Arithmetic Complexity Research Articles

On Complex Interval Arithmetic Using Polar Form

In this paper, complex arithmetic in polar form, briefly sectors arithmetic is discussed. The fact that the family of sectors is not closed under addition and subtraction gives rise to the need of an optimized approximation. Precisely, the problem discussed, is to determine the smallest possible sector that contains the sum of two sectors. An efficient algorithm that is solving this problem is introduced and compared with existing algorithms in the literature. The algorithms are implemented in MATLAB R2020a and tested for all cases. It is also shown by examples that the proposed algorithms perform much better than their ancestors and avoid their errors.

Efficient Cached 64 Point FFT Processor Using Floating Point Arithmetic for OFDM Application

Presently Fourth generation and other wireless systems are focused area for the research and development in the communication field. Fast Fourier Transform (FFT) & Inverse FFT (IFFT) are required for Orthogonal frequency division multiplexing in the integral part of modulation/demodulation modules that occupies more area and power. This paper presents low power and area efficient Cached memory for Fast Fourier Transform (FFT) processor using floating point arithmetic for OFDM application. To store computational permutations each butterfly unit needs one memory. So if considering higher radix of FFT processor, memory requirement increases, it yields to more power consumption and more density occupancy. In this proposed cached 64 point radix 2^6 SDF architecture for reducing the arithmetic hardware complexity of complex multipliers and complex adders present in butterfly structure to obtain low power and less memory requirement. The proposed system is synthesized by using CADENCE RTL COMPLIER and is implemented in ENCOUNTER RTL TO GDSII SYSTEM” using 90nm CMOS technology with a supply voltage of 1V. Synthesis results shows that the proposed design is efficient in terms of gate count, area and power consumption.

Efficient Design of Scaled Rectangular (Saramäki) Window

A rectangular-window sidelobe magnitude reduction method by widening the main lobe width was proposed by Saramäki. A technique for trading off main lobe width against sidelobe magnitude for any arbitrary window was reported by Lim et al.; a fast convergence algorithm for its implementation was proposed by the same authors in another article, where the derivatives of the window function are expressed in Chebyshev polynomials which have high arithmetic complexity. All the coefficients of a rectangular-window are equal; this special property is exploited, in this paper, for deriving the window function’s derivatives without the use of Chebyshev polynomials resulting in a great reduction in the arithmetic complexity.

Radix-$N$ Algorithm for Computing $N^{2^{n}}$-Point DFT Approximations

The ever increasing technological demand for the DFT computation poses several challenges both to theory and hardware realization. The design of usual fast Fourier transform (FFT) algorithms seems to have reached a stage of diminishing returns in terms of performance. Alternatively, approximate transform methods have been demonstrated to provide substantial gains in terms of energy-efficiency and performance by tolerating small inaccuracies in the results. In this paper, we present a transform scaling method variant of the Cooley-Tukey algorithm to obtain DFT approximations of large blocksize. The proposed method scales up a given <i>N</i>-point transformation to an <i>N</i>²-point transformation. Such scaling can be successively applied leading to <inline-formula><tex-math notation="LaTeX">$\mathop {{N}^2}\nolimits^n $</tex-math></inline-formula>-point transformations. We have fully presented the 32⁴-point DFT approximation which stems from a multiplierless 32-point DFT approximation. The proposed approximation is equipped with a fast algorithm; we also supply the arithmetic complexity assessment and an error analysis.

