Errors arise from approximations such as:
- Modeling due to problem simplification.
- During actual measurement
- As a result of previous computation in an iterative solution.
Data error and computation error
Total error=computational error + propagated data error.
Computational error is the difference between the method chosen, and that method that would give a correct result.
Data error is the difference between the current and exact input to a computational method.
Choice of the method/algorithm doesnt affect the propagated data error.
More on computation error
Can be divided further into:
- Rounding errors
- Truncation errors.
One or the other is the dominant during computation.
The difference is that data error affects the data result as it cant be exactly represented(Try representing 1/3 as a decimal) while the truncation error affects the method used. To understand truncation error think of cleaning the house. You don’t have to clean the entire house but you just need to clean most parts of the house for the house to be regarded as “clean enough”. You have thus truncated cleaning.
Rounding error affects the data and is the error that is as a result of finite precision thus the data can’t be exactly represented during computation.
Truncation error affects the computation method. Since most iterative algorithms are convergent on a well formed problem, the solution is stopped when the result is below a given error threshold. Thus the computation has been truncated.
Common error definitions
True error(Et)=True value-approximate value
It is difficult to ever know the true value thus in computation, the the true value is taken as the result of an analytical method(Pen and paper work out) while the approximate value is the final result using a numerical method.
True relative error=(True error)/True value
Approximate error(Ea)=Current approximation-previous approximation
Approximate relative error=Approximate error/current approximation