Computer Execution Time Research Articles

"Extending Amdahl's and Gustafson Baris's Laws by Adding Communication Overheads"

The extension of Amdahl's law and Gustafson-Barsis' law presented in this article provides insight into the relationship between communication costs and network topology in parallel computing. By considering communication latency in parallel computation execution time, these extensions can help researchers and developers optimize the interconnected network architecture of processing nodes and improve the performance of parallel computing systems. Today, parallel computer systems consisting of hundreds and thousands of processing nodes based on multiprocessor chip technology, high-speed optical transmission such as supercomputers are being researched, developed, and applied in many fields many areas. Although chip technology has progressed to the 3 nm process, the network architecture connecting processing nodes continues to be a problem that greatly affects the communication delay in the parallel computation time of the applications. In this paper, extensions of Amdahl's law and Gustafson-Barsis' law are presented with the addition of communication costs depending on the topology of the topology. These extensions provide insight into the relationship between communication costs and network topology in parallel computing. By considering communication latency in parallel computation execution time, these extensions can help researchers and developers optimize the interconnected network architecture of processing nodes and improve the performance of parallel computing systems.

ShaderPerFormer: Platform-independent Context-aware Shader Performance Predictor

The ability to model and predict the execution time of GPU computations is crucial for real-time graphics application development and optimization. While there are many existing methodologies for graphics programmers to provide such estimates, those methods are often vendor-dependent, require the platforms to be tested, or fail to capture the contextual influences among shader instructions. To address this challenge, we propose ShaderPerFormer, a platform-independent, context-aware deep-learning approach to model GPU performance and provide end-to-end performance predictions on a per-shader basis. To provide more accurate predictions, our method contains a separate stage to gather platform-independent shader program trace information. We also provide a dataset consisting of a total of 54,667 fragment shader performance samples on 5 different platforms. Compared to the PILR and SH baseline methods, our approach reduces the average MAPE across five platforms by 8.26% and 25.25%, respectively.

Improved Hybrid K-Nearest Neighbours Techniques in Segmentation of Low-Grade Tumor and Cerebrospinal Fluid

In image segmentation, identifying information or object detection in medical im-ages is crucial, particularly information that is harder to spot in magnetic resonance imaging (MRI) of low-grade tumors or cerebrospinal fluid (CSF). To address the aforementioned problems associated with missing data in MRI images and the low quality of MRI images that required longer processing times, this research is to seg-ment brain tumors or detect CSF in four-dimensional MRI images. A new hybrid k-nearest neighbors (k-NN) framework is also proposed, which consists of three tech-niques: correlation matrices of discrete Fourier transform (CM-DFT), Laplace Eigen maps of locally preserving projection (LELPP), and a hybrid GrabCut hidden Markov model of k-mean clustering (GCHMkC). The combination of the Hidden Markov Model (HMM) and the k-mean clustering technique is known as the Hidden Markov Model of the k-mean clustering method (HMMkC). To begin with, the Graph Cut and Support Vector Machine (GCSVM) and the GCHMkC approach are combined. The method increased the quality of the images suggested by the methodology, achieving an accuracy of 99.83%, a sensitivity of 99.99%, a specificity of 99.8%, and a computa-tional execution time of 14.9 seconds. Second, a technique called CM-DFT is sug-gested to improve MRI images while resolving the issue of missing imputation data. The accuracy of the MRI image datasets was improved to 99.84%, the time-lag in the hybrid k-NN algorithm was reduced to 99%, the missing data ratio was reduced to 0.9%, 10%, and 12%, and the correctness of the imputed data was improved to 1.533 seconds with computational execution. Thirdly, the nonlinear data is reduced and unnecessary features are eliminated using the Laplace Eigen maps of locally pre-serving projection (LELPP) approach. The hybrid k-NN algorithm used by the tech-nique yields results with 99% accuracy and an execution duration of 2.42 seconds.

