Question

We saw in class that adding two numbers of similar magnitude and opposing sign has the potential to cause large relative errors under floating-point arithmetic. This effect is commonly known as "cancellation."

You can proceed to address the effects of cancellation and explain or provide examples related to this phenomenon under floating-point arithmetic to earn the allocated marks.

Answer 1

Cancellation is the significant loss of precision that occurs when adding or subtracting two floating-point numbers of similar magnitude but opposite signs. It is important to retain as many significant figures as possible during calculations and to round only the final result to minimize errors.

Step-by-step explanation:

Understanding Cancellation in Floating-Point Arithmetic

Cancellation refers to the loss of precision that occurs when two numbers of similar magnitude but opposite signs are added together in floating-point arithmetic. This phenomenon can lead to large relative errors due to the way floating-point numbers are represented in computers. For example, consider the floating-point subtraction 5098 - (5.000 × 1010). If we process this calculation without rounding until the very end, we will retain all significant figures and the difference results in 48. However, rounding each number to two significant figures during the calculation will change the expression to 5100 - (5000 × 1000), which equals 100, a result that deviates significantly from the correct answer.

To minimize the impact of cancellation, it is essential to keep as many significant figures as possible during intermediate calculations and only round off the final answer to the required number of significant figures. In addition, checking that the final answer is reasonable helps to ensure that errors have not been compounded throughout the calculation process. By understanding and addressing the effects of cancellation, one can achieve more accurate results in computational tasks involving floating-point arithmetic.