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Rationale: SLAP is one of the two calibration materials for the isotopic water scale. By consensus the established δ18O value is −55.5‰, although several expert laboratories measure significantly more negative δ18OSLAP values. The real δ18OSLAP value as such does not influence the isotopic water scale; however, knowledge of the size of isotopic scale contraction in stable isotope measurements is vital for second‐order isotopes. This study describes the quantification of δ18OSLAP with respect to δ18OVSMOW. Methods: SLAP‐like water was quantitatively mixed with highly 18O‐enriched water to mimic VSMOW. The 18O concentration was determined using an electron ionization quadrupole mass spectrometer. The isotopic composition of the SLAP‐like and VSMOW‐like waters was measured using an optical spectrometer, alongside original VSMOW and SLAP. Results: This study resulted in a much more negative δ18O value for SLAP than expected. The averaged outcome of seven independent experiments is δ18OSLAP = −56.33 ± 0.03‰. There is a large discrepancy between the actual isotopic measurements of even the most carefully operating isotope laboratories and the true δ18O value. Conclusions: Although this finding as such does not influence the use of the VSMOW–SLAP scale, it raises the intriguing question of what we actually measure with our instruments and why even a fully corrected measurement can be so far off. Our result has consequences for issues like the transfer of δ18O from and to the VPDB scale, various fractionation factors, and Δ17O. The absolute 18O abundance for SLAP was calculated as (1887.98 ± 0.43) × 10−6 based on the absolute 18O abundance of VSMOW and the presented δ18OSLAP in this paper. [ABSTRACT FROM AUTHOR]

Lateral roots, including adventitious roots, are the main component of rapeseed roots with support, absorb, and synthesis functions and their morphological parameters directly affecting the plant's aboveground growth and yield. Root biomass, as a material base for lateral root growth, can be used as a link between plant phenotypes and their physiological processes, as well as to enhance root 3D growth model mechanisms and accuracy. To quantify the relationships between lateral root morphological indices and the corresponding organ biomass for rapeseed, we used two cultivars, NY 22 (conventional) and NZ 1818 (hybrid), and conducted cultivar and fertilizing cylindrical tube experiments during the 2016–2019, with two fertilizer levels, no fertilizer, and 180 kg N ha−1 fertilizer. The lateral root biomass and morphological parameters were determined during the whole growth period. The biomass‐based lateral root morphological parameter models were developed by analyzing the quantitative relationship between the lateral root morphological indices and their corresponding biomass, and the descriptive models were verified with independent experimental data. The results showed that the correlation (r) of simulated and observed values for the lateral root morphological parameters are all greater than 0.9 with significant levels at p < 0.001. The absolute values of the average absolute difference (da) of simulated and observed values for the lateral root length (LLR), lateral root average diameter (ADLR), lateral root surface area (SALR), and lateral root volume (VLR) are −30.408 cm, −0.003 mm, 12.902 cm2, and 0.039 cm3, respectively. The RMSE values are 175.183 cm, 0.010 mm, 59.710 cm2, and 1.513 cm3, respectively. The ratio of da to the average observed values (dap) for the LLR and VLR are all less than 5%, and the ADLR and SALR are all <6%. The models developed in this paper have good performance and reliability for predicting lateral root morphological parameters of rapeseed. The study provides a mechanistic method for linking the rapeseed growth model with the morphological model using corresponding organic biomass and laying a good foundation for establishing a 3D morphological model for rapeseed root system based on biomass. [ABSTRACT FROM AUTHOR]

In this paper, a simple, sensitive, and precise flow injection system was optimized and applied to ascorbic acid determination with a carbon paste electrode modified with the Schiff base copper complex. The modified sensor showed increased sensitivity toward ascorbic acid oxidation with a decrease of oxidation potential by 200 mV. Under optimized flow parameters, working potential, gasket thickness, sample and reaction coil volumes, and flow rate, the developed flow system gave a linear response from 5 to 75 μmol/L of ascorbic acid. The calculated limit of detection was 4.5 μmol/L, calculated as 3σ/s which corresponds to the absolute value of 40 ng for the 50 μL sample coil. The precision for six consecutive injections of 5 and 75 μmol/L ascorbic acid solutions was determined as a relative standard deviation of 5.8 % and 1.5 %, respectively. The flow injection system enabled up to 120 analyses per hour. An optimized flow system was applied to the determination of ascorbic acid in dietary supplements, after simple preparation. [ABSTRACT FROM AUTHOR]

