Divided differences: Difference between revisions

Revision as of 04:50, 2 May 2014

Definition

For a set of pairs of data points

Given $k + 1$ data points:

$(x_{0}, y_{0}), (x_{1}, y_{1}), \dots, (x_{k}, y_{k})$

The forward divided differences are defined as:

[y_{ν}] : = y_{ν}, ν \in {0, \dots, k}

[y_{ν}, \dots, y_{ν + j}] : = \frac{[y_{ν + 1}, \dots, y_{ν + j}] - [y_{ν}, \dots, y_{ν + j - 1}]}{x_{ν + j} - x_{ν}}, ν \in {0, \dots, k - j}, j \in {1, \dots, k} .

The backward divided differences are defined as:

[y_{ν}] : = y_{ν}, ν \in {0, \dots, k}

[y_{ν}, \dots, y_{ν - j}] : = \frac{[y_{ν}, \dots, y_{ν - j + 1}] - [y_{ν - 1}, \dots, y_{ν - j}]}{x_{ν} - x_{ν - j}}, ν \in {j, \dots, k}, j \in {1, \dots, k} .

For a function

Suppose $f$ is a function and $x_{0}, x_{1}, \dots, x_{k}$ are points in the domain of $f$ . The (forward or backward) divided difference of $f$ for these points, denoted in any of these ways: $[x_{0}, x_{1}, \dots, x_{k}] f$ , $[x_{0}, x_{1}, \dots, x_{k}; f]$ , $D [x_{0}, x_{1}, \dots, x_{k}] f$ , is defined as the (forward or backward respectively) divided difference for the set of pairs:

$(x_{0}, f (x_{0})), (x_{1}, f (x_{1})), \dots, (x_{k}, f (x_{k}))$

Definition as a function

Consider a function $f$ with domain a subset $S$ of $R$ . Suppose $k$ is a positive integer. Denote by $[S]_{k}$ the set of $k$ -tuples of pairwise distinct elements of $S$ . The $k$ -fold forward divided difference function is a function:

$[S]_{k} \to R$

defined as:

$(x_{0}, x_{1}, \dots, x_{k}) \mapsto [x_{0}, x_{1}, \dots, x_{k}; f]$

Relation with operations on functions

Method for constructing new functions from old	In symbols	Difference quotient in terms of the old functions and their difference quotients	Proof
pointwise sum	$f + g$ is the function $x \mapsto f (x) + g (x)$ $f_{1} + f_{2} + \dots + f_{n}$ is the function $x \mapsto f_{1} (x) + f_{2} (x) + \dots + f_{n} (x)$	Divided difference of sum is sum of divided differences	divided differences are linear
pointwise difference	$f - g$ is the function $x \mapsto f (x) - g (x)$	Divided difference of difference is difference of divided differences	divided differences are linear
scalar multiple by a constant	$a f$ is the function $x \mapsto a f (x)$ where $a$ is a real number	$x \mapsto a Δ f (x)$	divided differences are linear
pointwise product	$f \cdot g$ (sometimes denoted $f g$ ) is the function $x \mapsto f (x) g (x)$ $f_{1} \cdot f_{2} \cdot \dots f_{n}$ (sometimes denoted $f_{1} f_{2} \dots f_{n}$ is the function $x \mapsto f_{1} (x) f_{2} (x) \dots f_{n} (x)$	See product rule for divided differences	product rule for divided differences
pointwise quotient	$f / g$ is the function $x \mapsto f (x) / g (x)$	?	?
composite of two functions	$f \circ g$ is the function $x \mapsto f (g (x))$	?	chain rule for divided differences

@@ Line 22: / Line 22: @@
 <math>(x_0,f(x_0)), (x_1,f(x_1)), \dots, (x_k,f(x_k))</math>
+==Definition as a function==
+Consider a function <math>f</math> with domain a subset <math>S</math> of <math>\R</math>. Suppose <math>k</math> is a positive integer. Denote by <math>[S]_k</math> the set of <math>k</math>-tuples of pairwise distinct elements of <math>S</math>. The <math>k</math>-fold forward divided difference function is a function:
+<math>[S]_k \to \R</math>
+defined as:
+<math>(x_0,x_1,\dots,x_k) \mapsto [x_0,x_1,\dots,x_k;f]</math>
+==Relation with operations on functions==
+{| class="sortable" border="1"
+! Method for constructing new functions from old !! In symbols !! Difference quotient in terms of the old functions and their difference quotients !! Proof
+|-
+| [[pointwise sum of functions|pointwise sum]] || <math>f + g</math> is the function <math>x \mapsto f(x) + g(x)</math><br><math>f_1 + f_2 + \dots + f_n</math> is the function <math>x \mapsto f_1(x) + f_2(x) + \dots + f_n(x)</math> || Divided difference of sum is sum of divided differences  || [[divided differences are linear]]
+|-
+| [[pointwise difference of functions|pointwise difference]] || <math>f - g</math> is the function <math>x \mapsto f(x) - g(x)</math>|| Divided difference of difference is difference of divided differences  || [[divided differences are linear]]
+|-
+| [[scalar multiple of function|scalar multiple]] by a constant || <math>af</math> is the function <math>x \mapsto af(x)</math> where <math>a</math> is a real number || <math>x \mapsto a \Delta f(x)</math> || [[divided differences are linear]]
+|-
+| [[pointwise product of functions|pointwise product]] || <math>f \cdot g</math> (sometimes denoted <math>fg</math>) is the function <math>x \mapsto f(x)g(x)</math><br><math>f_1 \cdot f_2 \cdot \dots f_n</math> (sometimes denoted <math>f_1f_2\dots f_n</math> is the function <math>x \mapsto f_1(x)f_2(x) \dots f_n(x)</math> || See [[product rule for divided differences]] || [[product rule for divided differences]]
+|-
+| [[pointwise quotient of functions|pointwise quotient]] || <math>f/g</math> is the function <math>x \mapsto f(x)/g(x)</math> || ? || ?
+|-
+| [[composite of two functions]] || <math>f \circ g</math> is the function <math>x \mapsto f(g(x))</math> || ? || [[chain rule for divided differences]]
+|}