Retracted: Improved artificial bee colony algorithm with opposition‐based learning

IET Image ProcessingVolume 14, Issue 15 p. 3639-3650 THIS ARTICLE HAS BEEN RETRACTEDFree Access Retracted: Improved artificial bee colony algorithm with opposition-based learning Retraction(s) for this article Retraction: IPR Special Issue Retraction Statement Volume 17Issue 1IET Image Processing pages: 301-301 First Published online: August 22, 2022 Yongcun Cao, Yongcun Cao School of Information Engineering, Minzu University of China, Beijing, 100081 People's Republic of ChinaSearch for more papers by this authorSaisai Ji, Corresponding Author Saisai Ji saisai_ji@163.com School of Information Engineering, Minzu University of China, Beijing, 100081 People's Republic of ChinaSearch for more papers by this authorYong Lu, Yong Lu School of Information Engineering, Minzu University of China, Beijing, 100081 People's Republic of ChinaSearch for more papers by this author Yongcun Cao, Yongcun Cao School of Information Engineering, Minzu University of China, Beijing, 100081 People's Republic of ChinaSearch for more papers by this authorSaisai Ji, Corresponding Author Saisai Ji saisai_ji@163.com School of Information Engineering, Minzu University of China, Beijing, 100081 People's Republic of ChinaSearch for more papers by this authorYong Lu, Yong Lu School of Information Engineering, Minzu University of China, Beijing, 100081 People's Republic of ChinaSearch for more papers by this author First published: 02 October 2020 https://doi.org/10.1049/iet-ipr.2020.0111Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The artificial bee colony (ABC) algorithm is a biological-inspired optimisation algorithm proposed by Karaboga. Since its solution search equation is good at exploration but poor at exploitation, the ABC algorithm converges slowly and is easy to fall into local optimum. Inspired by opposition-based learning (OBL), the authors propose an improved ABC algorithm called opposition-based learning ABC (OLABC). In OLABC, firstly, the population would be initialised using OBL. Secondly, to ensure the diversity of the population during the iterative process, the solution search equation is employed to bee phase would be improved. Generate the opposite solution when the fitness value of the newly generated solution is smaller than the current solution, and then apply the greedy selection strategy to update the solution. Thirdly, the adaptive weight strategy is used to dynamically adjust the weight, balancing the global exploration and local exploitation capabilities of the algorithm. Experiments on a set of benchmark functions show that OLABC has better convergence speed and optimisation precision than the compared algorithms. 1 Introduction The artificial bee colony (ABC) algorithm is a heuristic algorithm inspired by the behaviour of the honey bee colony. It was proposed by the Turkish scholar Karaboga to optimise numerical function in 2005 [[1]]. ABC algorithm is an embodiment of swarm intelligence. By imitating the foraging behaviour of bees, the ABC algorithm finds the optimal solution of numerical functions. The main feature of the ABC algorithm is that it does not need to know the special information on the problem. It just needs to focus on comparing the quality of the solution to the problem. Through the local optimisation behaviour of the artificial bee individuals, the global optimal value emerges in the population finally. As a newly invented swarm intelligence optimisation algorithm, the ABC algorithm has become an important optimisation algorithm in the field of bionic intelligent computing. Due to its excellent performance in many aspects, its theoretical research and application have become a new research hotspot. The ABC algorithm is widely used in image processing [[2]], parameter optimisation [[3]], neural networks [[4], [5]], path planning [[6]], fuzzy control [[7]], engineering applications [[8], [9]] and other fields [[10]]. The performance of the ABC algorithm is comprehensively analysed in [[11]]. The ABC algorithm is compared with other classical swarm intelligence algorithms such as differential evolution (DE) algorithm [[12]-[14]], particle swarm optimisation (PSO) algorithm [[15], [16]], and genetic algorithm (GA) [[17]]. Experimental results show that the performance of the ABC algorithm is generally better than other similar algorithms. The ABC algorithm has a good ability to search globally and is less affected by the initial population. The ABC algorithm is more suitable for solving high-dimensional multi-mode problems. At the same time, because the ABC algorithm only generates a new solution through the parent solution, the amplitude of each search change is small, resulting in weak local search ability and slow convergence of the