The Relative Error of Calculation

Fig 8 The Relative Error of Calculation of < n by Approximate Methods as a Function of 0O and n (a) Zinn Method (b) Friedman Method (c) Boddington Method ... 

Table 5.1 shows the relative error of the single-point method compared to the multipoint method as a function of P/Pq and C as calculated from equation (5.6). 

If A is plotted versus 7 two branches occur. For 7 > Vi, A is a measure for the relative error of I-7. For 0<7SVi, Aisa measure for the relative error of 7. Since for 7 = Vi the values of I-7 and 7 are equal, the two branches meet for 7 = Vi. Thus if A is plotted versus 7 the resultant line will also be continuous for 7 = Vi. However, in general the line will be nondifferentiable for 7 = Vi. This is illustrated in Figure 6.14, where A is plotted versus 7 for first-order kinetics in a slab. Here, 7 was calculated with the formulae given in Tables 6.1 and 6.2, and 7 was calculated with either approximation 6.54 (the extended line),... 

Consider next the an estimation of computational errors for the recollision probability method. The relative error in calculation of the rate constant for an error e in calculation of the recollision probability (A ) is... 

Crystallinity index was calculated based on data of X-ray diffractometry [14], The degree of polymerization was determined by the viscosity measurements from cellulose cadoxen solutions (the relative error of the method was 5 %) [15]. 

The microactivity test (MAT) based on the ASTM-D-3907 [11] standard was used to determine activity and product selectivity of catalysts. MAT runs were performed in a X)Uel automated equipment with 4.0 g of catalyst using the same VGO as in the pilot plant runs. Unless otherwise specified catalyst samples were previously calcined at 853 K for three hours. Operating conditions were 793 K, CTO ratio of 4, 75 s injection time and WHSV of 15.7 h. Product analysis and conversion and selectivity calculations were done as in the pilot plant. The relative error of data was 5%. We analyzed coke bum products in-situ by IR analysis using an HORIBA VIA-510 analyzer. Product distribution was expressed in terms of produet yield/ activity ratio as defined in Table 2 as currently used for interpreting MAT numbers in equilibrium catalysts. 

Best-fit can be evaluated in a straightforward manner by fitting several regression methods to the data generated during the development phase. The relative error of the back-calculated standard calibrators should be calculated. As expected. 

The standard curve should be monitored during in-study validation with at least one set of calibrators per patch run. As for prestudy validation, the curve should be constructed from six concentrations in duplicate. Anchor points may be used. The final number of points used for curve lit must be either 75% of the total number or a minimum of six calibrator samples not including the anchor points. The relative error of the back-calculated samples should be <20% (<25% at the LLOQ). If either the high or low calibrator standards have to be deleted, the range for this particular run must be limited to the next standard point. Samples out of range must be repeated. 

The true mass of a glass bead is 0.1026 g. A student takes four measurements of the mass of the bead on an analytical balance and obtains the following results 0.1021 g, 0.1025 g, 0.1019 g, and 0.1023 g. Calculate the mean, the average deviation, the standard deviation, the percentage relative standard deviation, the absolute error of the mean, and the relative error of the mean. 

In Formula (12) and (13), the right of the equation is the nonlinear expression of the node pressure square, and thus the iterative approach should be used in the solution. The iterative method is that assume PJ = cP/ and calculate the number of the square root. The quasi linear equation system and a set of solutions can be obtained, according to which the number of the square root can be calculated, then the quasi linear equation system is solved again and a set of new solutions can be got. Do this recursion until the relative error of the two calculative solutions is less than a small value. 

The relative error of the calculations by these equations is found in the range of the accuracy of initial data for the proton (5 1%). 

Figure 16.5 shows a Kramers-Kronigvahdation for a typical impedance spectrum calculated with the KK Test for Windows software [12, 13]. For most of the spectrum, the relative errors of both real and imaginary data are <0.4%, confirming... 

In the above ERRj is some desired maximum relative error in the solution (sueh as l.e-5), is the old step size used in evaluation die relative error and Kew estimated step size needed to give a relative error equal to the desired maximum error of ERR. In the last equation it can be seen that if die estimated relative error is larger dian die desired maximum error the new step size will be appropriately smaller than the old step size. In this maimer the local step size can be adjusted as the solution progresses to achieve a desired relative error criterion. It should be noted that the power of 2 and 0.5 in the above equations is based upon the TP algorithm. For use with the RK algorithm die appropriate values would be 4 and 0.25 respectively since the error varies as the fourth power of the step size. For multiple variables widi several differential equations, a relative error can be associated widi each variable in die same form as die above equations. In such a case the criterion should use the largest calculated relative error among all the variables at each time step so that the relative error of all variables will satisfy some desired criterion. 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

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