scholarly journals Posit Arithmetic Hardware Implementations with The Minimum Cost Divider and SquareRoot

Electronics ◽  
2020 ◽  
Vol 9 (10) ◽  
pp. 1622
Author(s):  
Feibao Xiao ◽  
Feng Liang ◽  
Bin Wu ◽  
Junzhe Liang ◽  
Shuting Cheng ◽  
...  
Keyword(s):  
Hardware Implementation ◽  
Minimum Cost ◽  
Number System ◽  
Floating Point ◽  
Square Root ◽  
Subtraction Method ◽  
Embedded Devices ◽  
Hardware Implementations ◽  
Newton Raphson ◽  
Raphson Method

As a substitute for the IEEE 754-2008 floating-point standard, Posit, a new kind of number system for floating-point numbers, was put forward recently. Hitherto, some studies have proven that Posit is a better floating-point style than IEEE 754-2008 in some fields. However, most of these studies presented the advantages of Posit from the arithmetical aspect, but none of them suggested it had a better hardware implementation than that of IEEE 754-2008. In this paper, we propose several hardware implementations that contain the Posit adder/subtractor, multiplier, divider, and square root. Our goal is to achieve an arbitrary Posit format and exploit the minimum circuit area, which is required in embedded devices. To implement the minimum circuit area for the divider and square root, the alternating addition and subtraction method is used rather than the Newton–Raphson method. Compared with other works, the area of our divider is about 0.2×–0.7× (FPGA). Furthermore, this paper provides the synthesis results for each critical module with the Xilinx Virtex-7 FPGA VC709 platform.

scholarly journals Voltage Instability Analysis Using Mipower

2020 ◽  
pp. 220-225
Keyword(s):  
Power Factor ◽  
Power Flow ◽  
Flow Analysis ◽  
Minimum Cost ◽  
Load Flow ◽  
Instability Analysis ◽  
Voltage Instability ◽  
Load Flow Analysis ◽  
Newton Raphson ◽  
Raphson Method

Load Flow Analysis helps in error free operation of power system and also useful in forecasting the required equipment for expansion of the system. By forecasting the magnitude of the supply required along with effects caused by single or multiple defects in the system and calculating the magnitude of errors, it is very easy to compensate them using various techniques with minimum cost and effort. It means before installation the favorable sites and size of the infrastructure used are determined to maintain the power factor in the system. Here Power Flow Analysis is performed using Newton Raphson method. This method is used in solving power flow studies of various number of busesunder various conditions. In any network there will be undesired rise or drop or dissipation of voltage. Voltage instability decreases the efficiency of the system and also damages the equipment used. Hence voltage instability analysis is performed and magnitude of the instability is calculated and compensated using various techniques. Here we performed Load Flow Analysis on a 5bus system and Voltage Instability Analysis is also performed to the same with necessary outputs.[7]

scholarly journals Scalable Fixed Point QRD Core Using Dynamic Partial Reconfiguration

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Gayathri R. Prabhu ◽  
Bibin Johnson ◽  
J. Sheeba Rani
Keyword(s):  
Square Root ◽  
Partial Reconfiguration ◽  
Adaptive Equalizer ◽  
Dynamic Partial Reconfiguration ◽  
Givens Rotation ◽  
Look Up Table ◽  
Array Architecture ◽  
Boundary Cell ◽  
Newton Raphson ◽  
Raphson Method

A Givens rotation based scalable QRD core which utilizes an efficient pipelined and unfolded 2D multiply and accumulate (MAC) based systolic array architecture with dynamic partial reconfiguration (DPR) capability is proposed. The square root and inverse square root operations in the Givens rotation algorithm are handled using a modified look-up table (LUT) based Newton-Raphson method, thereby reducing the area by 71% and latency by 50% while operating at a frequency 49% higher than the existing boundary cell architectures. The proposed architecture is implemented on Xilinx Virtex-6 FPGA for any real matrices of sizem×n, where4≤n≤8andm≥nby dynamically inserting or removing the partial modules. The evaluation results demonstrate a significant reduction in latency, area, and power as compared to other existing architectures. The functionality of the proposed core is evaluated for a variable length adaptive equalizer.

scholarly journals A Modification of the Fast Inverse Square Root Algorithm

Computation ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 41 ◽  
Author(s):  
Cezary J. Walczyk ◽  
Leonid V. Moroz ◽  
Jan L. Cieśliński
Keyword(s):  
Analytical Approach ◽  
Floating Point ◽  
Square Root ◽  
Approximate Evaluation ◽  
Single Precision ◽  
Seed Solution ◽  
Numerical Tests ◽  
Newton Raphson ◽  
Relative Errors ◽  
Magic Constant

We present a new algorithm for the approximate evaluation of the inverse square root for single-precision floating-point numbers. This is a modification of the famous fast inverse square root code. We use the same “magic constant” to compute the seed solution, but then, we apply Newton–Raphson corrections with modified coefficients. As compared to the original fast inverse square root code, the new algorithm is two-times more accurate in the case of one Newton–Raphson correction and almost seven-times more accurate in the case of two corrections. We discuss relative errors within our analytical approach and perform numerical tests of our algorithm for all numbers of the type float.

