How can we derive the Interpolation formula for higher order?

Question

Let, $$y = mx + c \tag 1$$ is the equation of a straight line.

Let, it pass through the point $$(x_0, y_0).$$

So, from (1) we find, $$c = y_0 + mx_0 \tag 2$$ On the other hand, from the formula of slope we find $$m=\frac {y_1-y_0}{x_1-x_0} \tag 3$$ Putting (3) in (2) we find $$c = y_0 - {\frac {y_1-y_0}{x_1-x_0}} x_0 \tag 4$$ Putting (3) and (4) into (1) we find $$y = y_1 \frac {x-x_0}{x_1-x_0} + y_0 \frac {x-x_1}{x_0-x_1}$$ $$\Longrightarrow P(x) = y_1 \frac {x-x_0}{x_1-x_0} + y_0 \frac {x-x_1}{x_0-x_1} \tag 5$$

Putting $x = x_0$ into (5) we find, $$P(x_0) = y_0 \tag 6$$ Putting $x = x_1$ into (5) we find $$P(x_1) = y_1 \tag 7$$ Putting (6) and (7) into (5) we find $$\Longrightarrow P(x) = P(x_1) \frac {x-x_0}{x_1-x_0} + P(x_0) \frac {x-x_1}{x_0-x_1} \tag 8 $$ Ok. Now, how to find this:

$P(x)=$

$$=> P(x) = P(x_1) \frac {x-x_0}{x_1-x_0} + P(x_0) \frac {x-x_1}{x_0-x_1}$$" /> $\tag 9$

Answer 1

The idea, which works also for higher degree polynomial interpolation, comes from linear algebra (dual bases). Instead of solving directly the problem though systems of linear equations to find the coefficients, you first try to solve the simpler problem: find two polynomials $p_0(x)$ and $p_1(x)$ such that: $$\begin{cases} p_0(x_0)=1\\p_0(x_1)=0 \end{cases}\qquad \begin{cases} p_1(x_0)=0\\p_1(x_1)=1 \end{cases}$$ Once you get them, it is trivial that the initial problem has the solution: $$p(x)=a_0p_0(x)+a_1p_1(x).$$ Now, from the second condition on $p_0(x)$ we know $\;p_0(x)=c(x-x_1),\enspace c\in\mathbf R$. Normalising it with the first condition, we obtain $c=\dfrac1{x_0-x_1}$.

Similarly, $\;p_1(x)=\dfrac{x-x_0}{x_1-x_0}$, whence finally: $$p(x)=a_0\frac{x-x_1}{x_0-x_1}+a_1\frac{x-x_0}{x_1-x_0}.$$

The general case goes along the same lines: say, for instance, we want to find a quadratic polynomial that takes prescribed values $(y_0,y_1, y_2$ at given (distinct) points $x_0, x_1, x_2$. We find first three quadratic polynomials that satisfy each one of these conditions: $$\begin{cases} p_0(x_0)=1\\p_0(x_1)=0\\p_0(x_2)=0 \end{cases}\qquad \begin{cases} p_1(x_0)=0\\p_1(x_1)=1\\p_1(x_2)=0\end{cases}\qquad \begin{cases} p_2(x_0)=0\\p_2(x_1)=0\\p_2(x_2)=1 \end{cases}$$ Once we have them, the solution to the original problem is trivial: $$p(x)=y_0p_0(x)+y_1p_1(x)+y_2p_2(x).$$

Now each $p_i(x)$ must vanish at $x_j$ and $x_k$, hence it has the form: $$p_i(x)=c(x-x_j)(x-x_k),\quad c\in\mathbf R,$$ and it is normalised by the condition $p_i(x_i)=1$, hence $c=\dfrac1{(x_i-x_j)(x_i-x_k)}$, so that the final solution is: $$p(x)=y_0\frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}+y_1\frac{(x-x_2)(x-x_0)}{(x_1-x_2)(x_1-x_0)}+y_2\frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}.$$

Answer 2

For the case where $n = 2$: Take fixed numbers $x_0, x_1, a_0$, and $a_1$. Let's look at a few special cases.

Say we want a polynomial $Q$ such that $Q(x_0)$ is $a_0$ while $Q(x_1) = 0$.

Since $x_1$ is a root, we can say that $Q(x) = c(x-x_1)$ for some coefficient $c$. Then we have $a_0 = Q(x_0) = c(x_0-x_1)$. Solve this for $c$, and we find that $$Q(x) = c(x-x_1) = \frac{a_0}{x_0-x_1}(x-x_1) = a_0\frac{x-x_1}{x_0-x_1}.$$

Now say we want a polynomial $R$ such that $R(x_0)$ is $0$ while $R(x_1)$ is $a_1$. Through similar reasoning, we get that $$R(x) = a_1\frac{x-x_0}{x_1-x_0}.$$

Notice that we $Q(x) + R(x)$ satisfies the conditions that you put on your polynomial $P_1(x)$. So we finally have $$P_1(x) = Q(x) + R(x) = a_0\frac{x-x_1}{x_0-x_1} + a_1\frac{x-x_0}{x_1-x_0}.$$

(Furthermore, it's possible to prove that there is at most one polynomial that fits the chosen points, and hence this solution is unique.)

EDIT: For anyone reading this now, the author of this question had asked for a derivation of the formula for a Lagrange interpolating polynomial given two points. Now the author has edited his question to ask for a derivation of the general formula. The derivation I gave here can easily be extended to this situation.

Say we want a polynomial $P_0$ such that $P_0(x_0) = a_0$ while $P_0(x_1), P_0(x_2), \ldots , P_0(x_n)$ are roots. Knowing all these roots, we can say that $P_0(x) = c(x-x_1)(x-x_2)\ldots(x-x_n)$. Then we have $a_0 = P_0(x_0) = c(x_0-x_1)(x_0-x_2)\ldots(x_0-x_n)$. Solve for $c$ to find that $$P_0(x) = c(x-x_1)(x-x_2)\ldots(x-x_n) = a_0\frac{(x-x_1)(x-x_2)\ldots(x-x_n)}{(x_0-x_1)(x_0-x_2)\ldots(x_0-x_n)}.$$

Do the same for each of the other points -- that is, for each natural number $k$ where $0 \leq k \leq n$, find the equation for a polynomial $P_k$ such that $P_k(x_k) = a_k$ and all the other $x_0, x_1, \ldots, x_n$ points are roots. Then simply sum all these polynomials to get a final polynomial that hits each of the desired points.

Answer 3

The idea is:

• $y(x)=a+bx=c_0(x-x_0)+c_1(x-x_1)$

for some choice of $c_0$ and $c_1$ s.t. $y(x_i)=y_i, (i=0,1)$.

• $y(x)=a+bx+cx^2=c_0(x-x_1)(x-x_2)+c_1(x-x_0)(x-x_1)+c_2(x-x_0)(x-x_1)$ for some choice of $c_0,c_1$ and $c_2$ s.t. $y(x_i)=y_i, (i=0,1,2)$.

Now generalize this for $n$-degree polynomial.