Additively separable function: Difference between revisions

Revision as of 23:30, 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:

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

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 $x$ )	$F_{xy}(x,y)=0$ F_{yx}(x,y) = 0</math>
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 $F(x_{1},x_{2},\dots ,x_{n})=f_{A}({\mbox{only the variables }}x_{i},i\in A)+f_{B}({\mbox{only the variables }}x_{i},i\in B)$	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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 There is a weaker notion of ''partially additively separable'': if we express the set <math>\{ 1,2,\dots,n\}</math> as a union of two disjoint subsets <math>A,B</math>, <math>F</matH> is additively separable with respect to the partition if there exist functions <math>f_A,f_B</math> such that:
-<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>
+<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>
 ==Partial derivatives==
@@ Line 34: / Line 34: @@
 | 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) = 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.
 |}