What does truncation error refer to in numerical analysis?

Truncation error in numerical analysis specifically refers to the error that occurs when an infinite series is approximated by a finite number of terms. This type of error arises because certain numerical methods, such as those used in calculus or solving differential equations, rely on representing continuous functions with discrete approximations.

For instance, when using a Taylor series to approximate a function, truncating the series after a finite number of terms introduces truncation error. The higher the number of terms included in the approximation, the smaller the truncation error is likely to be. This is a fundamental concept in numerical methods, underscoring the importance of understanding how approximations impact calculations.

The other options mention different types of errors or concepts in numerical analysis. For example, numerical instability (the first choice) involves errors that arise from the way algorithms handle small changes in input or floating-point representations rather than from truncating series. Comparison of estimated and actual values pertains to rounding or measurement errors, while differences found between theoretical and experimental data primarily relate to experimental inaccuracies rather than the process of approximation itself. Thus, truncation error is distinct from these other categories by focusing specifically on the effects of limiting a series expansion.