ABC algorithm. Opposition-based learning (OBL), which has been proposed by Tizhoosh in 2005 [[18]], has been proven to be an effective way to improve the search ability of random search algorithms. The concept of OBL has been utilised to enhance the performance in many optimisation algorithms, such as DE [[19]], harmony search [[20]], PSO [[21]], firefly algorithm [[22]], etc. Mathematical theorems demonstrate that the probability close to the global optimal solution of the opposite solution is higher than the candidate solution generated randomly [[23]]. In the ABC algorithm, OBL can be used to select good individuals to enter the next generation by generating opposite solutions individuals in the current solutions interval and participating in the competition with the current population. Thereby ensure the diversity of the population during the iterative process, and avoid falling into local optimum. Inspired by the idea of OBL, this paper proposes an improved ABC algorithm with OBL (OLABC). Firstly, initialise the population by using the OBL method to improve the quality of the initial solution. Secondly, the idea of OBL is applied to the employed bee phase of the ABC algorithm to ensure the diversity of the population in each iteration. Finally, the adaptive weight strategy is adopted to dynamically adjust the weight to balance the ability of global search in the early stage and local search in the late stage. The experimental results show that the improved ABC algorithm proposed in this paper has better optimisation ability. In brief, the contributions of this paper are summarised as follows. (i) A novel solution search strategy is designed in the employed bee phase, in which the mechanism of OBL is incorporated to guarantee the diversity of the population during the process of iteration and avoid falling into local optimum. (ii) The adaptive weight strategy is used to adjust the weight of the solution search equation to balance the global exploration and local exploitation ability of the algorithm, which can improve the convergence speed of the algorithm to some extent. The remaining of this paper is organised as follows. Section 2 introduces the traditional ABC algorithm and the principle of OBL and gives a review of improved methods of the ABC algorithm. In Section 3, the proposed ABC algorithm with OBL is described in detail. The comparative experiments are conducted and the experimental results are analysed in Section 4. Finally, our work is concluded and the possible future work is discussed in Section 5. 2 Related work The ABC algorithm simulates the division of labour of the bee colony in the foraging process and the information-sharing mechanism between individuals. The bee colony can be divided into the employed bee, the onlooker bee and the scout bee according to the division of labour. For solving function optimisation problems, the position of the honey source represents the feasible solution of the function. The amount of nectar carried by bees represents the fitness value of the function. The process of solving the optimal value is the process in which bees search for the largest amount of nectar. When solving the objective function , SN solutions , (i = 1, 2, …, SN) are randomly initialised, and each solution is a D-dimensional vector. The working process of the algorithm is as follows: (i) Randomly initialise the population in the current solution space and calculate the fitness value of the initial solution as follows: (1) where , and is a random number between [0, 1]. (ii) In the employed phase, generate a new food source around the old one by using (2). And calculate the fitness value of the newly generated solution by using (3). Then the greedy selection strategy is applied to update the food source (2) where is a random number between [−1, 1], , and . (3) where fi is the objective function value of the food source xi, and fiti is the fitness value of the food source xi. (iii) The onlooker bee selects a food source according to the roulette probability calculated by (4). In (4), fiti can be calculated by (3). After selecting the food source, the onlooker bee performs a neighbourhood search by using (2) and updates the food source through a greedy selection strategy (4) (iv) In the phase of scout bee, the food source that has not been updated for a long time is abandoned, and the onlooker bee is converted into a scout bee. The scout bee searches for a new food source according to (5) randomly to replace the abandoned one. (5) In (5), ub and lb represent the upper and lower bounds of the solution, respectively, and represents a matrix of one row and d columns, in which . (v) Finally, judge the termination condition. And if it is satisfied, the optimal solution is outputted; otherwise, it returns to the second step. Aiming