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• ## Re: [PrimeNumbers] big numbers library

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• ... That s because Sun s Java implementation still uses horrible O(n^2) algorithms for multiplication. Their exponentiation routine already does
Message 1 of 9 , Aug 7, 2005
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fyatim wrote:
> I wrote a small method using a well known algorithm “Exponentiation by
> Squaring”, but it still take horrible execution time.

That's because Sun's Java implementation still uses horrible O(n^2)
algorithms for multiplication. Their exponentiation routine already
does exponentiation by squaring, so you probably can't improve on it
without fixing multiplication.

> Where can I find the algorithm you are mentioning i.e. “bit shifting” ??

"Bit shifting" works when your base contains powers of 2. This is
because you can do powers of 2 by simply left-shifting the binary
representation by the appropriate number of bits. If you're doing
something like calculating large Mersenne numbers, this makes it about a
thousand times faster or more than Sun's implementation. It's an easy
and obvious optimization that Sun missed.

In short, here's a code snippet that does it. The base is expected
to be in a BigInteger called "big", and the exponent in an int called
"exponent". It factors out powers of two quickly by the call to
getLowestSetBit(), and does the exponentiation for powers of two rapidly
with shiftLeft() and then multiplies it by the remaining part that isn't
a power of 2.

This only helps if your base contains powers of 2.

if (big.signum() > 0)
{
// Get factor of two
int bit = big.getLowestSetBit();

if (bit > 0)
{
big = big.shiftRight(bit);
BigInteger twoPower = FrinkBigInteger.ONE.shiftLeft(bit*exponent);
if (big.equals(FrinkBigInteger.ONE))
return FrinkInteger.construct(twoPower);
else
{
big = big.pow(exponent);
return FrinkInteger.construct(big.multiply(twoPower));
}
}
}

--
Alan Eliasen | "It is not enough to do your best;
eliasen@... | you must know what to do and THEN
http://futureboy.homeip.net/ | do your best." -- W. Edwards Deming
• If it wasn t clear (and it wasn t,) the algorithm I just posted was one to speed up exponentiation. I use it as a wrapper around BigInteger.pow(BigInteger
Message 2 of 9 , Aug 7, 2005
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If it wasn't clear (and it wasn't,) the algorithm I just posted was
one to speed up exponentiation. I use it as a wrapper around
BigInteger.pow(BigInteger big, int exponent) in Java.

--
Alan Eliasen | "It is not enough to do your best;
eliasen@... | you must know what to do and THEN
http://futureboy.homeip.net/ | do your best." -- W. Edwards Deming
• Jan, Alan, Thank you for your support. I wrote a small method using a well known algorithm Exponentiation by Squaring , but it still take horrible execution
Message 3 of 9 , Aug 20, 2005
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Jan, Alan,

I wrote a small method using a well known algorithm "Exponentiation by
Squaring", but it still take horrible execution time.

Where can I find the algorithm you are mentioning i.e. "bit shifting" ??

Faysal

_____

Behalf Of Jan van Oort
Sent: Friday, August 05, 2005 8:57 PM
To: Alan Eliasen
Cc: Prime Number
Subject: Re: [PrimeNumbers] big numbers library

Alan,
you are quite right. Anyway, if I were in Faysal's case, it probably would
boil down - and that rather quickly - to option 1, with an eye on option 2.
The next time I am in between two project, I am going to write a
BigInteger.pow( BigInteger _exponent ) method. Just for fun. On a rainy day.

:-D
Jan
PS I was not complaining about any limits. Faysal was.