scholarly journals A Modification of the Fast Inverse Square Root Algorithm

2019 ◽  
Author(s):  
Cezary J. Walczyk ◽  
Leonid V. Moroz ◽  
Jan L. Cieśliński
Keyword(s):  
Computational Cost ◽  
Floating Point ◽  
Square Root ◽  
Single Precision ◽  
Fast Calculation ◽  
Newton Raphson ◽  
Floating Point Numbers ◽  
Improved Algorithm

We present an improved algorithm for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithm is much more accurate than the famous fast inverse square root algorithm and has a similar computational cost. The presented modification concern Newton-Raphson corrections and can be applied when the distribution of these corrections is not symmetric (for instance, in our case they are always negative).

Axisymmetric flow near the critical point on the moving boundary

2010 ◽  
Vol 7 ◽  
pp. 182-190
Author(s):  
I.Sh. Nasibullayev ◽  
E.Sh. Nasibullaeva
Keyword(s):  
Fluid Flow ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Axial Velocity ◽  
Moving Boundary ◽  
Axisymmetric Flow ◽  
Boundary Motion ◽  
Newton Raphson ◽  
Raphson Method

In this paper the investigation of the axisymmetric flow of a liquid with a boundary perpendicular to the flow is considered. Analytical equations are derived for the radial and axial velocity and pressure components of fluid flow in a pipe of finite length with a movable right boundary, and boundary conditions on the moving boundary are also defined. A numerical solution of the problem on a finite-difference grid by the iterative Newton-Raphson method for various velocities of the boundary motion is obtained.

A Generalized Compositional Approach for Reservoir Simulation

1983 ◽  
Vol 23 (05) ◽  
pp. 727-742 ◽  
Author(s):  
Larry C. Young ◽  
Robert E. Stephenson
Keyword(s):  
Enhanced Recovery ◽  
Gas Condensate ◽  
Reservoir Modeling ◽  
Pressure Equation ◽  
Compositional Modeling ◽  
Fluid Property ◽  
Black Oil ◽  
Fluid Properties ◽  
Newton Raphson ◽  
Raphson Method

A procedure for solving compositional model equations is described. The procedure is based on the Newton Raphson iteration method. The equations and unknowns in the algorithm are ordered in such a way that different fluid property correlations can be accommodated leadily. Three different correlations have been implemented with the method. These include simplified correlations as well as a Redlich-Kwong equation of state (EOS). The example problems considered area conventional waterflood problem,displacement of oil by CO, andthe displacement of a gas condensate by nitrogen. These examples illustrate the utility of the different fluid-property correlations. The computing times reported are at least as low as for other methods that are specialized for a narrower class of problems. Introduction Black-oil models are used to study conventional recovery techniques in reservoirs for which fluid properties can be expressed as a function of pressure and bubble-point pressure. Compositional models are used when either the pressure. Compositional models are used when either the in-place or injected fluid causes fluid properties to be dependent on composition also. Examples of problems generally requiring compositional models are primary production or injection processes (such as primary production or injection processes (such as nitrogen injection) into gas condensate and volatile oil reservoirs and (2) enhanced recovery from oil reservoirs by CO or enriched gas injection. With deeper drilling, the frequency of gas condensate and volatile oil reservoir discoveries is increasing. The drive to increase domestic oil production has increased the importance of enhanced recovery by gas injection. These two factors suggest an increased need for compositional reservoir modeling. Conventional reservoir modeling is also likely to remain important for some time. In the past, two separate simulators have been developed and maintained for studying these two classes of problems. This result was dictated by the fact that compositional models have generally required substantially greater computing time than black-oil models. This paper describes a compositional modeling approach paper describes a compositional modeling approach useful for simulating both black-oil and compositional problems. The approach is based on the use of explicit problems. The approach is based on the use of explicit flow coefficients. For compositional modeling, two basic methods of solution have been proposed. We call these methods "Newton-Raphson" and "non-Newton-Raphson" methods. These methods differ in the manner in which a pressure equation is formed. In the Newton-Raphson method the iterative technique specifies how the pressure equation is formed. In the non-Newton-Raphson method, the composition dependence of certain ten-ns is neglected to form the pressure equation. With the non-Newton-Raphson pressure equation. With the non-Newton-Raphson methods, three to eight iterations have been reported per time step. Our experience with the Newton-Raphson method indicates that one to three iterations per tune step normally is sufficient. In the present study a Newton-Raphson iteration sequence is used. The calculations are organized in a manner which is both efficient and for which different fluid property descriptions can be accommodated readily. Early compositional simulators were based on K-values that were expressed as a function of pressure and convergence pressure. A number of potential difficulties are inherent in this approach. More recently, cubic equations of state such as the Redlich-Kwong, or Peng-Robinson appear to be more popular for the correlation Peng-Robinson appear to be more popular for the correlation of fluid properties. SPEJ p. 727

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