Significance of error analysis in numerical method?

Answer 1

Error analysis is absolutely critical to a successful numerical method. This is because, often, the method involves discrete numerical iteration, such as when solving a problem in integral calculus. Computers have errors in floating point representation, such as truncation and round-off. These errors can accumulate, and actually overwhelm the result.

For example, if your floating point format has 24 bit resolution (which is the size for a typical 32 bit float), adding 1 to 1×1025 will not change the result. If your program involves a loop, it could fail in this case. It is important to add and subtract numbers of comparable magnitude.

Another example is Taylor series, used for generating trignonometric functions such as sin(x). These series are most accurate between -pi/2 and +pi/2. If you were calculating sin(x) for large values of x, you would want to normalize x to be within that range by adding or subtracting 2 pi and then finally pi as needed. Problem is, that, at large values of x, 2 pi might only represent 1 or 2 bits of resolution, and your answer will be way off.

A third example is the solution of 3 equations in 3 unknowns. This represents three planes in 3-D space, which should, if not parallel, resolve to a single point. Error is measurement, and error in floating point representation, could easily (if two of the planes are nearly parallel) result in large error in the result.

The art is in balancing the accumulation of error against the increase in resolution as things become smaller and smaller, such as delta x when doing integration, a fourth example.