On 8/5/05, Alan Eliasen <eliasen@...> wrote:
>
> Jan van Oort wrote:
> > 2) take Colin Plum's source code, study it closely, and use to implement
> > your own method in a shared library ( .so or .dll compiled from C/C++ ).
> > Probably closely related to 1)
> > 3) download the sources of java.math.BigInteger and see if you can learn
> > anything from it, i.e. from the non-native ( Java ) part.
> > 4) look on the internet --- developer's forums etc --- if somebody has
> > already had the same idea / need as you, and developed something
> > 5) contact Colin Plum and ask him directly.
>
> I need to clarify that Sun's JVM hasn't used Colin Plumb's library
> since version 1.2. Since then, BigInteger has been implented in pure Java.
>
> If you use the free Kaffe JVM, it can be compiled to use the GMP
> library for BigInteger. You need to both compile with this and enable
> it at runtime with a command-line switch. Still, there's no API to have
> BigInteger exponents.
>
> In any case, BigInteger represents its bits as an int[] array. This
> limits the size of the numbers that can be represented to about
> 68 billion digits, or possibly less.
>
> But Sun's algorithms are horrible--they only use the most naive
> O(n^2) algorithm for multiplication, and equally horrible algorithms for
> time-bound. For example, it takes about 16 hours just to convert one of
> the larger Mersenne numbers, about 2^13000000, (which is *way, way*
> smaller than the 2^2147483647 limit that you're complaining about) to
> decimal on my computer. The exponentiation takes horribly long unless
> you write your own bit-shifting algorithms to fix Sun's inadequacies too.
>
> In short, if you're going to use numbers that big, you're going to
> have to write your own library. Or try Kaffe. NB: make sure you
> compile Kaffe with the option that intermediate numbers are allocated on
> the heap, not the stack, 'cause your stack isn't gigabytes in size.
>
> --
> Alan Eliasen | "It is not enough to do your best;
> eliasen@... | you must know what to do and THEN
> http://futureboy.homeip.net/ | do your best." -- W. Edwards Deming
>

--

Non sunt multiplicanda entia praeter necessitatem

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• Alan, Yes is clear .. Thanks. But, my problem still the same when the exponent is a prime 2... I m trying to find out the best (fastest) way to calculate it
Message 4 of 9 , Aug 21, 2005
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Alan,
Yes is clear .. Thanks.
But, my problem still the same when the exponent is a prime > 2...
I'm trying to find out the best (fastest) way to calculate it ...
I welcome any new ideas...
Regards
Faysal

-----Original Message-----
Behalf Of Alan Eliasen
Sent: Sunday, August 07, 2005 3:34 PM
To: fyatim
Cc: 'Jan van Oort'; 'Prime Number'
Subject: Re: [PrimeNumbers] big numbers library

fyatim wrote:
> I wrote a small method using a well known algorithm "Exponentiation by
> Squaring", but it still take horrible execution time.

That's because Sun's Java implementation still uses horrible O(n^2)
algorithms for multiplication. Their exponentiation routine already
does exponentiation by squaring, so you probably can't improve on it
without fixing multiplication.

> Where can I find the algorithm you are mentioning i.e. "bit shifting" ??

"Bit shifting" works when your base contains powers of 2. This is
because you can do powers of 2 by simply left-shifting the binary
representation by the appropriate number of bits. If you're doing
something like calculating large Mersenne numbers, this makes it about a
thousand times faster or more than Sun's implementation. It's an easy
and obvious optimization that Sun missed.

In short, here's a code snippet that does it. The base is expected
to be in a BigInteger called "big", and the exponent in an int called
"exponent". It factors out powers of two quickly by the call to
getLowestSetBit(), and does the exponentiation for powers of two rapidly
with shiftLeft() and then multiplies it by the remaining part that isn't
a power of 2.

This only helps if your base contains powers of 2.

if (big.signum() > 0)
{
// Get factor of two
int bit = big.getLowestSetBit();

if (bit > 0)
{
big = big.shiftRight(bit);
BigInteger twoPower = FrinkBigInteger.ONE.shiftLeft(bit*exponent);
if (big.equals(FrinkBigInteger.ONE))
return FrinkInteger.construct(twoPower);
else
{
big = big.pow(exponent);
return FrinkInteger.construct(big.multiply(twoPower));
}
}
}

--
Alan Eliasen | "It is not enough to do your best;
eliasen@... | you must know what to do and THEN
http://futureboy.homeip.net/ | do your best." -- W. Edwards Deming

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