Chapter 2 ERROR ANALYSIS

2.1.1 Definition of Error The arithmetic performed by a calculator or computer differs from the arithmetic that we use in our algebra and calculus courses Traditional mathematical world: numbers with an infinite number of nonperiodic digits Computational world: each representable number has only a fixed, finite number of digits. Those numbers which do not have a finite-digit representation, is given with an approximate representation within the machine, sufficiently close to the original number

A computing machine can only understand discrete values, so we always have to regard any continuous quantity as discrete one when calculating numerically. Example: we cannot find exact value of x with x2 = 2 , we can only determine the approximate value of it: x = 1.414…  approximation always differs from the real (or true) value.

The difference between the actual value and approximated one is called an error. If X is a real value, x is approximation, then the error e[x] is the function of x: Absolute error: the absolute value of difference between real and approximated values |e[x]| = |x-X| Relative error is provided that X≠0

The absolute value of error is When this inequality is satisfied, ε[x] is called error limit  The absolute value of relative error If this inequality is satisfied, er[x] is known here as relative error limit. If ε/x is small enough, then we can impose and relative error limit is expressed as

Figure 2.1 The range of real value

Accuracy expresses the exactness of conforming approximated value with real one Example: if actual value is 1.2345 and approximation is 1.2346, then the accuracy is of fourth order. Accuracy p is defined as The decimal accuracy can be calculated as log10 p

2.1.2Typical type of errors in numerical procedure A process of numerical modeling is usually carried out in the order represented on the figure below with errors occurring on every step. Physical development -> Mathematical model expression (model error) -> Computational algorithm (approximation error) -> Data input process (input error) -> Computation execution (computational error) - > Computational results output (output error)

Model error (equation error): occurs when formulating mathematical model, trying to compose the model more naturally and conveniently (for example, we can handle model linearly even if it is non-linear model) Approximation error: a result of that all the numerical models are usually calculated through approximate expressions Truncation error: error involved in using a truncated, or finite, summation to approximate the sum of an infinite series Input error: when inputting observed data, observation error occurs. Also, to express input data through finite number, approximation error occurs. These become input error in calculation process. Calculation error: occurs in computational procedure - rounding, terms elimination, error transmission, etc.

Round-off errors occurs when a calculator or computer is used to perform real-number calculations; results from replacing a number with it floating-point form Typical computer: only a relatively small subset of the real number system is used for the representation of all the real numbers This subset contains only rational numbers, both positive and negative, and stores a fractional part, called the mantissa, together with an exponential part, called the characteristic Underflow occurs when numbers have too small magnitude (of less than 16-65) – often set to zero Overflow occurs when numbers have too big magnitude (greater than 1663) – cause the computations to halt

(1) Model Error Ex: y = x2+1 and y = sin x Problem: to find y at the given x point. First equation is easy to solve by standard algebraic methods Second one should be expanded in Taylor series: Here n is infinitely large, but we need to limit it by finite number to use in practice.