In this paper, the adaptive tracking problem for a class of strictly feedback nonlinear time‐delay systems with output constraints, actuator failures, and input dead zones is studied. First, in the design of controller, the absolute value function and integral term are introduced into the candidate functions to deal with nonlinear factors and distributed time‐varying delays in the system. In addition, an adaptive neural network controller is designed to ensure the stability of the controlled system, and the system tracking error can converge to the origin without violating the constraints. Finally, simulation results verify the feasibility of the proposed control strategy. [ABSTRACT FROM AUTHOR]

In a previous paper, the evolution of certainty measured during a consensus‐based small‐group decision process was shown to oscillate to an equilibrium value for about two‐thirds of the participants in the experiment. Starting from the observation that experimental participants are split into two groups, those for whom the evolution of certainty oscillates and those for whom it does not, in this paper we perform an analysis of this bifurcation with a more accurate model and answer two main questions: what is the meaning of this bifurcation, and is this bifurcation amenable to the approximation method or numerical procedure? Firstly, we have to refine the mathematical model of the evolution of certainty to a function explicitly represented in terms of the model parameters and to verify its robustness to the variation of parameters, both analytically and by computer simulation. Then, using the previous group decision experimental data, and the model proposed in this paper, we run the curve‐fitting software on the experimental data. We also review a series of interpretations of the bifurcated behaviour. We obtain a refined mathematical model and show that the empirical results are not skewed by the initial conditions, when the proposed model is used. Thus, we reveal the analytical and empirical existence of the observed bifurcation. We then propose that sensitivity to the absolute value of certainty and to its rate of change are considered as potential interpretations of this split in behaviour, along with certainty/uncertainty orientation, uncertainty interpretation, and uncertainty/certainty‐related intuition and affect. [ABSTRACT FROM AUTHOR]

The present paper first establishes that an identity involving generalized fractional integrals is proved for differentiable functions by using two parameters. By utilizing this identity, we obtain several parameterized inequalities for the functions whose derivatives in absolute value are convex. Finally, we show that our main inequalities reduce to Ostrowski type inequalities, Simpson type inequalities and trapezoid type inequalities which are proved in earlier published papers. [ABSTRACT FROM AUTHOR]

In this paper, enlightened by the effectiveness of Anderson acceleration (AA), a well‐established technique for accelerating fixed‐point solvers, we first present the Anderson accelerating modulus‐based matrix splitting (AAMS) algorithms for a class of horizontal nonlinear complementarity problems. Then, by introducing the strong semi‐smoothness of the absolute value function, we establish the local convergence theory of the AAMS algorithms. More importantly, we provide the optimal parameter for the AAMS algorithms, which is independent of iteration and applicable to other reduced MS algorithms. Finally, the numerical experiments are conducted to clarify the efficiency and practicability of the AAMS algorithms and its optimal parameter, and the effects of two parameters (both derived from AA) of the AAMS algorithms are analyzed. [ABSTRACT FROM AUTHOR]

