MHB Uniqueness of Cubic Spline Interpolation: How Can We Prove It?

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The discussion focuses on proving the uniqueness of cubic spline interpolation given specific conditions. It establishes that a cubic spline consists of multiple segments, each with four coefficients, requiring a total of 4m independent equations for m segments. The necessary equations include continuity conditions at the segment boundaries and continuity of the first and second derivatives. The participants clarify that two additional conditions, either on the first or second derivatives at the endpoints, are needed to ensure a unique solution. Overall, the conversation emphasizes the importance of these conditions in achieving a well-defined cubic spline function.
mathmari
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Hey! 😊

Show that the interpolation exercise for cubic splines with $s(x_0), s(x_1), , \ldots , s(x_m)$ at the points $x_0<x_1<\ldots <x_m$, together with one of $s'(x_0)$ or $s''(x_0)$ and $s'(x_m)$ or $s''(x_m)$ has exactly one solution.

Could you give me a hint how we could show that? Do wwe have to assume that we have two different solutions annd get a contradiction? :unsure:
 
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A cubic spline for a segment has 4 coefficients. Since we have $m$ segments, we have $4m$ coefficients.
To find them, we need $4m$ independent equations.
Which equations do we have? 🤔
 
Klaas van Aarsen said:
A cubic spline for a segment has 4 coefficients. Since we have $m$ segments, we have $4m$ coefficients.
To find them, we need $4m$ independent equations.
Which equations do we have? 🤔

We have the equations $s(x_0), s(x_1), \ldots , s(x_m)$, then we have also conditions of continuity, or not? And we have also the conditions $s'(x_0)=s'(x_m)$ or $s''(x_0)=s''(x_m)$.

Is that correct? :unsure:
 
Yes, we have the conditions of continuity.
Each of the $m$ segments must have the given starting and ending points, which gives us $2m$ independent equations, doesn't it? 🤔

Don't we also have the condition that the derivative must be continuous? That is, no angles in the resulting spline? 🤔
 
Last edited:
Klaas van Aarsen said:
Yes, we have the conditions of continuity.
Each of the $m$ segments must have the given starting and ending points, which gives us $2m$ independent equations, doesn't it? 🤔

Don't we also have the condition that the derivative must be continuous? That is, no angles in the resulting spline? 🤔

I got stuck right now. We don't have an index at $s(x)$. Does this mean that at each segment we have the same function? Shouldn't it be $s_i(x)$ ? :unsure:
 
mathmari said:
I got stuck right now. We don't have an index at $s(x)$. Does this mean that at each segment we have the same function? Shouldn't it be $s_i(x)$ ?

Hmm... your OP says "the interpolation exercise for cubic splines". Which interpolation exercise is that? 🤔

Note that a "cubic" spline is a third order polynomial, which means that it has 4 coefficients.
Consequently if there are more than 4 points, we cannot find one that has all points on its curve.
So either the spline is actually of $m$-th order, or the spline consists of $m$ segments each with its own coefficients, or the spline is a best-fit function.
Btw, even if it is a best-fit function, a "cubic" spline should still have only at most 4 points given and not $m+1$ points.
And if it were an $m$-th order polynomial, then there is no need to set constraints on the first or second derivative at the end points.
So I assumed we have a set of $m$ cubic splines, which I guess may not be what is intended.

If not, can you tell that kind of spline we are talking about then? 🤔
There are many types after all.
 
mathmari said:
I got stuck right now. We don't have an index at $s(x)$. Does this mean that at each segment we have the same function? Shouldn't it be $s_i(x)$ ?

The spline function $s$ would be a composite function.
That is:
$$s(x)=\begin{cases}s_1(x)&\text{if } 0\le x < 1 \\ s_2(x-1) &\text{if } 1\le x < 2 \\\ldots\\ s_m(x-m+1) &\text{if } m-1\le x \le m\end{cases}$$
where $s_i(u)=a_{i,3} u^3 + a_{i,2} u^2 + a_{i,1} u + a_{i,0}$ for $i=1,\ldots,m$.

So we don't simply have $s_i(x)$ since it depends on the value of $x$, which $s_i$ we need. 🤔

Suppose we assume that this is indeed what was intended. Then a solution is a set of coefficients $a_{i,j}$.

