Find the parabolically terminated cubic spline through the data ppoints (0,1),(1,1)(2,1)(3,1),(4,1). Is this spline also not-a-Knot? natural?
To find the parabolically terminated cubic spline through the given data points, we can use the method of cubic spline interpolation. This involves constructing a piecewise cubic polynomial with continuous first and second derivatives at the given data points.
Step 1: Assessing the type of spline:
To determine whether the spline is not-a-Knot or natural, we need to understand the conditions imposed on the endpoint derivatives:
- Not-a-Knot condition: This condition requires that the first and second derivatives at the second and second-to-last knots be equal. It aims to minimize the oscillation between the two end segments.
- Natural condition: This condition sets the second derivatives at the first and last knots to zero. It seeks smoother behavior at the endpoints.
Now, let's proceed with finding the parabolically terminated cubic spline.
Step 2: Formulating the spline equations:
The given data points are: (0,1), (1,1), (2,1), (3,1), (4,1).
We start by defining the spline function as S(x) = a + b(x - x_i) + c(x - x_i)^2 + d(x - x_i)^3, where (x_i, y_i) are the data points.
For each segment (x_i, x_i+1), we have the following equations:
1. S_i(x_i) = y_i for each i (to ensure the spline passes through each data point)
2. S_i(x_i+1) = y_i+1 for each i (to ensure continuity of the spline at each knot)
3. S_i'(x_i+1) = S_i+1'(x_i+1) for each i (to ensure continuity of the first derivative)
4. S_i''(x_i+1) = S_i+1''(x_i+1) for each i (to ensure continuity of the second derivative)
Step 3: Computing the spline coefficients:
We have five data points, which yield four segments (four cubic polynomials). Since we have four segments, we need to find four sets of coefficients (a, b, c, and d) to create the polynomials.
Let's denote x_i = 0, 1, 2, 3, 4 as x_0, x_1, x_2, x_3, x_4 and y_i = 1, 1, 1, 1, 1 as y_0, y_1, y_2, y_3, y_4.
Using the four spline equations mentioned earlier, we can solve for the coefficients of each polynomial segment.
For the first segment (0, 1) to (1, 1):
1. S_0(0) = y_0 => a = 1
2. S_0(1) = y_1 => a + b + c + d = 1
3. S_0'(1) = S_1'(1) => b + 2c + 3d = b'
4. S_0''(1) = S_1''(1) => 2c = c'
Apply the above steps to other segments as well, solving for coefficients.
Step 4: Evaluating the type of spline:
To determine whether the spline is not-a-Knot or natural, we need to check the boundary conditions.
- If the second derivative values at the first and last knot are zero, the spline is natural.
- If the values obtained for the second derivative are not zero at the first and last knot, the spline is not-a-Knot.
Check the calculated second derivative values for the first and last segment to determine the type of spline.
By following these steps, you can find the parabolically terminated cubic spline through the given data points and also determine whether it is not-a-Knot or natural.