient iterative method. All operations are kept at a vector level and this renders the method capable of handling a large number of unknowns. Numerical results for a rectangular and a circular patch have been compared with measured results available in literature and a good agreement has been observed. Advantages and limitations are discussed.

In this paper we consider the problem of multiplying reasonably small integers using fewer counters than that required by straightforward partial product accumulation. Not surprisingly the method we use is based on the observation that integer multiplication can be formulated as aperiodic convolution. However, instead of using something like the Fast Fourier Transform to compute the aperiodic convolution, we use what are known as a fast convolution algorithms. In this way we can construct multipliers for as small as eighteen bit integers which use fewer counters than that required by straightforward partial product accumulation. Because of the perceived overhead involved with an aperiodic formulation of integer multiplication, the ability to do this goes somewhat against the conventional wisdom that aperiodic formulation of integer multiplication gains an advantage over a straightforward partial product formulation only for fairly large integers. >

Modern deep convolutional neural networks (CNNs) suffer from high computational complexity due to excessive convolution operations. Recently, fast convolution algorithms such as fast Fourier transform (FFT) and Winograd transform have gained attention to address this problem. They reduce the number of multiplications required in the convolution operation by replacing it with element-wise multiplication in the transform domain. However, fast convolution-based CNN accelerators have three major concerns: expensive domain transform, large memory overhead, and limited flexibility in kernel size. In this paper, we present a novel CNN accelerator based on number theoretic transform (NTT), which overcomes the existing limitations. We propose the low-cost NTT and inverse-NTT converter that only use adders and shifters for on-chip domain transform, which solves the inflated bandwidth problem and enables more parallel computations in the accelerator. We also propose the accelerator architecture that includes multiple tile engines with the optimized data flow and mapping. Finally, we implement the proposed NTT-based CNN accelerator on the Xilinx Alveo U50 FPGA and evaluate it for popular deep CNN models. As a result, the proposed accelerator achieves 2859.5, 990.3, and 805.6 GOPS throughput for VGG-16, GoogLeNet, and Darknet-19, respectively. It outperforms the existing fast convolution-based CNN accelerators up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$9.6\times $ </tex-math></inline-formula> .

This fourth part of a five-part series starts with the expansion of an analytic function of an arbitrary square matrix, involving real or complex numbers, in terms of its constituent matrices. The conjoint matrix and characteristic polynominal are computed by a convenient algorithm and are then used for calculating constituent matrices. Finally there is a discussion of the two-sided matrix equation AY + B = YC, and of the reduction of an arbitrary matrix to a similar Jordan matrix.

The prelims comprise: Definition and properties of Gibbs derivatives on finite non-Abelian groups Gibbs anti-derivative Partial Gibbs derivatives Gibbs differential equations Matrix interpretation of Gibbs derivatives Fast algorithms for calculation of Gibbs derivatives on finite groups Calculation of Gibbs derivatives through DDs References

In this letter, we present the dynamical equation and corresponding artificial recurrent neural network for computing the Drazin inverse for arbitrary square real matrix, without any restriction on its eigenvalues. Conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the Drazin inverse are considered. Several illustrative examples present the results of computer simulations.