Solving Block Low-Rank Matrix Eigenvalue Problems

To solve large-scale matrix eigenvalue problems (EVPs), a two-step tridiagonalization method using the block Householder transformation (HT) is often employed. Although the method based on dense matrix arithmetic requires a memory storage of O(N2) and an arithmetic operations of O(N3), in this study, these were reduced by approximating the method using block low-rank (BLR-) matrices. A special block HT for BLR-matrices and a two-step tridiagonalization method using it are proposed to solve an EVP with a real symmetric BLR-matrix. In the proposed block HT, block Householder vectors are also formed using BLR-matrices. It is demonstrated how the block size m in the BLR-matrix should be determined and confirmed that the memory and arithmetic complexities of the proposed method were O(N5/3) and O(N7/3), respectively, for typical cases when using an appropriate block size m ∝ N1/3. In numerical experiments of a string free vibration problem with known analytical solutions, for large eigenvalues, the calculated eigenvalues using the proposed method converge toward the analytical ones in accordance with the theoretical convergence curves. Owing to the reduced complexity, an EVP of a matrix was solved with about N =300,000, which is significantly larger than the limit of the conventional method with dense matrices, within a reasonable amount of time on a CPU core. For the calculation time, the proposed method was faster than the conventional method when the matrix size N was larger than a few tens of thousands.

Rusho’s Conjecture on Visualising Complex Arithmetic in Real Analysis

We all know Complex number or Imaginary Number can’t be seen in real analysis. Mathematicians made another field for this named complex analysis . But In this paper I am going to relate complex number with the real . At least We can take some feel from here . We will use nothing but simple known concept form High School Mathematics and Will imply it in a beautiful way . From headline It can be seen something boring or tough but if we dive in to it it will be understood by school student’s also . Analysis is the of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. . But for this paper one can should be known some basic tool like logarithmic function , Euler _lagrange Beautiful math equation , natural log series etc and beat more free time with enthusiastic mind.

Anxiety-Related Difficulties With Complex Arithmetic

Abstract. Human behavior depends on the interplay between cognition and emotion. Negative emotions like anxiety affect performance, particularly in complex tasks, by limiting cognitive resources – known as the anxiety–complexity effect. This study set out to replicate the anxiety–complexity effect in a web-based experiment. We investigated individual differences in math anxiety – a negative emotional response specific to math – and arithmetic performance ( N = 382). The mental arithmetic task consisted of a two-digit addition and subtraction, with/without carrying or borrowing, respectively. As expected and preregistered, higher math anxiety was related to poorer arithmetic performance, especially in complex tasks – indicating the anxiety–complexity effect. Consequently, the negative math anxiety-performance link is especially pronounced for complex arithmetic, which requires calculations across place-values and thus working memory resources. This successful replication of the anxiety–complexity effect suggests that math-anxious individuals have particular difficulties in complex arithmetic.

Recursion relations for hp‐FEM Element Matrices on quadrilaterals

AbstractIn this paper we consider higher order shape functions for finite elements on quadrilaterals. Using tensor products of suitable Jacobi polynomials, it can be proved that the corresponding mass and stiffness matrices are sparse with respect to polynomial degree p . Due to the orthogonal relations between Jacobi polynomials the exact values of the entries of mass and stiffness matrix can be determined. Using symbolic computation, we can find simple recurrence relations which allow us to compute the remaining nonzero entries in optimal arithmetic complexity. Besides the H1 case also the conformal basis functions for H(Div) and H(Curl) are investigated.

Precise Numerical Differentiation of Thermodynamic Functions with Multicomplex Variables.

The multicomplex finite-step method for numerical differentiation is an extension of the popular Squire–Trapp method, which uses complex arithmetics to compute first-order derivatives with almost machine precision. In contrast to this, the multicomplex method can be applied to higher-order derivatives. Furthermore, it can be applied to functions of more than one variable and obtain mixed derivatives. It is possible to compute various derivatives at the same time. This work demonstrates the numerical differentiation with multicomplex variables for some thermodynamic problems. The method can be easily implemented into existing computer programs, applied to equations of state of arbitrary complexity, and achieves almost machine precision for the derivatives. Alternative methods based on complex integration are discussed, too.