Mechanized Proofs of Adversarial Complexity and Application to Universal Composability

In this paper we enhance the EasyCrypt proof assistant to reason about computational complexity of adversaries. The key technical tool is a Hoare logic for reasoning about computational complexity (execution time and oracle calls) of adversarial computations. Our Hoare logic is built on top of the module system used by EasyCrypt for modeling adversaries. We prove that our logic is sound w.r.t. the semantics of EasyCrypt programs — we also provide full semantics for the EasyCrypt module system, which was previously lacking. We showcase (for the first time in EasyCrypt and in other computer-aided cryptographic tools) how our approach can express precise relationships between the probability of adversarial success and their execution time. In particular, we can quantify existentially over adversaries in a complexity class, and express general composition statements in simulation-based frameworks. Moreover, such statements can be composed to derive standard concrete security bounds for cryptographic constructions whose security is proved in a modular way. As a main benefit of our approach, we revisit security proofs of some well-known cryptographic constructions and we present a new formalization of Universal Composability (UC).

Digital twin of an absorption chiller for solar cooling

The aim of this study is to create a digital twin of a commercial absorption chiller for control and optimization purposes. The chiller is a complex system that is affected by solar intermittency and non-linearities. The authors use Adaptive Neuro-fuzzy Inference System (ANFIS) to model the chiller’s behavior during transients and part-load events. The chiller is divided into four sub-models, each modeled by ANFIS, and trained and validated using data from 15 days of operation. The ANFIS models are precise, accurate, and fast, with a worst-case Mean Absolute Percentage Error (MAPE) of 3.30% and reduced error dispersion (σE=0.88) and Standard Error (SE=0.01). The models outperformed literature models in terms of MAPE, with MAPEs of 1.12%, 2.21%, and 3.24% for the High Temperature Generator (HTG), absorber + condenser, and evaporator outlet temperatures, respectively. The computational execution time of the model is also a valuable asset, with an average simulation step taking less than 0.20 ms and a total simulation time of 8.9 s for three days of operation. The resulting digital twin is suitable for Model Predictive Control applications and fast what-if analysis and optimization due to its gray-box representation and computational speed.

Higher-Order Hexahedral Finite Elements for Structural Dynamics: A Comparative Review

The finite element method (FEM) is widely used in many engineering applications. The popularity of FEM led to the development of several variants of formulations, and hexahedral meshes surged as one of the most computationally effective. After briefly reviewing the reasons and advantages behind the formulation of increasing order elements, including the serendipity variants and the associated reduced integration schemes, a systematic comparison of the most common hexahedral formulations is presented. A numerical benchmark was used to assess convergency rates and computational efficiencies when solving the eigenvalue problem for linear dynamic analysis. The obtained results confirmed the superior performances of the higher-order brick element formulations. In terms of computational efficiency, defined as the ratio between achievable accuracy and computational execution time, quadratic or cubic formulations exhibited the best results for the stages of FE model assembly and solution computation, respectively.

Agent-Based Image Contrast Enhancement Algorithm

One crucial step in several image processing and computer vision applications is Image Contrast Enhancement (ICE), whose main objective is to improve the quality of the information contained in the processed images. Most of the proposed schemes attack the problem by redistributing the pixel intensities in a histogram, leading to undesirable effects such as noise amplification, over-saturation, and lousy human perception. On the other hand, Agent-Based Models (ABM) are computational models that allow describing the behavior and interactions of autonomous agents when they operate cooperatively. These agents follow behavioral rules rather than mathematical formulations. This mechanism allows the implementation of complex behavioral patterns in agents through their interactions. This paper proposes a two-step method where pixels in the processed image are considered agents whose behavioral rules permit to enhance significatively the contrast. In our approach, the interactions among the agents are characterized by the differences in intensity values among the pixels or agents. In the first step, pixels or agents that present enough high differences in their intensity are modified to increase even more their differences. In the second step, pixels or agents that maintain a very small difference are altered to assume a homogeneous intensity value. The proposed approach has been tested considering different public datasets commonly used in the literature. Its results are also compared with those produced by other well-known ICE techniques. Evaluation of the experimental results demonstrates that the proposed approach highlights the important details of the image taking a lower computational execution time.

Surrogate Model for Design Uncertainty Estimation of Nonlinear Electromagnetic Vibration Energy Harvester

The paper proposes a solution to the problem of estimating the uncertainty of the output power with respect to the design parameters for an electromagnetic vibration energy harvesting converter. Due to costly utilisation of time-domain mathematical models involved in the procedure of determination of the average output power of the system, an algorithm for developing the surrogate model that enables rapid estimation of this quantity within the prescribed frequency band limits is proposed. As a result, the metamodel sensitive to the most impactful design parameters is developed using Kriging with successive refinement of the design grid for gaining the accuracy. Under operational conditions with a constant magnitude of the acceleration signal and the prescribed frequency band limits, the surrogate model enables evaluation of the average output power of the system at 105 design points in less than 2 s of computer execution time. The consistency and accuracy of the results obtained from the surrogate model is confirmed by comparison of selected results of computations with measurements carried out on the manufactured prototype. Based on the latter and the surrogate model, the confidence intervals for the design procedure were determined and the most important spread quantities were estimated, providing quantitative information on the accuracy of the design procedure developed for the considered system.

