Number Of Arithmetic Operations Research Articles

Fast Radix-2 Algorithm for the Discrete Hartley Transform of Type II

We present a new efficient method for the computation of the discrete Hartley transform of type II and radix-2 length N=2n. This recursive method requires a reduced number of arithmetic operations compared with existing methods and can be easily implemented. A new efficient method for the direct computation of a length N type-II DHT from two adjacent DHT-II sequences of length N/2 is also presented.

Sturm root counting using chebyshev expansion

The Sturm's root counting method by the floating point calculation was studied to solve real roots of a real algebraic equation of high degree. To reduce the effects of round-off errors, polynomials are represented by the coefficients of Chebyshev polynomials instead of monomials. In the Sturm sequence calculation, the quotients are stored instead of remainders. The values of remainders at a given abscissa to count their sign variations are calculated from the stored sequence of quotients. This approach reduces the space and the number of arithmetic operations. Some experiments was made using the FP numbers.

Condition number based complexity estimate for solving polynomial systems

By modifying and combining algorithms in symbolic and numerical computation, we propose a real-root-counting based method for deciding the feasibility of systems of polynomial equations. Along with this method, we also use a modified Newton operator to efficiently approximate the real solutions when the systems are feasible. The complexity of our method can be measured by a number of arithmetic operations which is singly exponential in the number of variables.

On the Halley class of methods for unconstrainedoptimization problems

Third-order methods can be used to solve efficiently the unconstrained optimization problems, and they, in most cases, use fewer iterations but more computational cost per iteration than a second-order method to reach the same accuracy. Recently, it has been shown by an article that under some conditions the ratio of the number of arithmetic operations of a third-order method (the Halley class of methods) and Newton’s method is constant (at most 5) per iteration. Automatic differentiation (AD) can compute fast and accurate derivatives such as the Jacobian, Hessian matrix and the tensor of the function. The Halley class of methods includes these high-order derivatives. In this paper, we apply AD efficiently to the methods and investigate the computational complexity of them. The results show that under general conditions even including the computation of the function and its derivative terms, the upper bound of the ratio can be reduced to 3.5.

On the choice of a multiplication algorithm for polynomials and polynomial matrices

We investigate multiplication algorithms for dense and sparse polynomials and polynomial matrices over different numerical domains and obtain expressions for the complexity of multiplication of polynomials and polynomial matrices understood as the expectation of the number of arithmetic operations. These expressions for a set of parameters of practical interest are tabulated. The results of experiments with the corresponding programs are presented. Bibliography: 8 titles.

Calculating reliability of the recursively definable networks

The number of arithmetic operations required to determine the network reliability was estimated. Internet-like networks were considered by way of example. For the networks with edges of low and high reliability, the asymptotic formulas were derived. The calculations were based on approximating reliability by the sum of probabilities of operability of all acyclic paths or the sum of probabilities of failure of all network sections that are minimal in set-theoretical inclusion.

A Novel Split-Radix Fast Algorithm for 2-D Discrete Hartley Transform

This paper presents a fast split-radix- (2t2)/(8t8) algorithm for computing the 2-D discrete Hartley transform (DHT) of length N ×N with N = q · 2 m, where q is an odd integer. The proposed algorithm decomposes an N × N DHT into one N /2 × N /2 DHT and 48 N /8 × N /8 DHTs. It achieves an efficient reduction on the number of arithmetic operations, data transfers and twiddle factors compared to the split-radix-(2×2)/(4×4) algorithm. Moreover, the characteristic of expression in simple matrices leads to an easy implementation of the algorithm. If implementing the above two algorithms with fully parallel structure in hardware, it seems that the proposed algorithm can decrease the area complexity compared to the split-radix-(2×2)/(4×4) algorithm, but requires a little more time complexity. An application of the proposed algorithm to 2-D medical image compression is also provided.

Fast 2-D 8×8 discrete cosine transform algorithm for image coding

A new fast two-dimension 8×8 discrete cosine transform (2D 8×8 DCT) algorithm based on the characteristics of the basic images of 2D DCT is presented. The new algorithm computes each DCT coefficient in turn more independently. Hence, the new algorithm is suitable for 2D DCT pruning algorithm of pruning away any number of high-frequency components of 2D DCT. The proposed pruning algorithm is more efficient than the existing pruning 2D DCT algorithms in terms of the number of arithmetic operations, especially the number of multiplications required in the computation.

Improved Accuracy Factors for the Nonuniform Fast Fourier Transform (NUFFT) Algorithm

Based on the regular Fourier matrix, a new set of accuracy factors is proposed for the nonuniform fast Fourier transform algorithm to improve the accuracy of transformed data. It shows that the proposed factors can reduce the errors by three to more than ten times with almost the same number of arithmetic operations. Numerical examples are shown for the applications in computational electromagnetics.

