标签：right Theory mathscr divided Approximation part prod ldots left

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 \(x\).

误差估计:

\[f(x)-p_n(x) \approx\left\{\prod_{j=0}^n\left(x-x_j\right)\right\} f\left[x_0, x_1, \ldots, x_{n+1}\right] \]

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标签：right,Theory,mathscr,divided,Approximation,part,prod,ldots,left
From： https://www.cnblogs.com/kion/p/17340291.html