Interpolatsioonivea

Interpolatsioonivea, often translated as interpolation error, refers to the difference between the true value of a function and the value obtained through interpolation. When we use an interpolation method, such as linear or polynomial interpolation, we are approximating a function's behavior at points that are not explicitly known. The interpolation error quantifies how well our chosen interpolation method captures the underlying function's true behavior between the known data points.

The magnitude of the interpolation error depends on several factors. The smoothness of the original function plays a significant role; smoother functions generally lead to smaller interpolation errors. The degree of the interpolating polynomial is also crucial. Higher-degree polynomials can often fit the data points more closely, potentially reducing the error, but they can also introduce oscillations, particularly in regions far from the data points, leading to larger errors. The spacing of the data points used for interpolation is another important consideration. Unevenly spaced points can sometimes lead to larger errors compared to evenly spaced points, depending on the function's behavior.

Understanding and estimating interpolation error is vital in many fields, including numerical analysis, computer graphics, and data science. It helps in choosing the most appropriate interpolation method for a given problem and in assessing the reliability of the interpolated results. Various theorems, such as the remainder term of polynomial interpolation, provide bounds for the interpolation error, allowing for a quantitative assessment of its size.