Errors in Computation II

As I recall from one of my textbooks, Introductory Numerical Analysis by Aquino-Ruvivar and Cureg, this is how a prof of mine defined the types of error.

Errors that arises during computation:

Input error - error inherent in the source data. usually occur in experimental data or in the encoding stage or formulation of the mathematical computation.

Round-off error - same definition as above. It occurs because of the limited capacity of the computer.

Truncation error - errors that are due to the specific algorithm or formula that is used in a particular numerical method. Example of it is evaluating a specific number using Taylor Series. It is usually truncated at the first few terms of the series.

The prof also discussed that sizes of the error are actually generated when particular methods are used somewhat differ from the expected amount of errors. It is usually expressed in two forms:

Absolute Error: $|x-\tilde{x}|$

Relative Error:  $\frac{|x-\tilde{x}|}{|x|}$

Relative error, though is the better measure than the absolute error.

Errors in Computations

This is the preliminary in Basic Numerical Methods.

Computer and Errors

There are many types of error. In class, we discussed four.

Round-off error - errors made when rounding off numbers or truncating numbers.

 Example:

Given: 2.54674  
Computer rounding it off to the nearest hundreds: 2.55

Propagation - error usually occur in software programming, a normal error
                     - error within an error, what we call the butterfly effect

Example:    Given:  The machine can take in two numbers to the right of the decimal places.
                              $f(x)=x^2+7x, x=0.12$

Manual Solution:
$ f(0.12)=(0.12)^2+7(0.12)=0.0144+0.84=0.8544$
Machine's Solution:
$f(0.12)=(0.12)^2+7(0.12)=0.014+0.84=0.85$

Underflow and Overflow - too much error in the exponents, if the number is too small or to big.