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Re: Reducing binary representation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg57184] Re: Reducing binary representation
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Fri, 20 May 2005 04:43:16 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <d6heg9$d24$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <d6heg9$d24$1 at smc.vnet.net>,
Torsten Coym <torsten.coym at eas.iis.fraunhofer.de> wrote:
> I want to reduce the number of coefficients in the binary representation
> of arbitrary integer numbers. I managed to convert an integer number
> into a sum of powers of two in the following way:
>
> In[7]:=
> ToBinary[x_, n_] := Plus @@
> Sequence[MapThread[Times,
> {Table[(HoldForm[2^#1] & )[i], {i, n - 1, 0, -1}],
> IntegerDigits[x, 2, n]}]]
I don't think that n is required here. Here is simpler code for doing
what you want:
PowerSum[x_] := Reverse[x] . 2^HoldForm /@ (Range[Length[x]] - 1)
ToBinary[x_] := PowerSum[IntegerDigits[x, 2]]
> The sum of adjacent powers of two can be reduced as follows:
>
> In[9]:=
> Sum[2^i, {i, k, j}]
>
> Out[9]=
> 2^(1 + j) - 2^k
>
> I now want to apply that to the binary number representation, so that
> 121 will become
>
> 2^7-2^3+2^0
>
> but I cannont figure out how to do this. If I release the Hold[]
> Mathematica just evaluates all the terms containing "2" to get "121",
> which is not what I want ;)
>
> Unfortunately I have no idea how to tackle this kind of problem. Any
> suggestion would be appreciated.
One approach is to use pattern-matching:
ReducedSum[x_] := PowerSum[IntegerDigits[x, 2] /.
{1, 1, b___} :> {1, 0, -1, b} //.
{{a___, c_, d_, b___} :> {a, 0, c, b} /; c == -d != 0,
{a___, 0, c_, d_, b___} :> {a, c, 0, -c, b} /; c == d != 0}]
For example,
ToBinary[123451]
2^HoldForm[0] + 2^HoldForm[1] + 2^HoldForm[3] + 2^HoldForm[4] +
2^HoldForm[5] + 2^HoldForm[9] + 2^HoldForm[13] + 2^HoldForm[14] +
2^HoldForm[15] + 2^HoldForm[16]
ReleaseHold[%]
123451
ReducedSum[123451]
-2^HoldForm[0] - 2^HoldForm[2] + 2^HoldForm[6] + 2^HoldForm[9] -
2^HoldForm[13] + 2^HoldForm[17]
ReleaseHold[%]
123451
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
http://InternationalMathematicaSymposium.org/IMS2005/
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