JACC-FPGA: A hardware accelerator for Jaccard similarity estimation using FPGAs in the cloud

Genomic similarity is a key metric in genomics, used in important tasks such as genome clustering and metagenomic profiling. One commonly-used approach is to treat each genome as a set of k-mers and to compute the Jaccard coefficient between each genome pair. However, computing the Jaccard coefficient between genomes in a large dataset is a computationally-challenging task. In this paper, we present an algorithm and accelerator architecture that uses an FPGA-as-a-service paradigm to compute the Jaccard similarity between pairs of genomes in large datasets using sketches and hardware acceleration in the cloud. The algorithm can compute the similarity between all genome pairs in the dataset, or it can use a selection criterion to reduce the amount of computation when only genome pairs with a Jaccard coefficient above a user-supplied threshold are of interest. After building the sketches, our heterogeneous accelerator can compute more than 96 million Jaccard coefficients per second running on an AWS EC2 f1.2xlarge instance with a Xilinx XCVU9P FPGA, which is 58 times faster than a state-of-the art software implementation that exploits SIMD instructions and thread-level parallelism on a compute-optimized EC2 c5.9xlarge instance with 36 hardware threads. The accelerator also computes similarities 27 times faster than a straightforward GPU-accelerated implementation of the Jaccard coefficient using sketches, and 4 times faster than an optimized GPU implementation of our algorithm, both running on an EC2 g5.4xlarge instance tailored with an NVIDIA A10G GPU. Furthermore, using a Jaccard coefficient threshold of 0.8 reduces the execution time of similarity computation to approximately one third in the hardware-accelerated and parallel software implementations, compared to computing the complete similarity matrix.

A lightweight learning-based decoding algorithm for intraneural vagus nerve activity classification in pigs

Objective. Bioelectronic medicine is an emerging field that aims at developing closed-loop neuromodulation protocols for the autonomic nervous system (ANS) to treat a wide range of disorders. When designing a closed-loop protocol for real time modulation of the ANS, the computational execution time and the memory and power demands of the decoding step are important factors to consider. In the context of cardiovascular and respiratory diseases, these requirements may partially explain why closed-loop clinical neuromodulation protocols that adapt stimulation parameters on patient’s clinical characteristics are currently missing. Approach. Here, we developed a lightweight learning-based decoder for the classification of cardiovascular and respiratory functional challenges from neural signals acquired through intraneural electrodes implanted in the cervical vagus nerve (VN) of five anaesthetized pigs. Our algorithm is based on signal temporal windowing, nine handcrafted features, and random forest (RF) model for classification. Temporal windowing ranging from 50 ms to 1 s, compatible in duration with cardio-respiratory dynamics, was applied to the data in order to mimic a pseudo real-time scenario. Main results. We were able to achieve high balanced accuracy (BA) values over the whole range of temporal windowing duration. We identified 500 ms as the optimal temporal windowing duration for both BA values and computational execution time processing, achieving more than 86% for BA and a computational execution time of only ∼6.8 ms. Our algorithm outperformed in terms of BA and computational execution time a state of the art decoding algorithm tested on the same dataset (Vallone et al 2021 J. Neural Eng. 18 0460a2). We found that RF outperformed other machine learning models such as support vector machines, K-nearest neighbors, and multi-layer perceptrons. Significance. Our approach could represent an important step towards the implementation of a closed-loop neuromodulation protocol relying on a single intraneural interface able to perform real-time decoding tasks and selective modulation of the VN.

SPINNAKER: an R-based tool to highlight key RNA interactions in complex biological networks.

