29-05-2013, 12:18 PM
Types of Errors in Numerical Analysis
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In the world of math, the practice of numerical analysis is well known for focusing on algorithms as they are used to solve issues in continuous math. The practice is familiar territory for engineers and those who work with physical science, but it is beginning to expand further into liberal arts areas as well. This can be seen in astrology, stock portfolio analysis, data analysis and medicine. Part of the application of numerical analysis involves the use of errors. Specific errors are sought out and applied to arrive at mathematical conclusions.
The Round-Off Error
o The round-off error is used because it a representation of every number as a real number is not possible. So rounding is introduced adjust for this situation. A round-off error, represents the numerical amount between what a figure actually is versus its closest real number value, depending on how the round is applied. For instance, rounding to the nearest whole number means you round up or down to what is the closest whole figure. So if your result is 3.31 then you would round to 3. Rounding the highest amount would be a bit different. In this approach, if your figure is 3.31, your rounding would be to 4. In terms of numerical analysis the round-off error is an attempt to identify what the rounding distance is when it comes up in algorithms. It's also known as a quantization error.
The Truncation Error
o A truncation error occurs when approximation is involved in numerical analysis. The error factor is related to how much the approximate value is a variance from the actual value in a formula or math result. For example, take the formula of 3 times 3 plus 4. The calculation equals 28. Now, break it down and the root is close to 1.99. The truncation error value is equal to 0.01.
Errors
Infinite processes must be approximated in the real world by finite ones. Think of the problem of representing the square root of 2. If accuracy is not an issue, 1 might do. But 1.4 will be better. Better still is 1.41 and yet more accurate is 1.414. We might continue the process indefinitely, each time getting closer to the real value of the square root of 2. But there isn't time enough nor memory in all the computers in the universe to get it exactly. So we stop somewhere and decide that some representation, say, 1.414213562373 is close enough to the real value to be what we mean by the square root of 2. Of course, that means there will be some error inherent in our calculations. The error resulting from stopping an infinite process at some point is called "truncation error" and controlling and understanding it is a central issue in numerical analysis.
Numerical Stability or Algorithmic Errors
If an error stays at one point in an algorithm and doesn't aggregate further as the calculation continues, then it is considered a numerically stable error. This happens when the error causes only a very small variation in the formula result. If the opposite occurs, and the error propagates bigger as the calculation continues, then it is considered numerically unstable.
Conclusion
Math errors, unlike the inference of their name, come in useful in statistics, computer programming, advanced mathematics and much more. The error evaluation provides significantly useful information, especially when probability is required.