Double Precision Floating Point Numbers Research Articles

Rationalized diffraction calculations for high accuracy and high speed with few bits.

Diffraction calculations in few-bit formats, such as single-precision floating-point and fixed-point numbers, are important because they yield faster calculations and lower memory usage. However, these methods suffer from low accuracy owing to the loss of trailing digits. Fresnel diffraction is widely known to prevent the loss of trailing digits. However, it can only be used when the paraxial approximation is valid. In this study, a few-bit diffraction calculation method that achieves high accuracy without using any approximation is proposed. The proposed method is derived only by rationalizing the numerator of conventional formulas. Even for scenarios requiring double-precision floating-point numbers using conventional methods, the proposed method exhibits higher accuracy and faster computation time using single-precision floating-point numbers.

Periodic orbits in chaotic systems simulated at low precision

Non-periodic solutions are an essential property of chaotic dynamical systems. Simulations with deterministic finite-precision numbers, however, always yield orbits that are eventually periodic. With 64-bit double-precision floating-point numbers such periodic orbits are typically negligible due to very long periods. The emerging trend to accelerate simulations with low-precision numbers, such as 16-bit half-precision floats, raises questions on the fidelity of such simulations of chaotic systems. Here, we revisit the 1-variable logistic map and the generalised Bernoulli map with various number formats and precisions: floats, posits and logarithmic fixed-point. Simulations are improved with higher precision but stochastic rounding prevents periodic orbits even at low precision. For larger systems the performance gain from low-precision simulations is often reinvested in higher resolution or complexity, increasing the number of variables. In the Lorenz 1996 system, the period lengths of orbits increase exponentially with the number of variables. Moreover, invariant measures are better approximated with an increased number of variables than with increased precision. Extrapolating to large simulations of natural systems, such as million-variable climate models, periodic orbit lengths are far beyond reach of present-day computers. Such orbits are therefore not expected to be problematic compared to high-precision simulations but the deviation of both from the continuum solution remains unclear.

Accuracy Controllable SpMV Optimization on GPU

Sparse matrix vector multiplication (SpMV) is a key kernel widely used in a variety of fields, and mixed-precision calculation brings opportunities to SpMV optimization. Researchers have proposed to store nonzero elements in the interval (-1, 1) in single precision and calculate SpMV in mixed precision. Though it leads to high performance, it also brings loss of accuracy. This paper proposes an accuracy controllable optimization method for SpMV. By limiting the error caused by converting double-precision floating-point numbers in the interval (-1, 1) into single-precision format, the calculation accuracy of mixed-precision SpMV is effectively improved. We tested sparse matrices from the SuiteSparse Matrix Collection on Tesla V100. Compared with the existing mixed-precision MpSpMV kernel, the mixed-precision SpMV proposed in this paper achieves an accuracy improvement.

Reliability analysis ofN-equationsN-unknowns method for the solution of the finite-difference time-domain (FDTD) problems

AbstractFinite-difference time-domain (FDTD) solution of electromagnetic problems at high-frequency (HF) range is a challenging task. This is due to two requirements of accurate geometrical modeling for small objects/antennas and proper modeling of skin-depth effect. Generally, the fine meshes are used to overcome these problems. However, this leads to a large number of time iterations because of the small unit time step. To decrease the computational time,N-equationsN-unknowns (NE-NU) method is developed for multi-frequency sources. Therefore, utilizing wideband nature of the FDTD method, NE-NU method is valuable. However, the reliability analysis of the NE-NU method has not been investigated in the literature. This study is the first numerical application of the NE-NU method that the reliability analysis of the NE-NU method is performed by using condition number calculation and it is validated by time-domain signals of an illustrative numerical example of a HF radar problem. A successful case and a failing case of the NE-NU method are clearly revealed. The effect of using a double-precision floating-point number and a single-precision floating-point number is also discussed. It is proved that a crucial value of the condition number can be found for the reliable NE-NU results.

On fixpoints of Higham’s function

We study the strange behavior in floating-point arithmetic of a function proposed by Nicholas Higham, consisting of repeated square roots extraction followed by the same number of times squaring and find its fixpoints. For IEEE standard double precision floating point numbers the fixpoints have the form \[ x \in \left\{\left( 1+k\mathrm{eps}\right) ^{\frac{1}{\mathrm{eps}}},\quad k=\left[ -745:\frac{1}{2}:-\frac{1}{2},0:709\right]\right\} \cup \{0\} , \] where \mathrm{eps} is the machine epsilon."