Seismicity models are probabilistic forecasts of earthquake rates to support seismic hazard assessment. Physics‐based models allow extrapolating previously unsampled parameter ranges and enable conclusions on underlying tectonic or human‐induced processes. The Coulomb Failure (CF) and the rate‐and‐state (RS) models are two widely used physics‐based seismicity models both assuming pre‐existing populations of faults responding to Coulomb stress changes. The CF model depends on the absolute Coulomb stress and assumes instantaneous triggering if stress exceeds a threshold, while the RS model only depends on stress changes. Both models can predict background earthquake rates and time‐dependent stress effects, but the RS model with its three independent parameters can additionally explain delayed aftershock triggering. This study introduces a modified CF model where the instantaneous triggering is replaced by a mean time‐to‐failure depending on the absolute stress value. For the specific choice of an exponential dependence on stress and a stationary initial seismicity rate, we show that the model leads to identical results as the RS model and reproduces the Omori‐Utsu relation for aftershock decays as well stress‐shadowing effects. Thus, both CF and RS models can be seen as special cases of the new model. However, the new stress response model can also account for subcritical initial stress conditions and alternative functions of the mean time‐to‐failure depending on the problem and fracture mode. Plain Language Summary: One of the most pressing questions in earthquake physics is understanding where and when earthquakes occur and how seismicity is related to stress changes in the Earth's crust. This question is even more important today because humans are increasingly influencing stresses in the Earth by exploiting the subsurface. So far, two classes of physics‐based seismicity models have been used primarily. One assumes instantaneous earthquake occurrence when stress exceeds a threshold, and the other is based on the nucleation of earthquakes according to friction laws determined in the laboratory. Both models are very different in their approaches, have advantages and disadvantages, and are limited in their applicability. In this paper, we introduce a new concept of seismicity models, which is very simple and short to derive and combines the strengths of both previous models, as shown in various applications to human‐related seismicity. The forecasts of both traditional models turn out to be special cases of the new model. Key Points: We introduce a modified Coulomb Failure seismicity model in which a mean time‐to‐failure replaces instantaneous triggeringThe model explains the main features of time‐dependent seismicity, including aftershock activity and stress shadow effectsAs a special case, it includes the rate‐state model solutions but can also handle subcritical stresses and other fracture types [ABSTRACT FROM AUTHOR]

It is known that the Thresholded Lasso (TL), SCAD or MCP correct intrinsic estimation bias of the Lasso. In this paper we propose an alternative method of improving the Lasso for predictive models with general convex loss functions which encompass normal linear models, logistic regression, quantile regression, or support vector machines. For a given penalty we order the absolute values of the Lasso nonzero coefficients and then select the final model from a small nested family by the Generalized Information Criterion. We derive exponential upper bounds on the selection error of the method. These results confirm that, at least for normal linear models, our algorithm seems to be the benchmark for the theory of model selection as it is constructive, computationally efficient and leads to consistent model selection under weak assumptions. Constructivity of the algorithm means that, in contrast to the TL, SCAD or MCP, consistent selection does not rely on the unknown parameters as the cone invertibility factor. Instead, our algorithm only needs the sample size, the number of predictors and an upper bound on the noise parameter. We show in numerical experiments on synthetic and real‐world datasets that an implementation of our algorithm is more accurate than implementations of studied concave regularizations. Our procedure is included in the R package DMRnet and available in the CRAN repository. [ABSTRACT FROM AUTHOR]

This paper investigates the bifurcations of the following difference equations xn+1=axn+byne−xn,yn+1=cyn+dxne−yn,xn+1=ayn+bxne−yn,yn+1=cxn+dyne−xn,where a,b,c, and d are positive constants and the initial conditions x0 and y0 are positive numbers. Psarros, Papaschinopoulos, and Schinas (Math. Methods Appl. Sci., 2016, 39: 5216–5222) presented the semistability of the fixed point (0,0) when one eigenvalue is equal to 1 and the other eigenvalue has absolute value less than 1. In this paper, we consider another case: one eigenvalue is equal to −1. With the aid of the center manifold reduction theorem, we rigorously show that these two systems undergo flip bifurcations or generalized flip bifurcations. Moreover, the stability of the fixed point (0,0) and the existence of period‐two cycles are also given. [ABSTRACT FROM AUTHOR]

This paper presents a robust state estimation (SE) based on weighted least absolute value (WLAV) criterion for power systems having both traditional remote terminal units (RTU) and synchronizing phasor measurement units (PMU). It is known that WLAV‐based SE performs poorly when bad data occurred in leverage measurements. To improve the robustness of WLAV‐based SE, a technique for converting actual measurements to pseudo‐voltage measurements is proposed in this paper. Using these pseudo‐voltage measurements, a measurement model containing no leveraging measurement point can be formulated. The SE problem is formulated as a constrained nonlinear optimization based on WLAV criterion and solved by using a primal‐dual interior point algorithm. Several IEEE test systems with mixed measurements have been used to demonstrate the robustness of the proposed method. The results indicate that the proposed method yields superior SE accuracy even under extremely bad data conditions. © 2020 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

