1. **Stating the problem:** Define error, source of error, and type of error in numerical analysis. 2. **Definition of Error:** In numerical analysis, an error is the difference between the exact mathematical value and the approximate numerical value obtained by a computational method. 3. **Formula for Error:** $$\text{Error} = \text{Exact value} - \text{Approximate value}$$ 4. **Sources of Error:** - **Round-off error:** Caused by the finite precision of computer arithmetic. - **Truncation error:** Occurs when an infinite process is approximated by a finite one, such as truncating a Taylor series. - **Modeling error:** Arises when the mathematical model does not perfectly represent the real-world problem. 5. **Types of Error:** - **Absolute error:** The absolute difference between the exact and approximate values. $$\text{Absolute error} = |\text{Exact value} - \text{Approximate value}|$$ - **Relative error:** The absolute error divided by the magnitude of the exact value, giving a measure of error relative to the size of the exact value. $$\text{Relative error} = \frac{|\text{Exact value} - \text{Approximate value}|}{|\text{Exact value}|}$$ 6. **Explanation:** Errors are inevitable in numerical computations due to limitations in representation and approximation. Understanding the source and type of error helps in improving algorithms and estimating the accuracy of results. Final answer: Error is the difference between exact and approximate values, sources include round-off, truncation, and modeling errors, and types include absolute and relative errors.