Regular expression length via arithmetic formula complexity

We prove lower bounds on the length of regular expressions for finite languages by methods from arithmetic circuit complexity. First, we show a reduction: the length of a regular expression for a language L⊆{0,1}n is bounded from below by the minimum size of a monotone arithmetic formula computing a polynomial that has L as its set of exponent vectors. This result yields lower bounds for the language of all words with exactly k ones and n−k zeros and for the language of all Dyck words of length 2n. Second, we adapt a lower bound method for multilinear arithmetic formulas by so-called log-product polynomials to regular expressions. With this method we show almost tight lower bounds for the language of all n-bit binary numbers that are divisible by a given odd integer p and for the language of all permutations of {1,…,n}.

A short-memory operator splitting scheme for constant-Q viscoelastic wave equation

We propose a short-memory operator splitting scheme for solving the constant-Q wave equation, where the fractional stress-strain relation contains multiple Caputo fractional derivatives with order much smaller than 1. The key is to exploit its extension problem by converting the flat singular kernels into strongly localized ones, so that the major contribution of weakly singular integrals over a semi-infinite interval can be captured by a few Laguerre functions with proper asymptotic behavior. Despite its success in reducing both memory requirement and arithmetic complexity, we show that numerical accuracy under prescribed memory variables may deteriorate in time due to the dynamical increments of projection errors. Fortunately, it can be considerably alleviated by introducing a suitable scaling factor β>1 and pushing the collocation points closer to origin. An operator splitting scheme is introduced to solve the resulting set of equations, where the auxiliary dynamics can be solved exactly, so that it gets rid of the numerical stiffness and discretization errors. Numerical experiments on both 1-D diffusive wave equation and 2-D constant-Q P- and S-wave equations are presented to validate the accuracy and efficiency of the proposed scheme.

The Demands of Simple and Complex Arithmetic Word Problems on Language and Cognitive Resources.

Solving arithmetic word problems requires constructing a situation model based on the problem text and translating that into a mathematical model. As such, word problem solving makes demands on students’ language comprehension and their domain-general cognitive resources. These demands may decrease when students get more experienced and use strategies that do not require fully understanding the situation presented in the problem. The current study aims to address this hypothesis. Students (N=444) from third to sixth grade solved a paper-and-pencil task with 48 mathematics problems, comprising symbolic arithmetic problems and standard word problems, as well as more complex word problems that involve two arithmetic steps or include irrelevant numerical information. Their performance was analyzed with multilevel logistic regression analyses. Results showed that within each grade, performance on the different problem types did not differ, suggesting that already in third-grade students seem helped nor hindered by presenting arithmetic problems in a story, even if that story contains irrelevant numerical information. Non-verbal reasoning was more important in standard word problems than in arithmetic problems in symbolic format in one-step arithmetic, and reading comprehension was more important in solving two-step arithmetic word problems than in one-step arithmetic word problems.

A multi‐step finite‐state automaton for arbitrarily deterministic Tsetlin Machine learning

AbstractDue to the high arithmetic complexity and scalability challenges of deep learning, there is a critical need to shift research focus towards energy efficiency. Tsetlin Machines (TMs) are a recent approach to machine learning (ML) that has demonstrated significantly reduced energy compared to neural networks alike, while providing comparable accuracy on several benchmarks. However, TMs rely heavily on energy‐costly random number generation to stochastically guide a team of Tsetlin Automata (TA) in TM learning. In this paper, we propose a novel finite‐state learning automaton that can replace the TA in the TM, for increased determinism. The new automaton uses multi‐step deterministic state jumps to reinforce sub‐patterns, without resorting to randomization. A determinism parameter finely controls trading off the energy consumption of random number generation, against randomization for increased accuracy. Randomization is controlled by flipping a coin before every state jump, ignoring the state jump on tails. For example, makes every update random and makes the automaton completely deterministic. Both theoretically and empirically, we establish that the proposed automaton converges to the optimal action almost surely. Further, used together with the TM, only substantial degrees of determinism reduce accuracy. Energy‐wise, random number generation constitutes switching energy consumption of the TM, saving up to 11 mW power for larger datasets with high values. Our new learning automaton approach thus facilitates low‐energy ML.

