Approximation Theory and Method part 3

Basic properties of divided differences

Let $\left\{x_i ; i=0,1, \ldots, n\right\}$ be any $(n+1)$ distinct points of $[a, b]$, and let $f$ be a function in $\mathscr{C}[a, b]$. The coefficient of $x^n$ in the polynomial $p \in \mathscr{P}_n$ that satisfies the interpolation conditions

\[p\left(x_i\right)=f\left(x_i\right), \quad i=0,1, \ldots, n \]

is defined to be a divided difference of order $n$, and we use the notation $f\left[x_0, x_1, \ldots, x_n\right]$ for its value. We note that the order of a divided difference is one less than the number of arguments in the expression $f[., ., \ldots,.] . $ Hence $ f\left[x_0\right]$ is a divided difference of order zero, which, by definition, has the value $f\left(x_0\right)$. Moreover, when $n \geqslant 1$, it follows from equations (4.3) and (4.6) that the equation

\[f\left[x_0, x_1, \ldots, x_n\right]=\sum_{k=0}^n \frac{f\left(x_k\right)}{\prod_{\substack{i=0 \\ j \neq k}}^n\left(x_k-x_j\right)} \]

is satisfied. We see that the divided difference is linear in the function values $\left\{f\left(x_i\right) ; i=0,1, \ldots, n\right\}$, but formula (5.2) is not the best way of calculating the value of $f\left[x_0, x_1, \ldots, x_n\right]$. A better method is described in Section 5.3.

对于

\[l_k(x)=\prod_{\substack{j=0 \\ j \neq k}}^n\left(x-x_j\right) /\left(x_k-x_j\right), \quad a \leqslant x \leqslant b . \]

$x^n$ 的系数是 $\frac{1}{\prod_{\substack{i=0 \\ j \neq k}}^n\left(x_k-x_j\right)}$. 所以 $f\left[x_0, x_1, \ldots, x_n\right]=\sum_{k=0}^n \frac{f\left(x_k\right)}{\prod_{\substack{i=0 \\ j \neq k}}^n\left(x_k-x_j\right)}$.

Theorem 5.1
Let $f \in \mathscr{C}^{(n)}[a, b]$ and let $\left\{x_i ; i=0,1, \ldots, n\right\}$ be a set of distinct points in $[a, b]$. Then there exists a point $\xi$, in the smallest interval that contains the points $\left\{x_i ; i=0,1, \ldots, n\right\}$, at which the equation

\[f\left[x_0, x_1, \ldots, x_n\right]=f^{(n)}(\xi) / n ! \]

is satisfied.

Recall that

Theorem 4.2
For any set of distinct interpolation points $\left\{x_i ; i=0,1, \ldots, n\right\}$ in $[a, b]$ and for any $f \in \mathscr{C}^{(n+1)}[a, b]$, let $p$ be the element of $\mathscr{P}_n$ that satisfies the equations (4.2). Then, for any $x$ in $[a, b]$, the error (4.12) has the value

\[e(x)=\frac{1}{(n+1) !} \prod_{j=0}^n\left(x-x_j\right) f^{(n+1)}(\xi) \]
where $\xi$ is a point of $[a, b]$ that depends on $x$.