In this paper, a new interpretation and modification of the SWT function in terms of the total damaging energy density are proposed and discussed. The total damaging energy density is the sum of the damaging part of the strain energy density and complementary energy density corresponding to the first quadrant in damaging σD‐εD axes. For cyclic loading with a positive mean stress (σm ≥ 0), the proposed function reduces to the original SWT formulation. For cyclic loading with a negative mean stress (σm < 0), the maximum stress is augmented by 1/3 of absolute value of the mean stress. The proposed approach shows a consistent correlation of the mean stress effects for both positive and negative mean stresses. [ABSTRACT FROM AUTHOR]

An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch. [ABSTRACT FROM AUTHOR]

Given a graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between nu(e) and nv(e) over all edges e = uv of G, where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to u than to v and the number of vertices of G lying closer to v than to u. A tree‐like polyphenyl is a polycyclic aromatic hydrocarbon consisting of benzene rings, whose chemical graph will tend to a tree after contracting each hexagon into a vertex (the tree is called a contracted tree). A polyphenyl chain is a tree‐like polyphenyl whose contracted tree is a path. In this paper, those polyphenyl chains with n benzene rings having the least, the second least, the greatest and the second greatest Mostar indices are determined, respectively. Those tree‐like polyphenyls with n benzene rings having the least and the second least Mostar indices are also identified. What is more, some properties of those tree‐like polyphenyls with n benzene rings having the greatest Mostar index are obtained. At the end we state some further research problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the zero equilibrium and the occurrence of flip bifurcation of the following system of difference equations: xn+1=a1ynb1+yn+c1xnek1−d1xn1+ek1−d1xn,yn+1=a2znb2+zn+c2ynek2−d2yn1+ek2−d2yn,zn+1=a3xnb3+xn+c3znek3−d3zn1+ek3−d3zn,where ai, bi, ci, di, and ki, for i=1, 2, and 3, are real constants and the initial values x0, y0, and z0 are real numbers. We study the stability of this system in the special case when one of the eigenvalues is equal to −1 and the remaining eigenvalues have absolute value less than 1, using the center manifold theory. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical and semi‐analytical solutions of the time‐fractional Cahn–Allen equation, which describes the structure of dynamic for phase separation in Fe‐Cr‐X (X = Mo, Cu) ternary alloys. We apply a modified auxiliary equation method and the Adomian decomposition method to get distinct solutions to our suggested model. These solutions describe the dynamic of the phase separation in iron alloys and use in solidification and nucleation problems. The applications of this method arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. We apply a conformable fractional derivative to this fractional model to convert it into a nonlinear partial differential equation with integer order. We obtain many analytical wave solutions and also apply a semi‐analytical scheme to calculate the absolute value of error. All solutions are verified by using Mathematica software. [ABSTRACT FROM AUTHOR]

Aim: Ecoinformatics offer new opportunity to test islands as biogeographic and ecological models. In this paper we predicted three hypotheses: (1) plot‐based data issuing from vegetation surveys can be used to predict Island Species‐Area Relationships (ISARs) or island similarity; (2) the habitat area is an independent predictor of species richness patterns within island; (3) species richness and composition are more dependent on habitat type than island identity in land‐bridge islands. Area: Tuscan Archipelago, Italy. Methods: We assembled a database of all the vegetation plots available for the archipelago. For the first hypothesis we calculated ISARs, using Arrhenius model, and Beta Diversity, using Jaccard dissimilarity, on both published floras and cumulative plot data. For the second hypothesis, we modelled Habitat Species‐Area Relationships (HSARs), using Arrhenius model. For the third hypothesis, we used additive partitioning of species richness, NMDS and PERMANOVA. Results: Island Species‐Area Relationships based on plot data mirrored those on published floras, but absolute values of c and z parameters were different. Beta diversity based on plot data resembled those of published floras, but was higher. Species richness was significantly related to the habitat area. The total species richness of the archipelago was linked to large scale drivers, such as island identity, while plot species composition was driven by both habitat type and island identity. Conclusions: Data assembled issuing from vegetation surveys are useful to describe biogeographic patterns. Species richness in the archipelago is driven by spatial factors such as the amount of habitats and the differences among islands, while the species composition of local assemblages is largely driven by habitat filters rather than by island identity, as expected in land‐bridge islands. [ABSTRACT FROM AUTHOR]

