# What is a Rounding Error or a Tolerable Truncation Error?

This article explains what is a rounding or tolerable truncation error and how it affects life cycle assessment processes.

Occasionally, mistakes can lead to the mislabeling of a rounding error (the consequence of using finite-precision floating-point numbers in computers) as a truncation error, especially when the number is rounded by cutting. This is not the correct use of the truncation error; however, calling it by truncating a number may be acceptable. In life cycle assessment (LCA) processes, truncation errors occur when certain flows are cut off during boundary selection. The degree of cut-off varies from case to case in ongoing studies, making it difficult to compare different alternatives.

Hybrid analysis of consequential life cycle assessment (ACL) has a wider system boundary than process ACL, as it includes previously missing flows. It is often used to calibrate truncation errors in LCA processes (Mattila et al.). Truncation errors are not the only source of errors in numerical resolution of initial value problems using difference formats, as there is also a rounding error at each step of the calculation. The calculation of the initial value problem is carried out layer by layer.

Although the rounding error for each layer is very small, the sum of the rounding error accumulates and propagates as the number of layers increases, which can be very large. As this cumulative error grows larger and larger, the actual solution may be “submerged” by the cumulative error, resulting in instability of the numerical results. Thus, it is very important to analyze the stability of difference formats. Errors may also arise during the process of implementing the numerical method; these are known as procedural errors and are divided into two categories: rounding errors and truncation errors.

Truncation error is defined as the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation. For each increment (k+th (next)), the step size is usually estimated according to the truncation error in the kth (current) iteration. After integration, truncation error can be estimated at the (k+) th iteration, allowing for a decision to be made on whether to accept or reject the step size. For example, glycerin fractionation for PSA production has an estimated truncation error much larger than Cap-xylene production.

In Table 1, truncation errors estimated from upstream loss vary depending on hybridization methods and technologies used. However, care must be taken that rounding errors do not offset any reduction in truncation errors due to an increase in arithmetic operations. The “truncation error T (x, h) associated with the “corrective stage” of the “Milne-Hamming” method is given by. The truncation error can be reduced by using a better numerical model that usually increases the number of arithmetic operations.

The total numerical error in a process can be calculated as the sum of rounding errors and truncation errors in that process. Therefore, including tolerance as a constraint on the size of truncation error is ineffective, since each ODE would require a different value and it is difficult to define a minimum step size uniquely due to different units. The global truncation error is an accumulation of local truncation errors in all iterations, assuming perfect knowledge of true solution in initial time step. When a numerical process is truncated after a finite number of iterations for simplification purposes, a truncation error occurs. The “truncation error T (x, h) associated with the “predictor stage” of the “Milne—Hamming” method is given by. Therefore, including tolerance as a constraint on size of truncation error is ineffective since each ODE would require different value and minimum step size difficult to define uniquely due to problem of different units.

##### Charlotte Wilson

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