What error refers to a loss of arithmetic precision due to rounding or numerical instability?

What error refers to a loss of arithmetic precision due to rounding or numerical instability?

• A. Truncation error
• B. Logical error
• C. Semantic error
• D. Arithmetic error

Answer: (D)

Loss of arithmetic precision due to rounding and numerical instability is called arithmetic error. In floating-point computations, numbers can only be stored with finite precision, so many real numbers cannot be represented exactly. Each arithmetic operation may introduce a small rounding error, and when many operations are chained, these errors can accumulate or be amplified, leading to results that differ from the exact mathematical value. This is different from truncation error, which comes from approximating a process by a finite number of terms or steps, such as using a truncated series or a discretized algorithm; truncation error is about the inherent approximation in the method rather than rounding after each operation. Logical error and semantic error relate to the correctness of the algorithm's logic and meaning, not to numeric precision. Therefore, the described error aligns with arithmetic error.

Loss of arithmetic precision due to rounding and numerical instability is called arithmetic error. In floating-point computations, numbers can only be stored with finite precision, so many real numbers cannot be represented exactly. Each arithmetic operation may introduce a small rounding error, and when many operations are chained, these errors can accumulate or be amplified, leading to results that differ from the exact mathematical value. This is different from truncation error, which comes from approximating a process by a finite number of terms or steps, such as using a truncated series or a discretized algorithm; truncation error is about the inherent approximation in the method rather than rounding after each operation. Logical error and semantic error relate to the correctness of the algorithm's logic and meaning, not to numeric precision. Therefore, the described error aligns with arithmetic error.