Mastering Precision for Functions

We’ve previously spoken about the concepts of accuracy and precision and how they play a key role in determining the error range in mechanical engineering calculations. The main recommendations were to use the most specific input data possible and to note the accuracy, significant figures and precision used in the input data.

When dealing with functions and equations, such operations can prove exhausting and pricey to perform; for a given function y=f(x), the resulting significant figures will not only depend on the precision of x and its significant figures but also the behavior of f, the behavior being all the operations (summation and multiplication) and the smaller functions (tan, root, etc.) within its formulae. Needless to say, this can get very complex.

There are two approaches to simplify this process:

  1. Use as much precision as possible, then round up the result to answer the required precision.
  2. Know the significant figures and precision required for the result and compute the data accordingly.

The first approach will be exhausting, as well as an uneconomical use of available computing resources. The second one will require a study of its own to know exactly how the function’s behavior will alter the input’s precision as it goes through it.

There are some classic tricks that help solve this dilemma:

  • The precision can be loosely foreseen through f’(x) and f’’(x): If f’(x) is close to 0, there is usually a gain of precision and significant figures. The opposite case is true as well: if f’(x) and f’’(x) are relatively large, there will be a loss of precision and significant figures.
  • Significant figures are often lost by summation. So data involved in summation or subtraction should be given extra precision.
  • The precision of y should be approached by an interval [y1, y2] such asy1≤y≤ y2, so that calculating x within a range [x1, x2] can give a set of values that will enable an acceptable accuracy of the value of y.
  • Rearranging formulas, such as in calculus for limits evaluation, can help secure the significant figures.

To finish, let’s remember what renowned civil engineering professor Henry Petroski said about the error range:

“I emphasize that virtually every engineering calculation is ultimately a failure calculation, because without a failure criterion against which to measure the calculated result, it is a meaningless number.”

Therefore, the main point is to be aware you have a failure range and you have a proper idea how to assess it. This is the third and final installment of our articles’ series over accuracy and precision



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