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Round down for n

  1. Jun 21, 2013 #1
    I have 3 functions in n which are rounded down to the nearest integer, and I need a closed-form way of writing each function. n will be a positive integer in all cases.

    (Let the notation RD(f(n)) denote rounding-down the value of f(n) for a certain n, to the nearest integer.) What I have found with each function is that f(n)=RD(f(n)) for all even values of n, and RD(f(n))=f(n)-x where x is a fraction which varies from function to function but, for a given function, is constant across all values of n. For the first function, x=0.5, for the second, x=0.75, for the third, x=0.25.

    Knowing this, how can I rewrite RD(f(n)) as a closed-form function, k(n), of n and f(n)? The key lies in finding some g(n) so that g(n)=-1 if n is odd and g(n)=0 if n is even; then I could write k(n)=RD(f(n))=f(n)+g(n)*x.
     
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  3. Jun 21, 2013 #2

    HallsofIvy

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    Can't you use the "floor" function, [itex]f(x)= \lfloor x \rfloor[/itex] which is defined as "the largest integer less than or equal to x"?
     
  4. Jun 21, 2013 #3
    No, that's the very definition of my problem. I need a [itex]k(x)[/itex] such that [itex]k(x) = RD(f(x)) = \lfloor f(x) \rfloor[/itex] to use your notation.
     
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