Improving the Accuracy of the Fast Inverse Square Root by Modifying Newton-Raphson Corrections.

Direct computation of functions using low-complexity algorithms can be applied both for hardware constraints and in systems where storage capacity is a challenge for processing a large volume of data. We present improved algorithms for fast calculation of the inverse square root function for single-precision and double-precision floating-point numbers. Higher precision is also discussed. Our approach consists in minimizing maximal errors by finding optimal magic constants and modifying the Newton–Raphson coefficients. The obtained algorithms are much more accurate than the original fast inverse square root algorithm and have similar very low computational costs.

The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects

The paper addresses issues associated with implementing GPC controllers in systems with multiple input signals. Depending on the method of identification, the resulting models may be of a high order and when applied to a control/regulation law, may result in numerical errors due to the limitations of representing values in double-precision floating point numbers. This phenomenon is to be avoided, because even if the model is correct, the resulting numerical errors will lead to poor control performance. An effective way to identify, and at the same time eliminate, this unfavorable feature is to reduce the model order. A method of model order reduction is presented in this paper that effectively mitigates these issues. In this paper, the Generalized Predictive Control (GPC) algorithm is presented, followed by a discussion of the conditions that result in high order models. Examples are included where the discussed problem is demonstrated along with the subsequent results after the reduction. The obtained results and formulated conclusions are valuable for industry practitioners who implement a predictive control in industry.

A Latency-Effective Pipelined Divider for Double-Precision Floating-Point Numbers

In this article, we propose an effective algorithm of pipelined dividers for double-precision floating-point numbers. This reduces the latency of the previous pipelined dividers without increasing the lookup table size. The experimental results show that the proposed divider reduces the overall latency up to 16% compared to the previous divider.

A SYNTHETIC 3D SCENE FOR THE VALIDATION OF PHOTOGRAMMETRIC ALGORITHMS

Abstract. This paper describes the construction and composition of a synthetic test world for the validation of photogrammetric algorithms. Since its 3D objects are entirely generated by software, the geometric accuracy of the scene does not suffer from measurement errors which existing real-world ground truth is inherently afflicted with. The resulting data set covers an area of 13188 by 6144 length units and exposes positional residuals as small as the machine epsilon of the double-precision floating point numbers used exclusively for the coordinates. It is colored with high-resolution textures to accommodate the simulation of virtual flight campaigns with large optical sensors and laser scanners in both aerial and close-range scenarios. To specifically support the derivation of image samples and point clouds, the synthetic scene gets stored in the human-readable Alias/Wavefront OBJ and POV-Ray data formats. While conventional rasterization remains possible, using the open-source ray tracer as a render tool facilitates the creation of ideal pinhole bitmaps, consistent digital surface models (DSMs), true ortho-mosaics (TOMs) and orientation metadata without programming knowledge. To demonstrate the application of the constructed 3D scene, example validation recipes are discussed in detail for a state-of-the-art implementation of semi-global matching and a perspective-correct multi-source texture mapper. For the latter, beyond the visual assessment, a statistical evaluation of the achieved texture quality is given.

Conversion of Mersenne Twister to double-precision floating-point numbers

The 32-bit Mersenne Twister generator MT19937 is a widely used random number generator. To generate numbers with more than 32 bits in bit length, and particularly when converting into 53-bit double-precision floating-point numbers in [0,1) in the IEEE 754 format, the typical implementation concatenates two successive 32-bit integers and divides them by a power of 2. In this case, the 32-bit MT19937 is optimized in terms of its equidistribution properties (the so-called dimension of equidistribution with v-bit accuracy) under the assumption that one will mainly be using 32-bit output values, and hence the concatenation sometimes degrades the dimension of equidistribution compared with the simple use of 32-bit outputs. In this paper, we analyze such phenomena by investigating hidden F2-linear relations among the bits of high-dimensional outputs. Accordingly, we report that MT19937 with a specific lag set fails several statistical tests, such as the overlapping collision test, matrix rank test, and Hamming independence test.