On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix

Let \( \tilde \lambda \) be an approximate eigenvalue of multiplicity mc = n − r of an n × n real symmetric tridiagonal matrix T having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting mc rows of T−\( \tilde \lambda \)I so that the condition number of the r × n matrix B formed of the remaining r rows is as small as possible. A special basis of mc vectors with local supports is constructed for the subspace kerB. These vectors are approximate eigenvectors of T corresponding to \( \tilde \lambda \). Another method for deleting mc rows of T−\( \tilde \lambda \)I is also proposed. This method uses a rank-revealing QR decomposition; however, it requires a considerably larger number of arithmetic operations. For the latter algorithm, the condition number of B is estimated, and orthogonality estimates for vectors of the special basis of kerB are derived.

Power optimization of digital baseband WCDMA receiver components on algorithmic and architectural level

Abstract. High data rates combined with high mobility represent a challenge for the design of cellular devices. Advanced algorithms are required which result in higher complexity, more chip area and increased power consumption. However, this contrasts to the limited power supply of mobile devices. This presentation discusses the application of an HSDPA receiver which has been optimized regarding power consumption with the focus on the algorithmic and architectural level. On algorithmic level the Rake combiner, Prefilter-Rake equalizer and MMSE equalizer are compared regarding their BER performance. Both equalizer approaches provide a significant increase of performance for high data rates compared to the Rake combiner which is commonly used for lower data rates. For both equalizer approaches several adaptive algorithms are available which differ in complexity and convergence properties. To identify the algorithm which achieves the required performance with the lowest power consumption the algorithms have been investigated using SystemC models regarding their performance and arithmetic complexity. Additionally, for the Prefilter Rake equalizer the power estimations of a modified Griffith (LMS) and a Levinson (RLS) algorithm have been compared with the tool ORINOCO supplied by ChipVision. The accuracy of this tool has been verified with a scalable architecture of the UMTS channel estimation described both in SystemC and VHDL targeting a 130 nm CMOS standard cell library. An architecture combining all three approaches combined with an adaptive control unit is presented. The control unit monitors the current condition of the propagation channel and adjusts parameters for the receiver like filter size and oversampling ratio to minimize the power consumption while maintaining the required performance. The optimization strategies result in a reduction of the number of arithmetic operations up to 70% for single components which leads to an estimated power reduction of up to 40% while the BER performance is not affected. This work utilizes SystemC and ORINOCO for the first estimation of power consumption in an early step of the design flow. Thereby algorithms can be compared in different operating modes including the effects of control units. Here an algorithm having higher peak complexity and power consumption but providing more flexibility showed less consumption for normal operating modes compared to the algorithm which is optimized for peak performance.

An efficient local approach to convexity testing of piecewise-linear hypersurfaces

We prove the following criterion: a compact connected piecewise-linear hypersurface (without boundary) in R n ( n ⩾ 3 ) is the boundary of a convex body if and only if every point in the relative interior of each ( n − 3 ) -face has a neighborhood that lies on the boundary of some convex body. This criterion is derived from our theorem that any connected complete locally convex hypersurface in S n ( n ⩾ 3 ) is the boundary of a convex body in S n . We give an easy-to-implement convexity testing algorithm based on our criterion. This algorithm does not require any assumptions about the global topology of the input hypersurface. For R 3 the number of arithmetic operations used by our algorithm is at most linear in the number of vertices, while in general it is at most linear in the number of incidences between ( n − 2 ) -faces and ( n − 3 ) -faces. The algorithm still remains polynomial even when the dimension n is a variable and the bit complexity model for (exact) arithmetic operations is used. The suggested method works in more general situations than the convexity verification algorithms developed by [K. Mehlhorn, S. Näher, M. Seel, R. Seidel, T. Schilz, S. Schirra, U. Stefan, C. Uhrig, Checking geometric properties or verification of geometric structures, in: Proc. of 12th Annual ACM Sympos. Comput. Geom., Computational Geom., 1996, pp. 159–165] and [O. Devillers, G. Liotta, F. Preparata, R. Tamassia, Checking the convexity of polytopes and the planarity of subdivisions, Comput. Geom. 11 (1998) 187–208]—for example, our method does not require the input surface to be homeomorphic to the sphere, nor does it require the input data to include normal vectors to the facets that are oriented “in a coherent way”. For R 3 the complexity of our algorithm is the same as that of previously known algorithms; for higher dimensions there seems to be no clear winner, but our approach is the only one that easily treats surfaces of arbitrary topology without a preliminary topological computation aimed at verifying that the input is a topological sphere. Furthermore, our method can be easily extended to piecewise-polynomial surfaces of degree 2 and 3.

On the informative power of all possible linear functionals for the discretization of solutions of the Klein-Gordon equation in the metric of L 2,∞

For the Klein-Gordon equation with initial conditions in Nikol’skii classes, we obtain sharp orders of discretization errors with respect to precise information. In particular, these orders imply qualitatively and quantitatively that raising the smoothness of one of the initial conditions relative to the other does not improve the discretization characteristics of the solution of the equation. Bounds for information distortion are given under which the limit possible errors, i.e., discretization errors with respect to precise information, are still maintained. This is important because superfluous computational accuracy used when implementing an algorithm results in an unjustified increase in memory requirements and the number of arithmetic operations without improving the accuracy inherent to the algorithm. The quality of approximation aggregates is also estimated in mean square with respect to probability measures on Nikol’skii classes.

