The phrase "and the second approximation" is correct and usable in written English.
It can be used in contexts involving mathematics, science, or any field where approximations are discussed, particularly when referring to a specific level of precision.
Example: "In our calculations, we first used the first approximation, and the second approximation provided a more accurate result."
Alternatives: "and the subsequent estimate" or "and the next approximation".
In the sequential testing, with smaller stagewise sample size and type I error rate, the first approximation would result in unsatisfactory inflation of the type I error rate, and the second approximation could lead to excessive conservatism.
In Figure 2 we show the absolute error between the exact solution (2) and the second approximation (16) for different values of b.
In Figure 1, we show the surface of the exact solution (2) and the second approximation (16) for (alpha=0.7) in (b) at (a=1, b=0.01) and (c=1).
In the same manner, we can obtain further approximations (v_{3} x,t), ldots) Setting (beta=1) in the first approximation (24) and the second approximation (25) and after some simplifications, we obtain the same approximations as (15) and (16) on returning to the original variable.
where the first and the second approximations are obvious while the third one holds provided that M is sufficiently large and K is sufficiently small (e.g, for M=1024 and K=4, 4 log(M +4 log(K +2=50 while 4 log(M +4K−8=48).
The results show that the accuracy of the developed method can be considered suitable because the average percentage error, in comparison with the exact model, is in the order of 4% for the first approximation and 5% for the second approximation.
Figure 4 shows the surface of the exact solution (43) in (a) and the surface of the second approximation (34) for (alpha=0.7) and (beta=0.5) in (b) at (a=1), (b=1) and (c=1).
The second approximation is.
We design new approximation algorithms or new polynomial time approximation schemes (PTAS) for those problems and establish the first approximation hardness results for some of them.
Here we assumed a weakly scattering medium initially and, therefore, the first approximation to the transport coefficient was estimated without solving the Helmholz equation.
