Change the order of summation
- Thread starter raymond
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Harry_the_cat
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Depends on what the function is, surely?
Cubist
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In a computing context, if f(x,y) is a pure function then you can freely swap the order...
\(\displaystyle
\sum_{x=\mathit{c_1}}^{\mathit{c_2}}{ \left( \sum_{y=\mathit{c_3}}^{\mathit{c_4}}{f(x,y)} \right) }
=
\sum_{y=\mathit{c_3}}^{\mathit{c_4}}{ \left( \sum_{x=\mathit{c_1}}^{\mathit{c_2}}{f(x,y)} \right) }
\)
...but only if the variable that is holding the running sum does not overflow in either context.
In a mathematical context then I think a function is always pure? It might depend on the context and how the author has defined it.
\(\displaystyle
\sum_{x=\mathit{c_1}}^{\mathit{c_2}}{ \left( \sum_{y=\mathit{c_3}}^{\mathit{c_4}}{f(x,y)} \right) }
=
\sum_{y=\mathit{c_3}}^{\mathit{c_4}}{ \left( \sum_{x=\mathit{c_1}}^{\mathit{c_2}}{f(x,y)} \right) }
\)
...but only if the variable that is holding the running sum does not overflow in either context.
In a mathematical context then I think a function is always pure? It might depend on the context and how the author has defined it.
Cubist
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note that underflow (click) can also be a cause of different results if using floating point variables. If this might be happening then one solution is to use a variable type with extra precision.