Algorithm and Software Optimization of Variable Block Size Motion Estimation for H.264/AVC on a VLIW–SIMD DSP

We implemented the H.264/AVC variable block size motion estimation (VBSME) using a very long instruction word (VLIW)---single instruction multiple data (SIMD) digital signal processor (DSP). The SAD_Reuse method which has a regular structure is chosen for VBSME not only to remove redundant sum of absolute difference (SAD) operations but also to utilize the instruction level parallelism (ILP) and data level parallelism (DLP) of the architecture. A fast mode decision algorithm is developed to reduce the number of `compare and update' operations and simplify the rate distortion optimization (RDO). The developed fast mode decision uses the difference of motion vectors and the maximum a posteriori (MAP) estimation of the rate-distortion costs. Several advanced software techniques that include software pipelining and packed-data processing are employed. Especially, memory access overhead reduction schemes including the multi-block processing and the inter-procedural scheduling are used for the software optimization. In order to reduce the `write buffer full' in the quarter pixel ME, a 4 bit quantization scheme is developed, which increases the number of arithmetic operations but decreases the stall cycles very much. The implemented variable block size ME for H.264/AVC requires an average of 9 M and 78 Mcycles per frame for QCIF and CIF size video sequences, respectively, in the TMS320C64x DSP architecture.

Low-complexity 8×8 transform for image compression

An efficient 8 times 8 sparse orthogonal transform matrix is proposed for image compression by appropriately introducing some zeros in the 8 times 8 signed discrete cosine transform (SDCT) matrix. An algorithm for its fast computation is also developed. It is shown that the proposed transform provides a 25% reduction in the number of arithmetic operations with a performance in image compression that is much superior to that of the SDCT and comparable to that of the approximated discrete cosine transform.

DCT-IV computation

A new fast algorithm for computing a type of discrete cosine transform applied to compute modified cosine transform is described. The number of arithmetic operations of the algorithm is estimated and interpreted. The computational scheme with substitution of variables is presented, and methods for optimizing the code for digital signal processors are proposed.

A General Class of Split-Radix FFT Algorithms for the Computation of the DFT of Length-$2^{m}$

In this paper, a general class of split-radix fast Fourier transform (FFT) algorithms for computing the length-2m DFT is proposed by introducing a new recursive approach coupled with an efficient method for combining the twiddle factors. This enables the development of higher split-radix FFT algorithms from lower split-radix FFT algorithms without any increase in the arithmetic complexity. Specifically, an arbitrary radix-2/2s FFT algorithm for any value of s, 4les sles m, is proposed and its arithmetic complexity analyzed. It is shown that the number of arithmetic operations (multiplications plus additions) required by the proposed radix-2/2s FFT algorithm is independent of s and is (2m-3)2m+1+8 regardless of whether a complex multiplication is carried out using four multiplications and two additions or three multiplications and three additions. This paper thus provides a variety of choices and ways for computing the length-2m DFT with the same arithmetic complexity.

Design Equations for Jointly Optimized Frequency-Response Masking Filters

The introduction of nonlinear optimization techniques to the design of a frequency-response masking (FRM) filter has changed the way in which an FRM filter is synthesized. It allows all subfilters in an FRM structure to be optimized jointly, resulting in further savings in the number of arithmetic operations. Under the joint optimization, a new set of design equations is necessary, not only for a more computationally efficient filter, but also for the simplification of the design process and the reduction of the design time. In this paper, we present a set of design equations that estimates the filter lengths and optimum interpolation factor in an FRM filter under joint optimization. It is shown, by means of examples, that the proposed design equations lead to a better estimation of the optimum interpolation factor compared with existing design equations.

Comparison of two generalized methods for solving multi-peg towers of Hanoi puzzles

Two major generalized methods for solving the multi-peg tower of Hanoi problem are considered. These are the dynamic approach of the multi-peg problem as noted in Majumdar [8], and generalized recursive optimal solution for the multi-peg tower of Hanoi by Ikpotokin et al. [3]. It is also shown that the DP approach will utilize more storage space, more number of arithmetic operations and off course more time compare to the second method for the same number of peg t and disk, n.

COMMUNICATION COMPLEXITY AND SPEED-UP IN THE EXPLICIT DIFFERENCE METHOD

An earlier suggested parallel "ring" algorithm for solving the spatially one-dimensional initial-boundary-value problem (IBVP) for a parabolic equation using an explicit difference method is shortly described. Theoretical estimates for both of the speed-up function and the communication complexity of this parallel algorithm are studied. The speed-up function is determined as the ratio between times for realization of the algorithm in sequentional and parallel cases. Theoretical estimates of the speed-up function show the significant speed-up of the parallel algorithm in comparison with the serial one for a large number of values computed by one processor during one time level and it is shown that the the speed-up tends to the number of used processors. Communication complexity is determined as a ratio between the number of interchanges and the number of arithmetical operations. It has been proven that the coefficient of the communication complexity for spatially m-dimensional IBVP tends in general to ¾.