BackgroundRecently, we developed a mathematical model for identifying putative competing endogenous RNA (ceRNA) interactions. This methodology has aroused a broad acknowledgment within the scientific community thanks to the encouraging results achieved when applied to breast invasive carcinoma, leading to the identification of PVT1, a long non-coding RNA functioning as ceRNA for the miR-200 family. The main shortcoming of the model is that it is no freely available and implemented in MATLAB®, a proprietary programming platform requiring a paid license for installing, operating, manipulating, and running the software.ResultsBreaking through these model limitations demands to distribute it in an open-source, freely accessible environment, such as R, designed for an ordinary audience of users that are not able to afford a proprietary solution. Here, we present SPINNAKER (SPongeINteractionNetworkmAKER), the open-source version of our widely established mathematical model for predicting ceRNAs crosstalk, that is released as an exhaustive collection of R functions. SPINNAKER has been even designed for providing many additional features that facilitate its usability, make it more efficient in terms of further implementation and extension, and less intense in terms of computational execution time.ConclusionsSPINNAKER source code is freely available at https://github.com/sportingCode/SPINNAKER.git together with a thoroughgoing PPT-based guideline. In order to help users get the key points more conveniently, also a practical R-styled plain-text guideline is provided. Finally, a short movie is available to help the user to set the own directory, properly.Supplementary InformationThe online version contains supplementary material available at 10.1186/s12859-022-04695-x.

Reordering and Partitioning of Distributed Quantum Circuits

A new approach to reduce the teleportation cost and execution time in Distributed Quantum Circuits (DQCs) was proposed in the present paper. DQCs, a well-known solution, have been applied to solve the problem of maintaining a large number of qubits next to each other. In the distributed quantum system, the qubits are transferred to another subsystem by a quantum protocol like teleportation. Hence, a novel method was proposed to optimize the number of teleportation and to reduce the execution time for generating DQC. To this end, first, the quantum circuit was reordered according to the qubits placement to improve the computational execution time, and then the quantum circuit was modeled as a graph. Finally, we combined the genetic algorithm (GA) and the modified tabu search algorithm (MTS) to partition the graph model in order to obtain a distributed quantum circuit aimed at reducing the number of teleportation costs. A significant reduction in teleportation cost (TC) and execution time (ET) was obtained in benchmark circuits. In particular, we performed a more accurate optimization than the previous approaches, and the proposed approach yielded the best results for several benchmark circuits.

Adaptive Simulation-Based Optimization for Production Scheduling: A Comparative Study

Production planning and control is an essential activity for all production systems, as its performance directly affects important factors that are fundamental for business competitiveness. The flow shop scheduling problem is one of the most common problems involved in the planning of production systems, being often applied in the real world. The solution to this problem cannot be easily found due to its complexity, which makes it difficult to use methods that return optimal results due to their high computational execution time. Thus, Adaptive Simulation-Based Optimization is a tool with potential application as it considers the stochasticity of the problem, seeking solutions that are aligned with the real scenario through a simulation and optimizing the desired variables. The simulation-based optimization approach consists in a loop between an optimization heuristic/meta-heuristic and a simulation. The optimization inputs the simulation with a parameter set, while the simulation tests all the models generated by the optimization and returns the fitness function values. However, the literature presents several different heuristics, simulation approaches and optimization methodologies to solve the problem. The general objective of the present work is to comparatively evaluate different adaptive simulation-based optimization models for flow shop production systems. Two methods for comparison were considered, which apply a genetic algorithm integrated with a discrete events simulation. The first method seeks to find the best dispatching rule for each machine in the production system, while the second seeks to directly find the best sequencing of production orders without a predetermined rule. The results show that the rule-free sequencing of production orders returns a result 3.56% better than annual makespan and 5.62% better in execution time compared to the model based on dispatching rules.

A novel solution with rapid Voronoi-based coverage path planning in irregular environment for robotic mowing systems

This paper presents a novel solution of coverage path planning for robotic mowing applications. The planning algorithm is based on the Boustrophedon motions and the rapid Voronoi diagram. The coordinate conversion and the sweeping vector is defined by minimum bounding box and the Voronoi travel paths are designed to reduce the computational cost and execution time compared to conventional heuristic methods. The tracked path is controlled via dynamic feedback in standard lawn mowing robots under robot operating system (ROS). Within ROS, the information exchanging among different tasks of both extensive simulation cases and experimental field tests can be conducted easily. When meeting unknown obstacles, the proposed algorithm can re-plan its paths dynamically. The performance of the proposed algorithm is compared to several conventional coverage algorithms in terms of time efficiency, coverage, repetition, and robustness with respect to both concave and convex shapes. The field tests are conducted to demonstrate that the applicability of the sensor fusion and robustness of the proposed algorithm for the complete coverage tasks by robotic mowers.

