Two Types of Error
- Understanding error, its nature and sources, is a key part of numerical analysis.
- There are two main types of error we must contend with:
- Round-off (numeric)
- Stability (algorithmic
- They manifest in different ways and for different reasons.
- They can appear individually or in concert with each other.
Working With Error
- However, each type of error can be mitigated through defensive programming techniques.
- Also, we need to be aware that no matter how hard we try, error cannot be eliminated.
- For this reason, numerical analysts include error bound analysis in their work.
Double Precision and Real World Problems
- The use of double precision arithmetic allows us to save some cases of round-off error before they happen.
- Double precision representation is sufficient to provide most numbers we need and solve most problems we have.
Double Precision and Real World Problems
- Most numerical analysis problems represent some form of applied mathematical problem.
- Perhaps it is defining a curve for a wind turbine or estimating
wealth disparity.
- Applied mathematical problems generally involve some type
of measurement of real-world values.
- In practice, our measurement error will be a greater limiting factor for many of the calculations we will perform than precision limitations.
Data and Measurement Bias
- A standard ruler will only have a measurement resolution of 1/16th of
an inch, or so.
- This can be stored as a floating point number without
problems.
- However, with real-world data, each of the individual measurement
errors is a range that is added or subtracted to the given measurement.
- If we are lucky, these errors cancel each other out leaving us with a net-zero error.
- In practice, when measurement error is unbiased, then the error is centered at 0 and appears with an estimable frequency pattern.
Error Growth
- A measuring tool, such as a ruler or some mechanical or electrical device, may be biased, one way or the other.
- Consequently, especially in the realm of error propagation, the defined errors are maximum potential error, not minimum and certainly not estimates of the actual error.
- If we know what the error is (like on a HW problem), we can just adjust our results to accommodate it.
- But in reality, all we have is an upper bound, in most cases.
Upper Bounds for Error
- As we work with numerical analysis, we want to keep an eye on that upper bound of the error.
- We can use that to manage expectations on results.
- Many of the algorithms we work with are focused on minimizing that upper bound.
- When we fit that upper bound within a tolerance we have specified, we know that our answer was good enough, for whatever good enough means at that time.
A Look Ahead
- The rest of this book will introduce a series of algorithms for solving different types of mathematical problems commonly found in computational science.
- Some are general solutions and some are keyed to specific problems.
- Finally, some will be applied to specific real-world problems as we see how all of this connects back to applied mathematics.