Field Programmable Gate Arrays Based Real Time Robot Arm Inverse Kinematic Calculations and Visual Servoing

DOI: 10.26650/electrica.2018.49877 Reliability and precision are very important in space, medical, and industrial robot control applications. Recently, researchers have tried to increase the reliability and precision of the robot control implementations. High precision calculation of inverse kinematic, color based object recognition, and parallel robot control based on field programmable gate arrays (FPGA) are combined in the proposed system. The precision of the inverse kinematic solution is improved using the coordinate rotation digital computer (CORDIC) algorithm based on double precision floating point number format. Red, green, and blue (RGB) color space is converted to hue saturation value (HSV) color space, which is more convenient for recognizing the object in different illuminations. Moreover, to realize a smooth operation of the robot arm, a parallel pulse width modulation (PWM) generator is designed. All applications are simulated, synthesized, and loaded in a single FPGA chip, so that the reliability requirement is met. The proposed method was tested with different objects, and the results prove that the proposed inverse kinematic calculations have high precision and the color based object recognition is quite successful in finding coordinates of the objects.

Interval stabbing on the Automata Processor

The Automata Processor (AP) was designed for string-pattern matching. In this paper, we showcase its use to execute integer and floating-point comparisons and apply the same to accelerate interval stabbing queries. An interval stabbing query determines which of the intervals in a set overlap a query point. Such queries are often used in computational geometry, pattern matching, database management systems, and geographic information systems. The check for each interval is programmed as a single automaton and multiple automata are executed in parallel to provide significant performance gains. While handling 32-bit integers or single-precision floating-point numbers, up to 2.75 trillion comparisons can be executed per second, whereas 0.79 trillion comparisons per second can be completed for 64-bit integers or double-precision floating-point numbers. Additionally, our solution leaves the intervals in the set unordered allowing addition or deletion of an interval in constant time. This is not possible for contemporary solutions wherein the intervals are ordered, making update of intervals complex. Our automata designs are modular allowing them to become constituent parts of larger automata, where the numerical comparisons are part of the overall pattern matching operation. We have validated the designs on the hardware, and the routines to generate the necessary automata and execute them on the AP will be made available as software libraries shortly.

Analysis of Area and Delay for Floating Point Matrix Multiplication

In this paper, a new area and delay minimization architecture is proposed for rank-1 update matrix–matrix multiplication whose inputs are double precision floating point number. The architecture design is based on pipelined multiplication which handles matrix of arbitrary sizes; the processing data are stored in dedicated on-chip BRAM. This minimization is introduced to designers as a trade-off between bandwidth and local memory. Analysis is presented for the design parameters optimal choice. In addition, Vedic matrix multiplication architecture is given in order to explore the variations in power, delay and memory. The hardware architecture is described in Verilog HDL synthesized for a family virtex-6 and device xc6vlx240t FPGA, which scale more than 40 processing elements. Various parameters like LUTs, Slices, bonded IOBs, frequency, DSP48E1S, delay, Power, CPU Completion time and Memory usage are analysed. Rank 1 update consumes a power of 2.520 watt and has a delay of 5.277 ns respectively. Comparing other rank-1 multiplication methods our proposed algorithms uses 15% less area resources and improves the delay in 12%.

Development of a high-performance control system by decentralization with reflective memory on QUEST

The plasma control system (PCS) of QUEST was a centralized system, which lost its scalability because of the overload imposed on its central processing unit (CPU) of the PCS, making it impossible to add new functions. Thus, the PCS is distributed into a main workstation (WS) and subsystem (SS) with a reflective memory (RFM) in order to share data between these systems so as to mitigate the load on each system. As a result, 128 double-precision floating-point numbers (DBLs) can be transferred from the SS to the WS with a maximum latency of 250μs. The WS and the SS each have quad-core CPUs, and tasks are executed in parallel. Although one of the four cores is intermittently occupied by up to 90% by this transaction, the occupation is normally 60%. A time correction procedure is used to map the recorded data sets on the WS and the SS to a common time base by referring to the time difference between two systems.

Parallel Reproducible Summation

Reproducibility, i.e. getting bitwise identical floating point results from multiple runs of the same program, is a property that many users depend on either for debugging or correctness checking in many codes [10] . However, the combination of dynamic scheduling of parallel computing resources, and floating point nonassociativity, makes attaining reproducibility a challenge even for simple reduction operations like computing the sum of a vector of numbers in parallel. We propose a technique for floating point summation that is reproducible independent of the order of summation. Our technique uses Rump’s algorithm for error-free vector transformation [7] , and is much more efficient than using (possibly very) high precision arithmetic. Our algorithm reproducibly computes highly accurate results with an absolute error bound of $n \cdot 2^{-28} \cdot macheps \cdot \max _i |v_i|$ at a cost of $7n$ FLOPs and a small constant amount of extra memory usage. Higher accuracies are also possible by increasing the number of error-free transformations. As long as all operations are performed in to-nearest rounding mode, results computed by the proposed algorithms are reproducible for any run on any platform. In particular, our algorithm requires the minimum number of reductions, i.e. one reduction of an array of six double precision floating point numbers per sum, and hence is well suited for massively parallel environments.