Chio’s-like Method for Calculating the Rectangular (non-square) Determinants: Computer Algorithm Interpretation and Comparison

In this paper, we present an approach for the calculation of rectangular determinants, where in addition to the mathematical formula, we also provide a computer algorithm for their calculation. Firstly, we present a method similar to Sarrus method for calculating the rectangular determinant of the order 2 × 3. Secondly, we present an approach for calculating the rectangular determinants of order m ×n by adding a row with all elements equal to one (1) in any row, as well as an application of Chio’s rule for calculating the rectangular determinants. Thirdly, we find the time complexity and comparison of the computer execution time of calculation of the rectangular determinant based on the presented algorithms and comparing them with the algorithm based on the Laplace method.

Hybrid approaches for accelerated convergence of block-Jacobi iterative methods for solution of the neutron transport equation

The Parallel Block Jacobi (PBJ) [1,2] scheme is an iterative method with application in neutron transport solution which requires asynchronous sub-domain interfaces, where asynchronous here means flux continuity is not imposed across sub-domain interfaces. In this work, we present the development, analysis, and testing of hybrid methods which utilize both PBJ-type methods and synchronous sweep-based methods in a single iterative step. These methods are developed to reduce the penalty to iterative performance of the former incurred in problems containing optically thin cells, while maintaining the PBJ method's applicability to massively parallel solution on unstructured grids. As a prelude to this ultimate goal, this study is conducted in serial operation on 2-D Cartesian grids to facilitate implementation of the various tested iterative methods, to enable the reported spectral analysis, and for shorter computational execution times allowing the large amounts of performance data to be collected. The analysis consists of two Fourier analyses, demonstrating the sequential execution of a PBJ method and SI to be iteratively robust with respect to optical thickness. The results of these analyses and preliminary computational experimentation demonstrate the effectiveness of methods that execute PBJ and SI only in optically thick and thin regions of a problem, respectively. Further experimentation, including a parametric study, then demonstrates this approach to be effective when the thin region is solved using Inexact Parallel Block Jacobi (IPBJ), with sub-domains of a size such that the applicability to massively parallel solution on unstructured grids is not diminished. Finally, testing on realistic test problems containing large void regions confirms the viability of this approach. Performance comparisons include the traditional iterative acceleration methods Diffusion Synthetic Acceleration, Partial-current Nonlinear Diffusion Acceleration, and Adjacent-cell Preconditioner. The results show that while the hybrid PBJ methods generally consume more iterations than these traditional methods, our hybrid approach maintains an iteration count on the same order of magnitude as these acceleration methods, which are not fully developed for massively parallel solution on unstructured grids.

Energy and performance improvements in stencil computations on multi-node HPC systems with different network and communication topologies

Energy and performance improvements in stencil computations are relevant for both application developers and data center administrators. They appear as the fundamental scheme in many large-scale scientific simulations and workloads. Many research efforts have focused on some estimation techniques of the energy usage of HPC systems based on specific characteristics of parallel applications. In case of stencils, we have previously concentrated on detailed estimations of energy consumption and the energy-aware distribution of stencil computations on heterogeneous processors. However, we have restricted our comprehensive studies to a single heterogeneous computing node only. In this paper, we show how scheduling and optimization techniques can be applied for energy and performance improvements of stencil computations on multi-node HPC systems using different network topologies. We formulate a scheduling model together with a new Tabu Search algorithm, called Task Movement (TM), taking into account the communication hierarchies, to minimize the overall energy usage and the execution time of stencil computations. Experimental studies show that this algorithm solves the considered problem more efficiently comparing to other, simpler heuristics. We present computational experiments for a reference 7 point stencil computation pattern on three commonly used low-diameter network topologies: Fat-tree, Dragonfly, and Torus. According to our studies, the most promising multi-node HPC architecture for stencil computations is based on the Torus network concept. Finally, we argue that the proposed scheduling model and TM algorithm can be easily adopted within existing high-level parallel execution environments for stencils automatic performance tuning.

