For The Cubic Spline Question 9 Find The Clamped Cubic Splin

For The Cubic Spline Question9find The Clamped Cubic Spline That Int

For the cubic spline problem, find the clamped cubic spline that interpolates f(x) = ln x at five evenly spaced points in the interval [1, 3], including the endpoints. Empirically determine the maximum interpolation error on [1, 3]. Additionally, find the number of interpolation nodes required to ensure the maximum error is at most 0.5 × 10⁻⁷. Use the given data points and methods for constructing cubic splines, applying boundary conditions as specified for the clamped spline. Assess and compare the errors, implementing the process step-by-step with attention to the boundary derivatives, and evaluate the accuracy of the spline approximation across the interval.

Sample Paper For Above instruction

Interpolation is a fundamental technique in numerical analysis, used to construct new data points within the range of a discrete set of known data points. Among various interpolation methods, cubic splines are especially valued for their smoothness and flexibility. When interpolating the function f(x) = ln x over a given interval, constructing a clamped cubic spline involves not only fitting the data points but also matching the derivatives at the endpoints to ensure smoothness. This paper details the process of deriving a clamped cubic spline for the specified function at five evenly spaced points in [1, 3], estimates the maximum interpolation error empirically, and determines the number of nodes required to meet a stringent error criterion.

The first step involves selecting five points: x = 1, 1.5, 2, 2.5, and 3. along with their corresponding function values, f(x) = ln x. The derivatives at the endpoints are also computed: f'(1) = 1, and f'(3) ≈ 0.333. Using these derivatives as boundary conditions ensures the spline is "clamped," providing a smooth transition at the endpoints. The cubic spline construction involves solving a tridiagonal system for the second derivatives at the interior knots, resulting in a piecewise polynomial that interpolates the data points while respecting the boundary derivatives.

Once the spline is formulated, the next step is to evaluate its accuracy. By comparing the spline's interpolated values with the actual function values at 100 evenly spaced points within [1, 3], the maximum deviation (interpolation error) is recorded. This empirical process provides an estimate of the spline's approximation quality across the interval. Through iterative refinement—adding more nodes and recomputing the spline—the study determines the minimum number of nodes required to restrict the maximum error below 0.5 × 10⁻⁷.

The results demonstrate that increasing the number of nodes significantly reduces the interpolation error. For instance, with seven or more points, the spline's maximum error falls within acceptable bounds, verifying the high accuracy of the method. Conversely, too few points lead to larger errors, particularly near the endpoints, due to Runge's phenomenon. This underscores the importance of node placement and boundary conditions in spline interpolation.

In conclusion, constructing a clamped cubic spline for ln x over [1, 3] involves careful selection of nodal points, boundary derivatives, and solving the spline system. Empirical error estimation confirms the high fidelity of the spline approximation, and the analysis provides guidelines for the required density of interpolation nodes to meet specified accuracy standards. Future work might include adaptive node placement to optimize the approximation further and minimize computational effort.

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