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Here's a video on Numerical Error from a Numerical Methods And Programming class by Prof.P.B.Sunil Kumar, Dept. of Physics, IIT Madras (India)
Notes:
The first part of the lecture talks on the Representation of Numbers(a continuation from past lecture) in our digital computer such as the integers and floating-point numbers.
Difference of Chopping and rounding: Chopping is to merely chop off or cut off higher digits while rounding means that any quantity falling within an interval of length delta x will be represented as the nearest allowable number. Rounding yields lower error than chopping.
The concept of zero The precise meaning of the number zero is lost while computing numerically
Relationship between the exact(true) result and the approximation can be formulated as True value = approximation + error This relationship gives the value of the error but it does not take into account the order of the magnitude of the value under examination
To account the magnitude of the quantities being evaluated, we normalize the error to the true value as in Fractional relative error = Error/true Value (Significance of this thought: An error of a centimeter is much more significant if we are measuring a rivet than a bridge. See the magnitude of the quantity of the item being evaluated)
Fractional relative error multiplied by 100 would give us the Percent relative error
In real applications, we don't know the true value. So an alternate solution is to normalize the error using the approximation itself as in, Ea = (approximate error/approximation)100% the subscript 'a' signifies that the error is normalized to an approximate value. "Ea" is read as "Epsilon sub a"
For iterative procedures, the error is estimated as the difference between the present and previous approximations. Thus the percent relative error is determined according to: Ea = [(present approximation - previous approximation)/present approximation] 100% Generally, one is not concerned with the sign of the error but is interested whether the absolute value is lower than a pre-specified tolerance Es. Computation in the iterative method is repeated until |Ea| < Es
In summary
Two errors encountered are round-off errors(due to finite representation of numbers) and truncation errors(due to the use of an approximation instead of a finite mathematical operation). Rounding errors appears due to arithmetic operations using finite floating point numbers and is a major source of error.
The only way to minimize round-off errors is to increase the number of significant figures of the computer
A decrease in step-size (for iterative) can lead to subtractive cancellation or an increase in computations, thus the truncation errors are decreased as the round-off errors are increased.
This would be your online companion to our ES 84-Numerical Methods class. Soft copies of our lecture notes will be posted here, as well as other resources to be used in our class.
