What is the primary goal when interpolating a cubic spline through the points (0, 1), (2, 2), and (4, 0)?

The primary goal is to construct a smooth, continuous function composed of cubic polynomial segments that exactly pass through the given points, ensuring a smooth transition between them without oscillations.

How many cubic polynomial segments are needed to interpolate the points (0, 1), (2, 2), and (4, 0)?

Two cubic polynomial segments are needed: one between (0, 1) and (2, 2), and another between (2, 2) and (4, 0).

What boundary conditions are typically used when constructing a natural cubic spline for these points?

Natural cubic splines usually impose zero second derivatives at the endpoints, meaning the second derivative at (0, 1) and (4, 0) are set to zero to ensure a smooth, 'free' boundary.

Can you briefly outline the steps to interpolate these points using a cubic spline?

First, set up the system of equations enforcing the spline passes through the points and has continuous first and second derivatives at the internal point (2, 2). Then, apply boundary conditions (e.g., natural spline), solve for the spline coefficients, and finally, define the cubic polynomials for each segment.

What are the advantages of using cubic splines over simpler interpolation methods like linear interpolation for these points?

Cubic splines provide a smooth and visually appealing curve with continuous first and second derivatives, avoiding the sharp corners or oscillations common in linear interpolation, leading to more realistic and natural-looking interpolations.

Exercises 2.17 Interpolate A Cubic Spline Between The Threepoints (0, 1), (2, 2) And (4, 0).

Exercises 2.17 Interpolate A Cubic Spline Between The Threepoints (0, 1), (2, 2) And (4, 0).

Introduction to Cubic Spline Interpolation

Cubic spline interpolation is a powerful numerical method used to construct a smooth curve passing through a given set of points. Unlike simple polynomial interpolation, which can lead to oscillations between points (Runge's phenomenon), cubic splines ensure a smooth and stable fit by piecing together cubic polynomials with continuous first and second derivatives. This technique is particularly useful in data fitting, computer graphics, and numerical analysis, where smoothness and accuracy are essential.

In this article, we delve into the process of interpolating a cubic spline through three specific points: (0, 1), (2, 2), and (4, 0). We explore the mathematical formulation, derive the spline coefficients, and discuss the properties of the resulting spline.

Understanding the Data Points

Given Points

The data points provided for interpolation are:

(0, 1)

(2, 2)

(4, 0)

These points define the key locations through which the interpolating spline must pass. The goal is to find a piecewise cubic function \( S(x) \) such that:

\( S(x) \) passes through all three points.

\( S(x) \) is twice continuously differentiable over the interval \([0, 4]\).

The spline consists of two cubic polynomials: one from \( x=0 \) to \( x=2 \), and another from \( x=2 \) to \( x=4 \).

Formulating the Cubic Spline

Partitioning the Interval

Since we have three points, the interval \([0, 4]\) is divided into two subintervals:

\( [x0, x1] = [0, 2] \)

\( [x1, x2] = [2, 4] \)

Corresponding to these, we define two cubic polynomials:

\[
S1(x) = a1 + b1(x - x0) + c1(x - x0)^2 + d1(x - x0)^3
\]

\[
S2(x) = a2 + b2(x - x1) + c2(x - x1)^2 + d2(x - x1)^3
\]

where:

\( S_1(x) \) is valid for \( x \in [0, 2] \),

\( S_2(x) \) is valid for \( x \in [2, 4] \).

Conditions for the Spline

To ensure a smooth cubic spline, the following conditions must be met:

Interpolation Conditions:

\( S_1(0) = 1 \) (passes through (0, 1))

\( S_1(2) = 2 \) (passes through (2, 2))

\( S_2(2) = 2 \) (continuity at \( x=2 \))

\( S_2(4) = 0 \) (passes through (4, 0))

Smoothness Conditions:

Continuity of first derivatives at \( x=2 \):

\[
S1'(2) = S2'(2)
\]

Continuity of second derivatives at \( x=2 \):

\[
S1''(2) = S2''(2)
\]

Boundary Conditions:

Since no specific boundary conditions are given, a common choice is the "natural" spline, where second derivatives at endpoints are zero:

\[
S1''(0) = 0, \quad S2''(4) = 0
\]

Alternatively, for simplicity, in this case, we can proceed with these natural boundary conditions.

Deriving the Coefficients

Step 1: Expressing the Cubic Polynomials

Let’s set:

\( x0 = 0 \), \( x1 = 2 \), \( x_2 = 4 \).

The cubic polynomials:

\[
S1(x) = a1 + b1 x + c1 x^2 + d_1 x^3
\]

\[
S2(x) = a2 + b2 (x - 2) + c2 (x - 2)^2 + d_2 (x - 2)^3
\]

Note: Using shifted variables for \( S_2 \) simplifies the calculations.

Step 2: Applying the Interpolation Conditions

From \( S_1(0) = 1 \):

\[
a1 + b1 \cdot 0 + c1 \cdot 0 + d1 \cdot 0 = 1 \Rightarrow a_1 = 1
\]

From \( S_1(2) = 2 \):

\[
a1 + 2b1 + 4c1 + 8d1 = 2
\]

But since \( a_1 = 1 \):

\[
1 + 2b1 + 4c1 + 8d1 = 2 \Rightarrow 2b1 + 4c1 + 8d1 = 1
\]

From \( S_2(2) = 2 \):

At \( x=2 \), \( x - 2=0 \):

\[
a2 + b2 \cdot 0 + c2 \cdot 0 + d2 \cdot 0 = 2 \Rightarrow a_2=2
\]

From \( S_2(4) = 0 \):

At \( x=4 \), \( x - 2=2 \):

\[
a2 + b2 \cdot 2 + c2 \cdot 4 + d2 \cdot 8 = 0
\]

Using \( a_2=2 \):

\[
2 + 2b2 + 4c2 + 8d_2= 0
\]

Step 3: Derivative Conditions for Smoothness

First derivatives:

\[
S1'(x) = b1 + 2c1 x + 3d1 x^2
\]

\[
S2'(x) = b2 + 2c2 (x - 2) + 3d2 (x - 2)^2
\]

At \( x=2 \):

\[
S1'(2) = b1 + 4 c1 + 12 d1
\]

\[
S2'(2) = b2 + 0 + 0 = b_2
\]

Set equal for smoothness:

\[
b1 + 4 c1 + 12 d1 = b2
\]

Second derivatives:

\[
S1''(x) = 2 c1 + 6 d_1 x
\]

\[
S2''(x) = 2 c2 + 6 d_2 (x - 2)
\]

At \( x=2 \):

\[
S1''(2) = 2 c1 + 12 d_1
\]

\[
S2''(2) = 2 c2 + 0 = 2 c_2
\]

Set equal:

\[
2 c1 + 12 d1 = 2 c_2
\]

Step 4: Boundary Conditions for Natural Spline

At \( x=0 \):

\[
S1''(0) = 2 c1 + 0 = 0 \Rightarrow c_1=0
\]

At \( x=4 \):

\[
S2''(4) = 2 c2 + 6 d2 \cdot 2 = 2 c2 + 12 d_2=0
\]

Solving the System for Coefficients

Now, compile all the equations:

\( a_1=1 \)

\( 2b1 + 4 c1 + 8 d_1=1 \)

\( a_2=2 \)

\( 2 + 2b2 + 4 c2 + 8 d_2=0 \)

\( b2 = b1 + 4 c1 + 12 d1 \)

\( 2 c1 + 12 d1= 2 c_2 \)