at the insufficiency that the traditional ABC algorithm has a slow convergence speed and is easy to fall into local optimum solutions, many scholars have improved it from different aspects. As can be seen from the existing research, most of the improvements are focused on the solution search equation. Akay and Karaboga [[24]] proposed a classical improved algorithm named MrABC. By adding the vibration frequency MR, the number of dimensions searching for new honey sources is changed from 1 to MR, and add the vibration amplitude SF to make the range of Фij from [−1, + 1] change to [−SF, + SF] to speed up convergence. Cui et al. [[25]] proposed a ranking-based adaptive ABC (ARABC) algorithm, which selected the parent food sources used in the solution search equation according to their ranking, and the selection probability is proportional to the rank, thus improving the convergence speed of the ABC algorithm. Inspired by the gravity model, Xiang et al. [[26]] proposed an attractive model to select a better neighbour of the current individual to improve the exploitation ability of the ABC algorithm. Although most of the improvements are concentrated in the ABC algorithm search phase, in order to further improve the performance of the ABC algorithm, many researchers have made some improvements in other aspects of the ABC algorithm. Xiang and An [[27]] used the combination search equation to speed up the search process, and adopted the chaos algorithm in the phase of scout bees and population initialisation, and used the reverse selection of roulette in the onlooker bee phase to improve the performance of ABC algorithm. Aydin et al. [[28]] proposed a dynamic ABC algorithm, which uses the probability factor p to dynamically determine the size of the bee colony and applies it to the environmental economic dispatching problem of the power generation system. Cao et al. [[29]] proposed a variant of GABC [[30]] called IABC to further strengthen the performance of GABC, in which a modified solution search equation was used to improve the exploitation ability of the algorithm in the onlooker bee phase. In addition, many researchers have combined ABC algorithms with other optimisation algorithms to further improve optimisation performance. Inspired by PSO, Li et al. [[30]] proposed the classic GABC (gbest-guided ABC), which used the global optimal value to lead the onlooker bees to search for new honey sources, thus improving the exploitation ability of the ABC algorithm. Ozturk et al. [[31]] proposed the GB–ABC algorithm to deal with the binary optimisation problem by combining the genetic operators of GA with the neighbourhood search mechanism of the ABC algorithm. Cao et al. [[32]] proposed a binary ABC algorithm with DE (BABC-DE) to solve the 0–1 knapsack problem (KP01) by designing a new binary searching operator in the employed bee phase and adopting the mutation and crossover operations of DE. OBL is a new concept introduced in recent years. It has been successfully applied to neural networks, reinforcement learning, swarm intelligence, and evolutionary algorithms to improve the performance of algorithms. The main idea of OBL is to consider the current estimate value and its corresponding opposite simultaneously while looking for the optimal solution of the objective function, thereby finding the optimal solution more efficiently. If the current solution is far from the location of the optimal solution, the opposite solution of the current solution can direct the search process in the opposite direction, thereby having a greater probability of finding the region in which the optimal solution is located. The definition of opposite number is as follows: In the case where the search space is one-dimensional, let be a real number. The opposite of x is defined in [[18]] as follows: (6) In the case where the search space is D-dimensional, let be a point in the D-dimensional space, and . The opposite of is defined in [[18], [33]] as follows: (7) The idea of OBL is used in many optimisation algorithms, such as ABC algorithm, DE algorithm, PSO, firefly algorithm and so on. Babaeizadeh and Ahmad [[34]] proposed an enhanced constrained ABC algorithm called EC-ABC to balance the ability of exploration and exploitation, in which the OBL mechanism was used to initialise the population. Choi et al. [[35]] combined the self-adaptive DE algorithm with the OBL and proposed a variant of the DE algorithm. They used the opposition-based scheme to improve the population diversity and avoid the premature convergence of the algorithm. In order to solve the problem of premature convergence of the PSO algorithm, Wang et al. [[21]] proposed an enhanced PSO algorithm called GPSO, in which they used the OBL in population initialisation and generation jumping. Rajmohan and Natarajan [[22]] incorporated OBL to generate a quality initial population to improve the performance of the firefly algorithm. 