Evaluating FFT-based algorithms for strided convolutions on ARMv8 architectures

Convolutional Neural Networks (CNNs) have been widely adopted in many kinds of artificial intelligence applications. Most of the computational overhead of CNNs is spent on convolutions. An effective approach to reducing the overhead is transforming convolutions in the time domain into multiplications in the frequency domain by means of Fast Fourier Transform (FFT) algorithms, known as FFT-based fast algorithms for convolutions. However, current FFT-based fast implementations only work for unit-strided convolutions with stride as 1, and cannot be directly applied to strided convolutions with stride size greater than 1, which are usually used as the first layer of CNNs and as an effective alternative to the pooling layers for downsampling.In this paper, we first introduce rearrangement- and sampling-based methods for applying FFT-based fast algorithms to strided convolutions, and the arithmetic complexities of these two methods and the direct method are compared in detail. Then, the highly optimized parallel implementations of the two methods on ARMv8-based many-core CPU are presented. Lastly, we benchmark these implementations against two GEMM-based implementations on this ARMv8 CPU. Our experimental results with convolutions of different kernels, feature maps, and batch sizes show that the rearrangement-based method generally exceeds the sampling-based one under the same optimizations in most cases, and these two methods can get much better performance than GEMM-based ones when the kernels, feature maps and batch sizes are large. The experimental results on the convolutional layers in popular CNNs further demonstrate the conclusions above.

Effective randomness for continuous measures

We investigate which infinite binary sequences (reals) are effectively random with respect to some continuous (i.e., non-atomic) probability measure. We prove that for every n n , all but countably many reals are n n -random for such a measure, where n n indicates the arithmetical complexity of the Martin-Löf tests allowed. The proof rests upon an application of Borel determinacy. Therefore, the proof presupposes the existence of infinitely many iterates of the power set of the natural numbers. In the second part of the paper we present a metamathematical analysis showing that this assumption is indeed necessary. More precisely, there exists a computable function G G such that, for any n n , the statement “All but countably many reals are G ( n ) G(n) -random with respect to a continuous probability measure” cannot be proved in Z F C n − \mathsf {ZFC}^-_n . Here Z F C n − \mathsf {ZFC}^-_n stands for Zermelo-Fraenkel set theory with the Axiom of Choice, where the Power Set Axiom is replaced by the existence of n n -many iterates of the power set of the natural numbers. The proof of the latter fact rests on a very general obstruction to randomness, namely the presence of an internal definability structure.

Developmental fronto-parietal shift of brain activation during mental arithmetic across the lifespan: A registered report protocol.

Arithmetic processing is represented in a fronto-parietal network of the brain. However, activation within this network undergoes a shift from domain-general cognitive processing in the frontal cortex towards domain-specific magnitude processing in the parietal cortex. This is at least what is known about development from findings in children and young adults. In this registered report, we set out to replicate the fronto-parietal activation shift for arithmetic processing and explore for the first time how neural development of arithmetic continues during aging. This study focuses on the behavioral and neural correlates of arithmetic and arithmetic complexity across the lifespan, i.e., childhood, where arithmetic is first learned, young adulthood, when arithmetic skills are already established, and old age, when there is lifelong arithmetic experience. Therefore, brain activation during mental arithmetic will be measured in children, young adults, and the elderly using functional near-infrared spectroscopy (fNIRS). Arithmetic complexity will be manipulated by the carry and borrow operations in two-digit addition and subtraction. The findings of this study will inform educational practice, since the carry and borrow operations are considered as obstacles in math achievement, and serve as a basis for developing interventions in the elderly, since arithmetic skills are important for an independent daily life.