To ensure continuity, we need that $s_i(0)=s(i-1)$ and $s_i(1)=s(i)$ for $i=1,\ldots,m$. These are $2m$ independent equations.
To ensure continuity of the derivative, we need $s_i'(1)=s_{i+1}'(0)$ for $i=1,\ldots,m-1$. These are $2m-2$ independent equations.
So we are 2 equations short to find all coefficients.
If we add equations with the values of $s_1'(0)$ and $s_m'(1)$ we have a complete independent set with a unique solution.
Alternatively we can add equations with the values of $s_1''(0)$ and $s_m''(1)$ and we have again a complete independent set with a unique solution. 🤔

In practice the boundary condition $s_1''(0)=s_m''(1)=0$ is a typical choice.
 
Klaas van Aarsen said:
The spline function $s$ would be a composite function.
That is:
$$s(x)=\begin{cases}s_1(x)&\text{if } 0\le x < 1 \\ s_2(x-1) &\text{if } 1\le x < 2 \\\ldots\\ s_m(x-m+1) &\text{if } m-1\le x \le m\end{cases}$$
where $s_i(u)=a_{i,3} u^3 + a_{i,2} u^2 + a_{i,1} u + a_{i,0}$ for $i=1,\ldots,m$.

Why is the function defined in that form, I mean the argument is "x-something" and not $$s(x)=\begin{cases}s_1(x)&\text{if } 0\le x < 1 \\ s_2(x) &\text{if } 1\le x < 2 \\\ldots\\ s_m(x) &\text{if } m-1\le x \le m\end{cases}$$
Klaas van Aarsen said:
To ensure continuity, we need that $s_i(0)=s(i-1)$ and $s_i(1)=s(i)$ for $i=1,\ldots,m$. These are $2m$ independent equations.

Do we not get from the continuity the condition $s_i(x_i)=s_{i+1}(x_i)$ ? :unsure:
 
mathmari said:
Why is the function defined in that form, I mean the argument is "x-something" and not $$s(x)=\begin{cases}s_1(x)&\text{if } 0\le x < 1 \\ s_2(x) &\text{if } 1\le x < 2 \\\ldots\\ s_m(x) &\text{if } m-1\le x \le m\end{cases}$$
My mistake, it should be:
$$s(x)=\begin{cases}s_1(x)&\text{if } x_0\le x < x_1 \\ s_2(x) &\text{if } x_1\le x < x_2 \\\ldots\\ s_m(x) &\text{if } x_{m-1}\le x \le x_m\end{cases}$$
🤔

Do we not get from the continuity the condition $s_i(x_i)=s_{i+1}(x_i)$ ?
Yes, with the new version of the function $s$ that is indeed the continuity condition. 🤔
 
  • #10
Klaas van Aarsen said:
My mistake, it should be:
$$s(x)=\begin{cases}s_1(x)&\text{if } x_0\le x < x_1 \\ s_2(x) &\text{if } x_1\le x < x_2 \\\ldots\\ s_m(x) &\text{if } x_{m-1}\le x \le x_m\end{cases}$$
🤔

Yes, with the new version of the function $s$ that is indeed the continuity condition. 🤔

So we have the following:
\begin{equation*}s(x)=\begin{cases}s_1(x)&\text{wenn } x_0\le x < x_1 \\ s_2(x) &\text{wenn } x_1\le x < x_2 \\\ldots\\ s_m(x) &\text{wenn } x_{m-1}\le x \le x_m\end{cases}\end{equation*}
with $s_i(x)=a_{i,3} x^3 + a_{i,2} x^2 + a_{i,1} x + a_{i,0}$ for $i=1,\ldots,m$.

A cibic spline is $C^2$, so the second derivative must be continuous.

So that the function is continuous we need $s_i(x_i)=s_{i+1}(x_i)$ for $i=1,\ldots,m-1$ and $s_i(x_{i-1})=s(x_{i-1})$ for $i=1,\ldots,m+1$.
So we have $(m-1)+(m+1)=2m$ independent equations.

So that the first derivative is continuous we need $s_i'(x_i)=s_{i+1}'(x_i)$ for $i=1,\ldots,m-1$.
So we have $m-1$ independent equations.

So that the second derivative is continuous we need $s_i''(x_i)=s_{i+1}''(x_i)$ for $i=1,\ldots,m-1$.
So we have $m-1$ independent equations.

Till now we have $2m+(m-1)+(m-1)=4m-2$ conditions (equations).

The remaining two conditions are the two possible pairs given at the statement. Is everything correct? :unsure:
 
  • #11
Looks good enough to me. (Nod)
 
  • #12
Klaas van Aarsen said:
Looks good enough to me. (Nod)

Great! Thank you very much! (Smile)
 

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