3 Improved OLABC 3.1 Population initialisation The population initialisation affects the global convergence speed and the quality of the solution of the evolutionary algorithm. In the absence of any prior information, the traditional ABC algorithm generally uses the random initialisation method to generate the initial solution. This method cannot guarantee the generated initial population can effectively extract the information of the solution space, which limits the optimisation efficiency of the algorithm to some extent. In order to improve the quality of the initial solution of the algorithm, in the population initialisation phase, this paper uses the initialisation operation with OBL to construct the random initial population and its opposite population, calculates the fitness value, and uses the solution with better fitness value as the initial population, thereby increasing the diversity of the population and the quality of the initial solution. The steps to initialise population using opposite learning are given in Fig. 1. Fig. 1Open in figure viewerPowerPoint Algorithm 1: Population initialisation strategy based on opposite learning The M individuals with better fitness values are selected from the union of and as the initial population. 3.2 Improved search strategy in the employed bee phase OBL is an optimisation strategy for machine learning. At each iteration of the algorithm, all opposite solutions of these current solutions are detected simultaneously, and select the solution which is more favourable to algorithm evolution from the current solution set and the opposite solution set, thereby reducing the blindness of the algorithm. In order to further improve the global search ability of the ABC algorithm and ensure the diversity of the population during the iterations, the idea of OBL is applied to the employed bee phase. When the fitness value of the newly generated solution is smaller than the current solution, generate the opposite solution of the current solution, and calculate its fitness value. Then apply the greedy selection strategy to update the food source. Fig. 2 is a schematic diagram of the search strategy with OBL. The blue dotted circle in the figure represents the neighbourhood range of the search. The circle with a solid line represents the solution obtained by the honey bees. And the pentagram represents the optimal solution. We assume that the distance of each solution to the optimal solution is inversely proportional to the quality of the solution. For the traditional search strategy, the corresponding bee in the solution a searches in the neighbourhood, and a new solution c is generated. As the new solution c is closer to the optimal solution, the new solution c is better than the original solution a. As a result, the solution a is replaced by the solution c. For the search strategy with OBL, the bee corresponding to the solution d searches in its neighbourhood to generate a new solution f. As the new solution f is farther away from the optimal solution, the quality of solution f is worse than the solution d. Therefore, the opposite solution g of the solution d is generated. Due to the solution g is better than the solution d, replace the solution d with the solution g. As can be seen from the figure, the operation of generating the opposite solution is beneficial to jump out of the original search range and avoid falling into the local optimum. Fig. 2Open in figure viewerPowerPoint Search strategy with OBL Define the optimisation problems as follows: (8) where and D is the dimensional size. In the traditional ABC algorithm, in the phase of employed bee, (2) is used to generate a new solution vi, and the fitness value of the new solution is calculated. Then the greedy selection strategy is applied to update the solution. In our proposed methods, if the fitness value of the solution vi is less than or equal to , the opposite solution of the current solution is generated using (9), where is the maximum value of the jth dimension of the current solution, and is the minimum value of the jth dimension of the current solution (9) If the opposite solution is outside the range of the parameters, use (10) to make adjustments (10) After generating the opposite solution, (11) is used to choose between the opposite solution and the current solution (11) The main process is provided in Algorithm 2 (see Fig. 3). Fig. 3Open in figure viewerPowerPoint Algorithm 2: Employed bee search strategy with OBL 3.3 Adaptive decreasing strategy As an important parameter of the ABC algorithm, the weight plays a crucial role in balancing the exploration ability in the early stage and exploitation ability