Problem 3 - An Introduction to Numerical Analysis

Question

Suppose that the natural cubic spline \(s_{2}\) interpolates the function \(f: x \mapsto x^{3}\) on the interval \([0,1]\), the knots being equally spaced, so that \(x_{i}=i h, i=0,1, \ldots, m\), with \(h=1 / m, m \geq 2\). Write down the equations which determine the quantities \(\sigma_{i} .\) If the two additional conditions are \(\sigma_{0}=\sigma_{m}=0\), show that these equations are not satisfied by \(\sigma_{i}=f^{\prime \prime}\left(x_{i}\right), i=1, \ldots, m-1\), so that \(s_{2}\) and \(f\) are not identical. If, however, these two additional conditions are replaced by \(\sigma_{0}=f^{\prime \prime}(0), \sigma_{m}=f^{\prime \prime}(1)\), show that \(\sigma_{i}=f^{\prime \prime}\left(x_{i}\right), i=0,1, \ldots, m\), and deduce that \(s_{2}\) and \(f\) are identical.

Step-by-Step Solution

Answer

Using \(\sigma_0 = f''(0)\) and \(\sigma_m = f''(1)\), \(s_2\) and \(f(x) = x^3\) are identical.

Step 1: Understand the Problem

We need to determine the cubic spline that interpolates the function \(f(x) = x^3\) on \([0,1]\) with equally spaced knots \(x_i = ih\), where \(h = \frac{1}{m}\). Our task includes writing the equations that determine \(\sigma_i\) for the spline and proving certain conditions for \(\sigma\).

Step 2: Setup the Cubic Spline Equations

For a natural cubic spline on \([a, b]\), the second derivatives \(\sigma_i\) at the internal knots must satisfy: \[ 2(\sigma_{i-1} + 4\sigma_i + \sigma_{i+1}) = \frac{6}{h^2}(f_i - 2f_{i+1} + f_{i+2}) \] for each interior knot \(i=1,2,...,m-1\). Here, \(f_i = f(x_i)\) and \(f_{i+2} = (ih)^3\).

Step 3: Establish Boundary Conditions

We consider two scenarios for boundary conditions. First, we use \(\sigma_0 = \sigma_m = 0\). Then check if \(\sigma_i = f''(x_i)\). The second scenario uses \(\sigma_0 = f''(0)\) and \(\sigma_m = f''(1)\). We then verify if these conditions allow \(\sigma_i = f''(x_i)\).

Step 4: Verify for \(\sigma_0 = \sigma_m = 0\)

Calculate \(f''(x) = 6x\). Thus, \(f''(0) = 0\) and \(f''(1) = 6\), thus for all interior \(\sigma_i = 6x_i\), which for natural boundary conditions should be zero at both ends but zero only at \(\sigma_0\). Hence, \(\sigma_i eq 6x_i\), failing the requirement of a natural spline.

Step 5: Verify for \(\sigma_0 = f''(0), \sigma_m = f''(1)\)

Take \(\sigma_0 = f''(0) = 0\), \(\sigma_m = f''(1) = 6\). These satisfy all \(\sigma_i = 6x_i\) since derived from kernel 0 to the known output \(f''(1) = 6\), thereby aligning \(s_2\) to \(f\) over the endpoint constraints.

Step 6: Conclusion and Deductions

The assumptions are aligned. Therefore, \(\sigma_i = f''(x_i)\) is consistent across known endpoints with adjusted boundary conditions. Thus, \(s_2\) equals \(f\) within \([0, 1]\), showing identically aligned results when using correctly chosen boundary conditions, and equations mirror desired properties of \(f(x) = x^3\).

Key Concepts

Natural Cubic Spline

A natural cubic spline is a smooth curve used to approximate known data points. This type of spline gets its name due to the way it naturally "settles" to a straight line beyond the boundary points. For any function interpolated by a natural cubic spline on a specified interval, the condition is that the second derivatives at the boundary knots are zero. This ensures that the spline blends into a linear slope, mimicking natural occurrences of bending objects like beams.
In essence, a natural cubic spline aims to create the most relaxed curve possible given the constraints, resembling how a beam bends under its own weight. Its primary purpose is to preserve smoothness, lowering the chances of creating overly complex curves.

Boundary Conditions

Boundary conditions are crucial assumptions needed for constructing a cubic spline. They dictate the behavior of the spline at the edges of the interpolation interval. In the context of natural cubic splines, the boundary conditions ensure that the second derivatives at the boundary knots are zero. This condition helps in achieving the natural "straightening" of the spline at the ends.
In practice, different boundary conditions can significantly change the form of the spline. If the standard natural spline conditions (\(\sigma_0 = \sigma_m = 0\)) cause a mismatch with the function you're interpolating, it's sometimes necessary to adjust them to match the known values of the second derivatives, like setting \(\sigma_0\) and \(\sigma_m\) to known derivatives of the function at the endpoints.

Second Derivative

The second derivative of a function is an important concept in cubic spline interpolation, particularly in determining the curvature of the spline. In our context, the second derivative \(f''(x)\) represents the amount of bending at each point of the spline. For a natural cubic spline, ensuring the second derivatives are continuous across the interval is key to achieving a smooth curve.
The second derivative condition is critical because it not only links the function's curvature but also influences the selection of boundary conditions. In the exercise you're studying, it's important to compute these derivatives accurately to ensure that the interpolated spline correctly reflects the function's properties.

Equally Spaced Knots

Equally spaced knots are significant in cubic spline interpolation as they simplify the calculations and ensure consistent segment lengths across the interval. When interpolating a function, defining the knots at regular intervals (equal spacing) ensures that the spline is closer to the actual function in terms of distribution, which is essential for maintaining accuracy across the spline.
The idea is to create a uniform grid over the interpolation interval, making calculations more straightforward and balancing accuracy across all segments. For the function \(f(x) = x^3\) in the exercise example, the knots were placed at \(x_i = i \cdot h\) where \(h = \frac{1}{m}\), providing a systematic approach to creating the spline and ensuring that no segment is unnecessarily extended or shortened.