Additively separable function: Difference between revisions

Revision as of 23:33, 10 April 2012

Definition

For a function of two variables

Suppose $F$ is a function of two variables $x$ and $y$ . We say that $F$ is additively separable if there exist functions $f, g$ of one variable such that:

$F (x, y) = f (x) + g (y)$

on the entire domain of $F$ .

Note that the concept of additively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally additively separable need not remain additively separable.

For a function of many variables

Suppose $F$ is a function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . We say that $F$ is completely additively separable if there exist functions $f_{1}, f_{2}, \dots, f_{n}$ , each a function of one variable, such that:

$F (x_{1}, x_{2}, \dots, x_{n}) = f_{1} (x_{1}) + f_{2} (x_{2}) + \dots + f_{n} (x_{n})$

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially additively separable: if we express the set ${1, 2, \dots, n}$ as a union of two disjoint subsets $A, B$ , $F$ is additively separable with respect to the partition if there exist functions $f_{A}, f_{B}$ such that:

$F (x_{1}, x_{2}, \dots, x_{n}) = f_{A} (only the variables x_{i}, i \in A) + f_{B} (only the variables x_{i}, i \in B)$

Partial derivatives

Additively separable functions are the exceptions to the general rule that value of partial derivative depends on all inputs.

Version type	Statement about first-order partial derivatives	Statement about second-order mixed partial derivatives
additively separable function $F$ of two variables $x, y$ , both pieces are differentiable functions, written as $F (x, y) = f (x) + g (y)$	$F_{x} (x, y) = f^{'} (x)$ (independent of $y$ ) $F_{y} (x, y) = g^{'} (y)$ (independent of $x$ )	$F_{x y} (x, y) = 0$ $F_{y x} (x, y) = 0$
completely additively separable function $F$ of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ , written as $f_{1} (x_{1}) + \dots + f_{n} (x_{n})$	$F_{x_{i}} (x_{1}, x_{2}, \dots, x_{n}) = {f_{i^{'}}}^{'} (x_{i})$ for each $i$ . Note that each first-order partial depends only on that variable and not on the others.	$F_{x_{i} x_{j}} (x_{1}, x_{2}, \dots, x_{n}) = 0$ for each $i, j$ .
partially additively separable function $F (x_{1}, x_{2}, \dots, x_{n})$ equals $f_{A} (only the variables x_{i}, i \in A) + f_{B} (only the variables x_{i}, i \in B)$	Each first-order partial of $F$ with respect to a variable in $A$ equals the corresponding first-order partial of $f_{A}$ , and in particular depends only on the variables within $A$ . Each first-order partial of $F$ with respect to a variable in $B$ equals the corresponding first-order partial of $f_{B}$ , and in particular depends only on the variables within $B$ .	Any second-order mixed partial involving a variable in $A$ and a variable in $B$ is zero.

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 | completely additively separable function <math>F</math> of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>, written as <math>f_1(x_1) + \dots + f_n(x_n)</math> || <math>F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i)</math> for each <math>i</math>. Note that each first-order partial depends only on that variable and not on the others.|| <math>F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0</math> for each <math>i,j</math>.
 |-
-| partially additively separable function <math>F(x_1,x_2,\dots,x_n) = f_A(\mbox{only the variables } x_i, i \in A) + f_B(\mbox{only the variables } x_i, i \in B)</math> || Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. || Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.
+| partially additively separable function <math>F(x_1,x_2,\dots,x_n)</math> equals <math>f_A(\mbox{only the variables } x_i, i \in A) + f_B(\mbox{only the variables } x_i, i \in B)</math> || Each first-order partial of <math>F</math> with respect to a variable in <math>A</math> equals the corresponding first-order partial of <math>f_A</math>, and in particular depends only on the variables within <matH>A</math>.<br>Each first-order partial of <math>F</math> with respect to a variable in <math>B</math> equals the corresponding first-order partial of <math>f_B</math>, and in particular depends only on the variables within <matH>B</math>. || Any second-order mixed partial involving a variable in <math>A</matH> and a variable in <math>B</math> is zero.
 |}