A Pipelined Fused Processing Unit for DSP Applications

This paper designs a processing element for FFT processor capable of operating on 32-bit double precision floating point numbers. Pipelining is performed on the computational elements of the DSP processor to enhance the throughput. The performance of the Processing unit is increased by using the concept of fused architecture on the sub modules – the dot product unit and the add sub unit. Pipelining increases the speed of the CE of the processor while fused operations claim area optimization. The DSP applications involve FFT Processors that make use of the butterfly operations consisting of multiplications, additions, and subtractions of complex valued data (data is split into real part and the imaginary part). The radix-2 and radix-4 butterflies are designed using fused architecture. The fused FFT butterflies are to be 20 percent speedier and 30 percent smaller in area compared with the conventional method. The processing unit covers almost all the computations necessary for the processor.

More reliable forecasts with less precise computations: a fast-track route to cloud-resolved weather and climate simulators?

This paper sets out a new methodological approach to solving the equations for simulating and predicting weather and climate. In this approach, the conventionally hard boundary between the dynamical core and the sub-grid parametrizations is blurred. This approach is motivated by the relatively shallow power-law spectrum for atmospheric energy on scales of hundreds of kilometres and less. It is first argued that, because of this, the closure schemes for weather and climate simulators should be based on stochastic–dynamic systems rather than deterministic formulae. Second, as high-wavenumber elements of the dynamical core will necessarily inherit this stochasticity during time integration, it is argued that the dynamical core will be significantly over-engineered if all computations, regardless of scale, are performed completely deterministically and if all variables are represented with maximum numerical precision (in practice using double-precision floating-point numbers). As the era of exascale computing is approached, an energy- and computationally efficient approach to cloud-resolved weather and climate simulation is described where determinism and numerical precision are focused on the largest scales only.

Double precision rational approximation algorithm for the inverse standard normal second order loss function

We present a double precision algorithm based upon rational approximations for the inverse standard normal second order loss function. This function is used frequently in inventory management. No direct approximation or closed formulation exists for the inverse standard normal second order loss function. Calculations are currently based on root-finding methods and intermediate computations of the cumulative normal distribution or tabulations. Results then depend on the accuracy and valid range of that underlying function. We deal with these issues and present a direct, double precision accurate algorithm valid in the full range of double precision floating point numbers.

Exploring Rounding Errors in Matlab Using Extended Precision

We describe a simple package of Matlab programs which implements an extended-precision class in Matlab. We give some examples of how this class can be used to demonstrate the effects of rounding errors and truncation errors in scientific computing. The package is based on a representation called Double-Double, which represents each floating-point real as an unevalu- ated sum of IEEE double-precision floating point numbers. This allows Matlab computations that are accurate to 30 decimal digits. The data structure, basic arithmetic and elementary functions are implemented as a Matlab class, entirely using the Matlab programming language.

CUDA-enabled Sparse Matrix–Vector Multiplication on GPUs using atomic operations

Existing formats for Sparse Matrix–Vector Multiplication (SpMV) on the GPU are outperforming their corresponding implementations on multi-core CPUs. In this paper, we present a new format called Sliced COO (SCOO) and an efficient CUDA implementation to perform SpMV on the GPU using atomic operations. We compare SCOO performance to existing formats of the NVIDIA Cusp library using large sparse matrices. Our results for single-precision floating-point matrices show that SCOO outperforms the COO and CSR format for all tested matrices and the HYB format for all tested unstructured matrices on a single GPU. Furthermore, our dual-GPU implementation achieves an efficiency of 94% on average. Due to the lower performance of existing CUDA-enabled GPUs for atomic operations on double-precision floating-point numbers the SCOO implementation for double-precision does not consistently outperform the other formats for every unstructured matrix. Overall, the average speedup of SCOO for the tested benchmark dataset is 3.33 (1.56) compared to CSR, 5.25 (2.42) compared to COO, 2.39 (1.37) compared to HYB for single (double) precision on a Tesla C2075. Furthermore, comparison to a Sandy-Bridge CPU shows that SCOO on a Fermi GPU outperforms the multi-threaded CSR implementation of the Intel MKL Library on an i7-2700 K by a factor between 5.5 (2.3) and 18 (12.7) for single (double) precision. Source code is available at https://github.com/danghvu/cudaSpmv.