In this paper, first, we prove the weighted Hermite–Hadamard–Mercer inequalities for convex functions, after we establish some new weighted inequalities connected with the right‐sides of weighted Hermite–Hadamard–Mercer type inequalities for differentiable functions whose derivatives in absolute value at certain powers are convex. The results presented here would provide extensions of those given in earlier works. [ABSTRACT FROM AUTHOR]

Rolled steel plates and sections are often applied in structures in such a way that the principal load direction corresponds with the rolling direction. Examples are beams, arches, or pylons of bridges, supporting beams of ship decks, and the main elements of crane structures. However, some types of structure are subjected to a multiaxial stress state or are loaded with the main load direction perpendicular to rolling. The orientation may influence the mechanical properties. This paper studies the influence of anisotropy observed in the microstructure of rolled C–Mn steels on the tensile properties, Charpy impact values and particularly the fatigue crack growth rates. The influence of anisotropy is determined through tests performed at different orientations with respect to the rolling direction, namely, L‐T, T‐L and T‐S orientations. Samples were taken from structures that were constructed between 25 and 50 years ago from steel grades Fe510C or St52.3 (modern equivalences S355J2 or S355N). The orientation appears to have a statistically relevant influence on Charpy impact value and fatigue crack growth rate. The anisotropy ratio, defined as the ratio between the mechanical property in a certain orientation with that of the L‐T orientation, ranged between 0.30 and 0.53 for Charpy impact values. The anisotropy ratios appear correlated with the absolute Charpy value, with a correlation coefficient of ρ = −0.8. The anisotropy ratios of the crack growth in T‐L and T‐S orientations were 1.19 and 0.43, respectively. Anisotropy ratios for crack growth appear uncorrelated with anisotropy ratios for Charpy impact. The observed anisotropy may partially explain the difference between uniaxial and multiaxial fatigue crack growth as determined by others. [ABSTRACT FROM AUTHOR]

Protective coal layer (PCL) mining is a preferred regional method for preventing coal and gas outbursts; stress‐relief of PCL mining is the prerequisite of pressure‐relief gas drainage. In order to study the stress‐relief effect of PCL mining on protected coal seams and determine the protection scope of PCL mining under different inclinations, this paper uses physical simulations to study the stress‐relief law and deformation characteristics of the strata during upper PCL mining with a gentle inclination (25°), an intermediate inclination (45°), and a steep inclination (65°). The results show that as the coal seam dip increases, the degree of deformation and failure of the overburden strata gradually decreases, the severe subsidence and expansion area of the overburden strata tends to gradually develop in the uphill direction of the working face, and the floor strata underlying the protective seam significantly expand toward the goaf. However, the absolute value of the upward movement of the floor strata is much smaller than that of the roof strata; and the greater the dip angle, the greater the ratio of floor displacement to roof subsidence. The stress‐relief curve of the protected layer is approximately U‐shaped. As the inclination of the coal seam increases, the stress‐relief range decreases gradually. The maximum relief‐stress and the stress release coefficient of the protected layer initially increase and then decrease. The stress concentration coefficient of the protected coal seam initially increases first and then decreases in both uphill and downhill of the working face in the PCL. As the inclination increases, the stress‐relief angle of the downhill boundary initially increases and then decreases, while the stress‐relief angle of the uphill boundary decreases gradually. Our research results provide guidance for the delimitation of the protected scope of PCLs with different dip angles. [ABSTRACT FROM AUTHOR]