Ad Hoc Vehicular Fog Enabling Cooperative Low-Latency Intrusion Detection

Internet of Vehicles and vehicular networks have been compelling targets for malicious security attacks where several intrusion detection solutions have been proposed for protecting them. Nonetheless, their main problem lies in their heavy computation, which makes them unsuitable for next-generation artificial intelligence-powered self-driving vehicles whose computational power needs to be primarily reserved for real-time driving decisions. To address this challenge, several approaches have been lately presented to take advantage of the cloud computing for offloading intrusion detection tasks to central cloud servers, thus reducing storage and processing costs on vehicles. However, centralized cloud computing entails high latency on intrusion detection related data transmission and plays against its adoption in delay-critical intelligent applications. In this context, this article proposes a vehicular-edge computing (VEC) fog-enabled scheme allowing offloading intrusion detection tasks to federated vehicle nodes located within nearby formed ad hoc vehicular fog to be cooperatively executed with minimal latency. The problem has been formulated as a multiobjective optimization model and solved using a genetic algorithm maximizing offloading survivability in the presence of high mobility and minimizing computation execution time and energy consumption. Experiments performed on resource-constrained devices within actual ad hoc fog environment illustrate that our solution significantly reduces the execution time of the detection process while maximizing the offloading survivability under different real-life scenarios.

A mechanism for balancing accuracy and scope in cross-machine black-box GPU performance modeling

The ability to model, analyze, and predict execution time of computations is an important building block that supports numerous efforts, such as load balancing, benchmarking, job scheduling, developer-guided performance optimization, and the automation of performance tuning for high performance, parallel applications. In today’s increasingly heterogeneous computing environment, this task must be accomplished efficiently across multiple architectures, including massively parallel coprocessors like GPUs, which are increasingly prevalent in the world’s fastest supercomputers. To address this challenge, we present an approach for constructing customizable, cross-machine performance models for GPU kernels, including a mechanism to automatically and symbolically gather performance-relevant kernel operation counts, a tool for formulating mathematical models using these counts, and a customizable parameterized collection of benchmark kernels used to calibrate models to GPUs in a black-box fashion. With this approach, we empower the user to manage trade-offs between model accuracy, evaluation speed, and generalizability. A user can define their own model and customize the calibration process, making it as simple or complex as desired, and as application-targeted or general as desired. As application examples of our approach, we demonstrate both linear and nonlinear models; these examples are designed to predict execution times for multiple variants of a particular computation: two matrix-matrix multiplication variants, four discontinuous Galerkin differentiation operation variants, and two 2D five-point finite difference stencil variants. For each variant, we present accuracy results on GPUs from multiple vendors and hardware generations. We view this highly user-customizable approach as a response to a central question arising in GPU performance modeling: how can we model GPU performance in a cost-explanatory fashion while maintaining accuracy, evaluation speed, portability, and ease of use, an attribute we believe precludes approaches requiring manual collection of kernel or hardware statistics.

Efficient Routines for Obtaining Radiation View-Factor for Non-Uniform Horizons

There are a large number of engineering applications wherein estimation of radiation exchange between surfaces is required. One important part of the latter procedure is to obtain the so-called ‘view factor’ of the emitter–receiver combination. The principal aim of this article is to present finite-element based procedures to obtain view factor with good time efficiency. As such routines presented here compute the view factor between a non-uniform emitter field and the receiving surface. An example of this is ground-reflected solar radiation to thermal or photovoltaic collectors, the foreground itself being composed of surfaces that have wide-ranging reflectivities. Four routines have been presented here: one based on a uniform grid for emitting and receiving surfaces using a brute force approach (Uniform Populous Grid (UPG)), and another that used non-uniform grid for the receiving surface, where cells’ sizes increased in arithmetic progression as one withdraws from the common edge (Non Uniform Grid Populous (NUGP)). The last two routines used combinations of the first two approaches with Monte-Carlo approach (Uniform Grid Monte-Carlo (UGMC) and Non Uniform Grid Monte-Carlo (NUGMC)). It was found that the NUGP algorithm was the most efficient to reduce the calculation error for the same number of computations, it was about 450 times (430 for non-uniform reflectivity) more accurate than UPG, 160 (125) times more than UGMC and 70 (60) times more than NUGMC. Finally a comparison of advantages and disadvantages of all four considered routines was added, using the following criteria: ease of programming, computational execution time, accuracy of results obtained, and predictability of the errors.