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The aim of this section is to study the numerical contour integral methods (NCIMs) for solving the free-boundary PDEs (2)-(6).
In the following, we aim to solve [phi](V) exactly that allows us to avoid using the iterative procedure in  for approximating [[tau].sub.f](V) and optimizing NCIMs.
In this section, we use NCIMs for solving the free-boundary PDEs (2)-(6) under four stochastic volatility processes: GBMP, MRGP, MRSRP, and MRLP.
In the test, we set [V.sub.max] = 4K and T = 2, the space mesh partition number N = 4000 for FDMs and NCIMs. Moreover the FDMs, when time mesh sizes equal T/2000, are taken as the benchmark methods.
This paper studied the NCIMs for solving free-boundary PDEs from American volatility options pricing.
The numerics in Table 5 show that the prices obtained by NCIM and FDM benchmark are very close and the relative errors are less than 0.25% in most cases, whereas the CPU time for NCIM is less than the FDM, and the reasons for NCIM achieving good performance have two aspects: on the one hand, the NCIM has the exponential-order convergence rate with respect to the number of the quadrature nodes L for the hyperbolic contour integral (see formula (83)); on the other hand, there is no iterative procedure for numerical finding the free-boundary value of the corresponding perpetual American volatility put; thus it is not like .
The numerics in Table 6 show that the prices obtained by NCIM and FDM benchmark are very close and the relative errors are less than 0.8% in most cases, whereas the CPU time for NCIM is much less than the FDM.
The numerics in Table 7 show that the prices obtained by NCIM and FDM benchmark are very close and the relative errors are less than 0.8% in most cases, whereas the CPU time for NCIM is less than the FDM.
Therefore, the NCIMS Goat Milk Committee voted in 1991 to allow the maximum somatic cell count to remain at 1 million cells per ml, thus allowing for natural seasonal variation.
In addition to these methods, 27.3 percent of survey respondents reported attending at least one training course provided by FDA and NCIMS. Understanding food safety concerns is imperative for dairy processors (like all food manufacturers) because of the regulations--both public and industry--governing food businesses.
Of those who reported membership to the NCIMS, however, 85.7 percent are processing according to the standards outlined in the Dairy Grade A Voluntary HACCP program, suggesting greater buy-in among members of the organization which helped develop the program.
Many facilities have found this assurance through the voluntary HACCP program developed in collaboration with the NCIMS, and others continue to find success with the standard PMO that has been around for decades.
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