Age-related differences in how negative emotions influence arithmetic performance

ABSTRACT In this study, we investigated the role of negative emotions on arithmetic and whether this role changes with aging during adulthood. Young and older adults were asked to verify one-digit addition problems (Experiment 1) and to estimate the results of two-digit multiplication problems (Experiment 2). In both experiments, easier and harder problems were displayed superimposed on emotionally neutral (e.g. mushrooms) or emotionally negative (e.g. a corpse) pictures. In both simple and complex arithmetic, young and older adults obtained poorer arithmetic performance under negative emotion conditions, especially while solving harder problems. Most interesting, deleterious effects of negative emotions on arithmetic performance were larger in young than in older adults. These findings have important implications for further our understanding of the role of negative emotions in the domain of arithmetic and age-related differences in this role.

A Fast Algorithm to Solve Delay Vandermonde Systems in Phased-Array Digital Receivers

Phased-array multibeam RF beamformers require calibration of receivers used in an array of elements before the signals can be applied to an analog beamforming network. This article presents a fast <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) algorithm for solving linear systems having a delay Vandermonde matrix (DVM) for the analog beamforming matrix. The structure of the DVM enables to obtain a fast algorithm to efficiently reverse multibeams at the DVM output such that calibration of the input low noise amplifiers can be achieved. The arithmetic complexity of the proposed DVM system solving algorithm is preferable over the standard matrix inversion consuming <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations. Numerical experiments are presented for forward accuracy of the proposed algorithm computed at the calibration frequency. Moreover, numerical results are shown to compare the order of the arithmetic complexity and the execution time of the proposed algorithm. Signal flow graphs are presented for four- and eight-element multibeam phased arrays.

On lattice point counting in $$\varDelta $$-modular polyhedra

Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $\Delta$ in absolute values. We show that the short rational generating function for the power series $$ \sum\limits_{m \in P \cap Z^n} x^m $$ can be computed with the arithmetic complexity $ O\left(T_{SNF}(d) \cdot d^{k} \cdot d^{\log_2 \Delta}\right), $ where $k$ and $\Delta$ are fixed, $d = \dim P$, and $T_{SNF}(m)$ is the complexity to compute the Smith Normal Form for $m \times m$ integer matrix. In particular, $d = n$ for the case (i) and $d = n-k$ for the case (ii). The simplest examples of polyhedra that meet conditions (i) or (ii) are the simplicies, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. We apply these results to parametric polytopes, and show that the step polynomial representation of the function $c_P(y) = |P_{y} \cap Z^n|$, where $P_{y}$ is parametric polytope, can be computed by a polynomial time even in varying dimension if $P_{y}$ has a close structure to the cases (i) or (ii). As another consequence, we show that the coefficients $e_i(P,m)$ of the Ehrhart quasi-polynomial $$ \left| mP \cap Z^n\right| = \sum\limits_{j = 0}^n e_i(P,m)m^j $$ can be computed by a polynomial time algorithm for fixed $k$ and $\Delta$.

Optimised Floating Point FFT Core for Improved OMP CS System

ABSTRACT The overall network effectiveness in any real time application is defined by the amount of precision used in computing and the capacity to sustain low cost hardware. Within this context optimisation method for floating point computing is starting to emerge gradually due to the enhanced precision level. Particularly with regard to FFT floating point arithmetic model calculation, hardware sophistication is forced to minimise the uncertainty penalty gap between fixed points – FFT floating point prototype. Hardware fusion systems are designed primarily for increasing the use rate of hardware throughout FPU arithmetic calculations. The proposed system to compute mantissa is actually achieved using which hardware counts are reduced in a complex multiplication process and afterwards the signed digit recoding can be reconfigured by a digital integer. Eventually, the technique of radix factorisation is integrated to minimise complex arithmetic associated with phases of FFT. The results show that the algorithm proposed shows decreased field, elevated frequency of operation and an increased level of accuracy. Ultimately, it is contrasted with efficient FPGA implementation that uses the capital and robustness of the proposed fused CSD FP models.