Error Analysis & The Taylor Series

Vocabulary flashcards covering key terms from the lecture on error analysis and the Taylor series, including definitions of various error metrics, stopping rules, major error types, and fundamental components of the Taylor polynomial.

Error Analysis

The study of discrepancies that arise when numerical methods approximate exact mathematical results.

Absolute Error (Et)

The magnitude of the difference between the true value and an approximation, |True − Approximation|.

True Fractional Relative Error

The ratio of the absolute error to the true value, |Et| / (True value).

Percent True Relative Error

The true fractional relative error expressed as a percentage: (|Et| / True value) × 100%.

Approximate Error (ea)

The difference between successive numerical approximations when the true value is unknown.

Percent Relative Approximate Error

(|Current approximation − Previous approximation| / Current approximation) × 100%.

Stopping Criterion

A rule for terminating an iterative method when ea < es, where es is a preset tolerance.

es (Prespecified Tolerance)

The acceptable percent error used in stopping criteria; calculated by es = 0.5 × 10^(2−n)% for n significant figures.

Significant Figures (n)

Digits that carry meaningful information about a number’s precision; used to set es.

Round-off Error

Error caused by storing or manipulating numbers with a limited number of significant figures in a computer.

Truncation Error

Error that results from approximating an exact mathematical process (e.g., cutting off a series).

Total Numerical Error

Combined effect of round-off and truncation errors in a numerical computation.

Blunder

A gross human mistake, such as mis-typing data, that introduces error into results.

Formulation Error

Error stemming from simplifying or misrepresenting the physical problem in mathematical form.

Data Uncertainty

Error introduced because measured input data cannot be known with perfect precision.

Taylor Series

A polynomial expansion that approximates a smooth function about a reference point x₀ using its derivatives.

Reference Point (x₀)

The point about which a Taylor series is expanded.

Step Size (h)

The distance from x₀ to the point where the function is approximated (x₀ + h).

Order of Approximation (n)

The highest derivative (and power of h) retained in a truncated Taylor series.

Remainder Term (Rn)

The error term that accounts for the difference between the true function and its nth-order Taylor polynomial.