in the later stage. is a random number between [ − 1, 1], and it controls the generation of food sources near xij. As can be seen from (2), as the difference between xij and xkj decreases, the disturbance near the xij position also reduces. The larger is conducive to global search and increase the diversity of the population. The smaller can improve the local exploitation ability of the algorithm. Therefore, the adaptive weight strategy is used to dynamically adjust , which can ensure a better global search and local search ability of the algorithm. In the phase of onlooker bees, the adaptive weight strategy is used to search for new food sources. The way to search for new food sources is as follows: (12) (13) First, determine whether the number of iterations is less than half of the maximum number of iterations. If the number of iterations is less than half of the maximum number of iterations, a new solution is generated using (12). If the number of iterations is greater than or equal to half of the maximum number of iterations, use (13) to generate a new solution. And then, a random number r between [0,1] is generated. r determines the direction of the new solution relative to the current solution. If the random number r is >0.5, generate a new solution in the same direction as the current solution. Otherwise, generate a new solution in the opposite direction to the current solution. 3.4 Proposed OLABC algorithm In order to maintain the diversity of the population in the iterative process and balance the early exploration and later exploitation ability of the algorithm, the idea of OBL and adaptive weight strategy is integrated into the ABC algorithm. By doing so, we can improve the convergence speed of the algorithm and prevent it from falling into local optimum (see Fig. 4). The pseudo-code of the ABC algorithm based on the opposite learning is given in Algorithm 3 (see Fig. 5). Fig. 4Open in figure viewerPowerPoint OLABC algorithm Fig. 5Open in figure viewerPowerPoint Algorithm 3: Procedure of OLABC 4 Experiments and discussion 4.1 Benchmark functions In order to analyse the optimisation performance of the OLABC algorithm, ten commonly used classical functions are selected for testing [[36]]. These functions have different characteristics and can fully reflect the optimisation ability of the algorithm. In this paper, the OLABC algorithm proposed is compared with the traditional ABC algorithm. Table 1 gives the function names, the function definition, the parameter ranges and the theoretical optimal value of the ten test functions. D represents the dimension of the test function solution space. Table 1. Benchmark test functions in experiments Name Function Search space Optimum Sphere [−100, 100]D 0 Rosenbrock [−50, 50]D 0 Griewank [−600, 600]D 0 Rastrigin [−5.12, 5.12]D 0 SumSquare [−10, 10]D 0 SumPower [−1, 1]D 0 Schwefel 2.22 [−10, 10]D 0 Alpine [−10, 10]D 0 Quartic [−1.28, 1.28]D 0 Ackley [−32, 32]D 0 In these ten test functions, Sphere, Rosenbrock, Sumsquare, Schwefel 2.22 and Quartic are unimodal functions that can be used to test the optimisation accuracy and convergence speed of the algorithm. Rastrigin, Griewank, Ackley, Sumpower, and Alpine are non-linear multimodal functions with many local minima that can be used to test the ability of an algorithm to jump out of local extrema and avoid falling into local optimum. 4.2 Experiments settings In the experiment, the population size NP is 40. The number of food sources is equal to half of the population, which means that the SN is 20. The control parameter limit is set to 300. If the food source has not been improved after the ‘limit’ times, it will be abandoned by its employed bee. The maximum number of iterations of the algorithm maxCycle is set to 3000. For each algorithm, experiments are performed with dimensions of 30 and 50, respectively. At the same time, in order to ensure the generality of the algorithm, each group of experiments runs independently 30 times, and calculate optimal value, worst value, mean value and standard deviation. The optimal value represents the function value of the optimal solution obtained by the algorithm in the iterative process. The maximum value represents the function value of the worst solution obtained by the algorithm during the iterative process. The mean and optimal value are used to evaluate the convergence precision of each algorithm. The standard deviation (Std.) can reflect the stability of the algorithm. We can use these values to evaluate the performance of the algorithm comprehensively. 