The global food system is a major energy user and a relevant contributor to climate change. To date, the literature on the energy profile of food systems addresses individual countries and/or food products, and therefore a comparable assessment across regions is still missing. This paper uses a global multi‐regional environmentally extended input–output database in combination with newly constructed net energy‐use accounts to provide a production and consumption‐based stock‐take of energy use in the food system across different world regions for the period 2000–2015. Overall, the ratio between energy use in the food system and the economy is slowly decreasing. Likewise, the absolute values point toward a relative decoupling between energy use and food production, as well as to relevant differences in energy types, users, and consumption patterns across world regions. The use of (inefficient) traditional biomass for cooking substantially reduces the expected gap between per capita figures in high‐ and low‐income countries. The variety of energy profiles and the higher exposure to energy security issues compared to the total economy in some regions suggests that interventions in the system should consider the geographical context. Reducing energy use and decarbonizing the supply chains of food products will require a combination of technological measures and behavioral changes in consumption patterns. Interventions should consider the effects beyond the direct effects on energy use, because changing production and consumption patterns in the food system can lead to positive spillovers in the social and environmental dimensions outlined in the Sustainable Development Goals. [ABSTRACT FROM AUTHOR]

Intelligent wells equipped with the interval control valve (ICV), which have the potential applications to significantly improve oil production or control the water production of wells and fields, are more and more widely utilized in the oilfields. The radial metal seal of an ICV is intended to isolate the flow of produced fluids between the casing annulus and the well tubing. The contact mechanical behaviors of the radial metal seal are essential to the structural design and reliability of ICVs in the intelligent well. In this paper, the two‐dimensional axisymmetric assembly of the radial metal seal and the closure member is abstracted as a combination of interference fit model and cantilever beam model. Based on that, the related theoretical model, between the radial metal seal's contact stress and its independent variables, has been derived. Next, numerically simulations of contact stress between the radial metal seal and the closure member for different independent variables are conducted by the finite element method (FEM) to validate the theoretical model. It is found that the mean absolute value of relative error between the numerical and theoretical results of the contact stress is approximately 10%. Additionally, the structural parameters of the radial metal seal are proposed with the absolute value of relative error <10%. Furthermore, the value of the contact stress based on the cantilever beam model can be used to calculate an approximation theoretical solution. In summary, the proposed theoretical model provides a foundation for the design, improvement, and selection of radial metal seal under the known oilfield conditions. [ABSTRACT FROM AUTHOR]

The well‐known Siegel Lemma gives an upper bound cUm/(n−m) for the size of the smallest non‐zero integral solution of a linear system of m⩾1 equations in n>m unknowns whose coefficients are integers of absolute value at most U⩾1; here c=c(m,n)⩾1. In this paper, we show that a better upper bound Um/(n−m)/B is relatively rare for large B⩾1; for example, there are θ=θ(m,n)>0 and c′=c′(m,n) such that this happens for at most c′Umn/Bθ out of the roughly (2U)mn possible such systems. [ABSTRACT FROM AUTHOR]

In this Full Paper, mistakes were found in Figure 1, Figure 5, and in the caption of Figure S14 in the Supporting Information. The corrected caption fo Figure S14 reads: "(a) g-Factor value of BSDA/NaYF4 UCNPs hybrid film upon irradiation with (i) right-handed, (ii) left-handed 980 nm CPL or (iii) unpolarized 980 nm laser respectively. (c) UV/Vis absorption spectra and the UCL spectrum of the NaYF4 nanoparticles in CHCl3 upon irradiation with a 980 nm laser. [Extracted from the article]

Summary: Heavily damped quadratic eigenvalue problem (QEP) is a special type of QEPs. It has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize the original problem via linearizations. Previous work on the accuracy of eigenpairs of not heavily damped QEP focuses on analyzing the backward error of eigenpairs relative to linearizations. The objective of this paper is to explain why different linearizations lead to different errors when computing small and large eigenpairs. To obtain this goal, we bound the backward error of eigenpairs relative to the linearization methods. Using these bounds, we build upper bounds of growth factors for the backward error. We present results of numerical experiments that support the predictions of the proposed methods. [ABSTRACT FROM AUTHOR]