Why can rounding errors affect numerical solutions found by computers?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In numerical analysis, computers rely on finite precision arithmetic to perform calculations. Because computers cannot represent all real numbers with infinite precision, they must approximate values, which introduces rounding errors.1

In the context of linear algebra, these errors are particularly significant when solving large systems of equations. Because these problems often involve a vast number of individual arithmetic operations, small rounding errors can accumulate throughout the computation.1

Furthermore, the impact of these errors is closely tied to the stability of the problem. If a problem is ill-conditioned, it is inherently sensitive to small changes in data or parameters. When such a problem is processed using finite precision, the rounding errors act as perturbations that can lead to significantly inaccurate results. Consequently, numerical analysts must carefully evaluate the stability of both the original problem and the chosen algorithm—such as Gaussian elimination—to ensure that the final solution remains reliable.1