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Re: [PrimeNumbers] big numbers library

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  • Jan van Oort
    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.
    Message 1 of 9 , Aug 5, 2005
      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
      > radix conversion and exponentiation, so your program would be horribly
      > 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


      [Non-text portions of this message have been removed]
    • Alan Eliasen
      ... That s because Sun s Java implementation still uses horrible O(n^2) algorithms for multiplication. Their exponentiation routine already does
      Message 2 of 9 , Aug 7, 2005
        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
      • Alan Eliasen
        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 3 of 9 , Aug 7, 2005
          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
        • fyatim
          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 4 of 9 , Aug 20, 2005
            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 time.

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

            Faysal





            _____

            From: primenumbers@yahoogroups.com [mailto:primenumbers@yahoogroups.com] On
            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
            > radix conversion and exponentiation, so your program would be horribly
            > 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


            [Non-text portions of this message have been removed]



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          • fyatim
            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 5 of 9 , Aug 21, 2005
              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-----
              From: primenumbers@yahoogroups.com [mailto:primenumbers@yahoogroups.com] On
              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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