4.3 Effectiveness of the proposed algorithm components Table 2 shows the comparison of the optimisation performance of ABC and OLABC when the dimension is 30. The better results are marked in bold. As can be seen from Table 2, in the case of the dimension of 30, the optimal values and mean obtained by using the OLABC algorithm are superior to the original ABC algorithm for the test functions f1, f3, f5, f6, f7, f8, f9 and f10. For the test function f2, the optimal value and mean obtained by the traditional ABC algorithm are better than OLABC. For the test function f3, the original ABC algorithm and OLABC algorithm converge to the theoretical optimal value under a certain number of iterations. It can be concluded that for most of the test functions, the OLABC algorithm has higher optimisation precision than the original ABC algorithm. Therefore, the optimisation performance of the OLABC algorithm is better. Table 2. Comparison of optimisation performance between ABC and OLABC at D = 30 Function ABC OLABC Optimal Mean Max Min Std. Optimal Mean Max Min Std. f1(x) 4.30 × 10−16 6.01 × 10−16 7.64 × 10−16 4.30 × 10−16 1.05 × 10−16 1.77 × 10−17 1.56 × 10−16 4.33 × 10−16 1.77 × 10−17 9.43 × 10−17 f2(x) 3.78 × 10−3 7.66 × 10−1 6.10 × 101 3.78 × 10−3 1.25 × 101 1.49 × 101 2.78 × 101 8.60 × 101 1.49 × 101 1.13 × 101 f3(x) 4.44 × 10−16 5.51 × 10−16 6.66 × 10−16 4.44 × 10−16 4.59 × 10−17 0.00 × 101 9.62 × 10−17 1.11 × 10−16 0.00 × 101 3.84 × 10−17 f4(x) 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 f5(x) 3.00 × 10−16 5.38 × 10−16 7.50 × 10−16 3.00 × 10−16 1.05 × 10−16 3.16 × 10−17 1.14 × 10−16 2.50 × 10−16 3.16 × 10−17 6.79 × 10−17 f6(x) 5.72 × 10−18 3.33 × 10−17 6.35 × 10−17 5.72 × 10−18 1.47 × 10−17 1.77 × 10−20 1.36 × 10−18 8.72 × 10−18 1.77 × 10−20 2.11 × 10−18 f7(x) 9.95 × 10−16 1.33 × 10−15 1.64 × 10−15 9.95 × 10−16 1.50 × 10−16 6.52 × 10−17 4.00 × 10−16 6.90 × 10−16 6.52 × 10−17 1.56 × 10−16 f8(x) 1.89 × 10−11 4.78 × 10−8 3.31 × 10−7 1.89 × 10−11 9.68 × 10−8 7.73 × 10−17 2.17 × 10−15 2.70 × 10−14 7.73 × 10−17 5.23 × 10−15 f9(x) 5.38 × 10−2 8.97 × 10−2 1.47 × 10−1 5.38 × 10−2 2.31 × 10−2 1.90 × 10−4 5.07 × 10−3 1.08 × 10−2 1.90 × 10−4 2.53 × 10−3 f10(x) 2.80 × 10−14 3.98 × 10−14 4.93 × 10−14 2.80 × 10−14 5.22 × 10−15 3.11 × 10−15 6.66 × 10−15 1.38 × 10−14 3.11 × 10−15 3.09 × 10−15 From the results of the standard deviation, for the test functions f1, f3, f5, f6, f8, f9 and f10, the standard deviation of the OLABC algorithm is smaller than the traditional ABC algorithm. On the test function f2, the standard deviation of OLABC and ABC is equal. On the test function f7, the standard deviation of the ABC algorithm is slightly smaller than OLABC. Overall, it can be concluded that the stability of the OLABC algorithm is higher than that of the ABC algorithm for most of the test functions. When the dimension is increased to 50 dimensions, the search space becomes more complex and more capable of testing the algorithm's ability to search the optimal solution. Table 3 shows the comparison of the optimisation performance of ABC and OLABC when the dimension is 50. The better results are marked in bold. It can be seen from Table 3 that for the test functions f1, f3, f5, f6, f7, f8, f9 and f10, the convergence precision of the OLABC algorithm is better than the traditional ABC algorithm. For f4, when the dimension is 30, both the traditional ABC algorithm and OLABC can converge to the theoretical optimal value, but when the dimension is increased to 50 dimensions, the mean and standard deviation of the ABC algorithm are significantly larger than the OLABC algorithm. It can be concluded that when the dimension increases, the OLABC algorithm can still converge to the optimal value quickly and the stability is better. Thus, the optimisation performance of the OLABC algorithm is significantly higher than the traditional ABC algorithm for most of the test functions. Table 3. Comparison of optimisation performance between ABC and OLABC at D = 50 Function ABC OLABC Optimal Mean Max Min Std Optimal Mean Max Min Std f1(x) 9.11 × 10−16 1.22 × 10−15 1.64 × 10−15 9.11 × 10−16 1.64 × 10−16 6.63 × 10−17 3.87 × 10−16 8.62 × 10−16 6.63 × 10−17 1.67 × 10−16 f2(x) 2.48 × 10−2 1.68 × 101 2.00 × 101 2.48 × 10−2 3.69 × 101 4.42 × 101 4.67 × 101 4.75 × 101 4.42 × 101 7.44 × 10−1 f3(x) 9.99 × 10−16 1.20 × 10−15 1.44 × 10−15 9.99 × 10−16 1.88 × 10−16 1.11 × 10−16 3.85 × 10−16 6.66 × 10−16 1.11 × 10−16 1.86 × 10−16 f4(x) 0.00 × 10 7.30 × 10−14 5.52 × 10−13 0.00 × 101 1.36 × 10−13 0.00 × 10 0.00 × 101 0.00 × 101 0.00 × 101 0.00 × 101 f5(x) 9.31 × 10−16 1.14 × 10−15 1.44 × 10−15 9.31 × 10−16 1.08 × 10−16 8.33 × 10−17 3.58 × 10−16 6.88 × 10−16 8.33 × 10−17 1.45 × 10−16 f6(x) 3.01 × 10−17 1.95 × 10−16 1.06 × 10−15 3.01 × 10−17 2.49 × 10−16 1.69 × 10−19 4.33 × 10−18 1.19 × 10−17 1.69 × 10−19 3.41 × 10−18 f7(x) 2.76 × 10−13 5.87 × 10−13 1.35 × 10−12 2.76 × 10−13 2.70 × 10−13 4.76 × 10−17 1.97 × 10−14 1.26 × 10−13 4.76 × 10−17 3.65 × 10−14 f8(x) 6.97 × 10−7 6.72 × 10−5 5.41 × 10−4 6.97 × 10−7 1.10 × 10−4 4.43 × 10−16 6.11 × 10−4