Error always plays a major role when you’re dealing with approximations. The numerical analysis provides approximations but accurate results, but these results are not 100% accurate and you don’t need them to completely accurate as well.
Depending upon your usage your degree of error can vary. Also fining very accurate results where you don’t need them is a waste of resources and time. As the level of complication rises, the amount of time required to process a result increases and in certain scenarios, you need quick and accurate answers.
Now, these errors can propagate while rounding off to get the desired number of significant digits. Also, you can use either absolute error or relative error to base your calculation upon. It is usually advised to use relative error especially when you’re dealing with values of high or low scale.
Say take two number 0.000345 and 0.000487. Now the absolute error between them is 0.000142 which does not seem like much. The relative error between them is 41.16% which is quite a lot.
It is said that the sum operation is inexact on calculators or computers. Since error propagates, a+b+c will be even more inexact. This also signifies that If you take a number of the order (-10) then there is a chance that when using the sum operation on such numbers you will get a different answer when calculating using a computer and a calculator. Answers vary even in different models of the calculator.
When you use numerical analysis to find the value of integration then it leads to erroneous answers. Since integration is the sum of infinite trapezoids whereas numerical analysis could only find the sum of a finite number of trapezoids. This specific type of error is known as Truncated error
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