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version 1.1.1.1, 2000/01/10 15:35:21 version 1.1.1.2, 2000/09/09 14:12:18
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 This is Info file gmp.info, produced by Makeinfo-1.64 from the input  This is gmp.info, produced by makeinfo version 4.0 from gmp.texi.
 file gmp.texi.  
   
   INFO-DIR-SECTION GNU libraries
 START-INFO-DIR-ENTRY  START-INFO-DIR-ENTRY
 * gmp: (gmp.info).               GNU Multiple Precision Arithmetic Library.  * gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
 END-INFO-DIR-ENTRY  END-INFO-DIR-ENTRY
   
    This file documents GNU MP, a library for arbitrary-precision     This file documents GNU MP, a library for arbitrary-precision
 arithmetic.  arithmetic.
   
    Copyright (C) 1991, 1993, 1994, 1995, 1996 Free Software Foundation,     Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000
 Inc.  Free Software Foundation, Inc.
   
    Permission is granted to make and distribute verbatim copies of this     Permission is granted to make and distribute verbatim copies of this
 manual provided the copyright notice and this permission notice are  manual provided the copyright notice and this permission notice are
Line 26  versions, except that this permission notice may be st
Line 26  versions, except that this permission notice may be st
 translation approved by the Foundation.  translation approved by the Foundation.
   
   
   File: gmp.info,  Node: Integer Exponentiation,  Next: Integer Roots,  Prev: Integer Division,  Up: Integer Functions
   
   Exponentiation Functions
   ========================
   
    - Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t MOD)
    - Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long int
             EXP, mpz_t MOD)
        Set ROP to (BASE raised to EXP) `mod' MOD.  If EXP is negative,
        the result is undefined.
   
   
    - Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
             EXP)
    - Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
             unsigned long int EXP)
        Set ROP to BASE raised to EXP.  The case of 0^0 yields 1.
   
   
   File: gmp.info,  Node: Integer Roots,  Next: Number Theoretic Functions,  Prev: Integer Exponentiation,  Up: Integer Functions
   
   Root Extraction Functions
   =========================
   
    - Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
        Set ROP to the truncated integer part of the Nth root of OP.
        Return non-zero if the computation was exact, i.e., if OP is ROP
        to the Nth power.
   
    - Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
        Set ROP to the truncated integer part of the square root of OP.
   
    - Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
        Set ROP1 to the truncated integer part of the square root of OP,
        like `mpz_sqrt'.  Set ROP2 to OP-ROP1*ROP1, (i.e., zero if OP is a
        perfect square).
   
        If ROP1 and ROP2 are the same variable, the results are undefined.
   
    - Function: int mpz_perfect_power_p (mpz_t OP)
        Return non-zero if OP is a perfect power, i.e., if there exist
        integers A and B, with B > 1, such that OP equals a raised to b.
        Return zero otherwise.
   
    - Function: int mpz_perfect_square_p (mpz_t OP)
        Return non-zero if OP is a perfect square, i.e., if the square
        root of OP is an integer.  Return zero otherwise.
   
   
   File: gmp.info,  Node: Number Theoretic Functions,  Next: Integer Comparisons,  Prev: Integer Roots,  Up: Integer Functions
   
   Number Theoretic Functions
   ==========================
   
    - Function: int mpz_probab_prime_p (mpz_t N, int REPS)
        If this function returns 0, N is definitely not prime.  If it
        returns 1, then N is `probably' prime.  If it returns 2, then N is
        surely prime.  Reasonable values of reps vary from 5 to 10; a
        higher value lowers the probability for a non-prime to pass as a
        `probable' prime.
   
        The function uses Miller-Rabin's probabilistic test.
   
    - Function: int mpz_nextprime (mpz_t ROP, mpz_t OP)
        Set ROP to the next prime greater than OP.
   
        This function uses a probabilistic algorithm to identify primes,
        but for for practical purposes it's adequate, since the chance of
        a composite passing will be extremely small.
   
    - Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
        Set ROP to the greatest common divisor of OP1 and OP2.  The result
        is always positive even if either of or both input operands are
        negative.
   
    - Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
             unsigned long int OP2)
        Compute the greatest common divisor of OP1 and OP2.  If ROP is not
        `NULL', store the result there.
   
        If the result is small enough to fit in an `unsigned long int', it
        is returned.  If the result does not fit, 0 is returned, and the
        result is equal to the argument OP1.  Note that the result will
        always fit if OP2 is non-zero.
   
    - Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t
             B)
        Compute G, S, and T, such that AS + BT = G = `gcd'(A, B).  If T is
        `NULL', that argument is not computed.
   
    - Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
        Set ROP to the least common multiple of OP1 and OP2.
   
    - Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
        Compute the inverse of OP1 modulo OP2 and put the result in ROP.
        Return non-zero if an inverse exists, zero otherwise.  When the
        function returns zero, ROP is undefined.
   
    - Function: int mpz_jacobi (mpz_t OP1, mpz_t OP2)
    - Function: int mpz_legendre (mpz_t OP1, mpz_t OP2)
        Compute the Jacobi and Legendre symbols, respectively.  OP2 should
        be odd and must be positive.
   
    - Function: int mpz_si_kronecker (long A, mpz_t B);
    - Function: int mpz_ui_kronecker (unsigned long A, mpz_t B);
    - Function: int mpz_kronecker_si (mpz_t A, long B);
    - Function: int mpz_kronecker_ui (mpz_t A, unsigned long B);
        Calculate the value of the Kronecker/Jacobi symbol (A/B), with the
        Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
        All values of A and B give a well-defined result.  See Henri
        Cohen, section 1.4.2, for more information (*note References::).
        See also the example program `demos/qcn.c' which uses
        `mpz_kronecker_ui'.
   
    - Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
        Remove all occurrences of the factor F from OP and store the
        result in ROP.  Return the multiplicity of F in OP.
   
    - Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
        Set ROP to OP!, the factorial of OP.
   
    - Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
    - Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
             unsigned long int K)
        Compute the binomial coefficient N over K and store the result in
        ROP.  Negative values of N are supported by `mpz_bin_ui', using
        the identity bin(-n,k) = (-1)^k * bin(n+k-1,k) (see Knuth volume 1
        section 1.2.6 part G).
   
    - Function: void mpz_fib_ui (mpz_t ROP, unsigned long int N)
        Compute the Nth Fibonacci number and store the result in ROP.
   
   
   File: gmp.info,  Node: Integer Comparisons,  Next: Integer Logic and Bit Fiddling,  Prev: Number Theoretic Functions,  Up: Integer Functions
   
   Comparison Functions
   ====================
   
    - Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
        Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero
        if OP1 = OP2, and a negative value if OP1 < OP2.
   
    - Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
    - Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
        Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero
        if OP1 = OP2, and a negative value if OP1 < OP2.
   
        These functions are actually implemented as macros.  They evaluate
        their arguments multiple times.
   
    - Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
    - Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
        Compare the absolute values of OP1 and OP2.  Return a positive
        value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1
        < OP2.
   
    - Macro: int mpz_sgn (mpz_t OP)
        Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
   
        This function is actually implemented as a macro.  It evaluates its
        arguments multiple times.
   
   
   File: gmp.info,  Node: Integer Logic and Bit Fiddling,  Next: I/O of Integers,  Prev: Integer Comparisons,  Up: Integer Functions
   
   Logical and Bit Manipulation Functions
   ======================================
   
      These functions behave as if two's complement arithmetic were used
   (although sign-magnitude is used by the actual implementation).
   
    - Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
        Set ROP to OP1 logical-and OP2.
   
    - Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
        Set ROP to OP1 inclusive-or OP2.
   
    - Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
        Set ROP to OP1 exclusive-or OP2.
   
    - Function: void mpz_com (mpz_t ROP, mpz_t OP)
        Set ROP to the one's complement of OP.
   
    - Function: unsigned long int mpz_popcount (mpz_t OP)
        For non-negative numbers, return the population count of OP.  For
        negative numbers, return the largest possible value (MAX_ULONG).
   
    - Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
        If OP1 and OP2 are both non-negative, return the hamming distance
        between the two operands.  Otherwise, return the largest possible
        value (MAX_ULONG).
   
        It is possible to extend this function to return a useful value
        when the operands are both negative, but the current
        implementation returns MAX_ULONG in this case.  *Do not depend on
        this behavior, since it will change in a future release.*
   
    - Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int
             STARTING_BIT)
        Scan OP, starting with bit STARTING_BIT, towards more significant
        bits, until the first clear bit is found.  Return the index of the
        found bit.
   
    - Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int
             STARTING_BIT)
        Scan OP, starting with bit STARTING_BIT, towards more significant
        bits, until the first set bit is found.  Return the index of the
        found bit.
   
    - Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)
        Set bit BIT_INDEX in ROP.
   
    - Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
        Clear bit BIT_INDEX in ROP.
   
    - Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)
        Check bit BIT_INDEX in OP and return 0 or 1 accordingly.
   
   
   File: gmp.info,  Node: I/O of Integers,  Next: Integer Random Numbers,  Prev: Integer Logic and Bit Fiddling,  Up: Integer Functions
   
   Input and Output Functions
   ==========================
   
      Functions that perform input from a stdio stream, and functions that
   output to a stdio stream.  Passing a `NULL' pointer for a STREAM
   argument to any of these functions will make them read from `stdin' and
   write to `stdout', respectively.
   
      When using any of these functions, it is a good idea to include
   `stdio.h' before `gmp.h', since that will allow `gmp.h' to define
   prototypes for these functions.
   
    - Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
        Output OP on stdio stream STREAM, as a string of digits in base
        BASE.  The base may vary from 2 to 36.
   
        Return the number of bytes written, or if an error occurred,
        return 0.
   
    - Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
        Input a possibly white-space preceded string in base BASE from
        stdio stream STREAM, and put the read integer in ROP.  The base
        may vary from 2 to 36.  If BASE is 0, the actual base is
        determined from the leading characters: if the first two
        characters are `0x' or `0X', hexadecimal is assumed, otherwise if
        the first character is `0', octal is assumed, otherwise decimal is
        assumed.
   
        Return the number of bytes read, or if an error occurred, return 0.
   
    - Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
        Output OP on stdio stream STREAM, in raw binary format.  The
        integer is written in a portable format, with 4 bytes of size
        information, and that many bytes of limbs.  Both the size and the
        limbs are written in decreasing significance order (i.e., in
        big-endian).
   
        The output can be read with `mpz_inp_raw'.
   
        Return the number of bytes written, or if an error occurred,
        return 0.
   
        The output of this can not be read by `mpz_inp_raw' from GMP 1,
        because of changes necessary for compatibility between 32-bit and
        64-bit machines.
   
    - Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
        Input from stdio stream STREAM in the format written by
        `mpz_out_raw', and put the result in ROP.  Return the number of
        bytes read, or if an error occurred, return 0.
   
        This routine can read the output from `mpz_out_raw' also from GMP
        1, in spite of changes necessary for compatibility between 32-bit
        and 64-bit machines.
   
   
   File: gmp.info,  Node: Integer Random Numbers,  Next: Miscellaneous Integer Functions,  Prev: I/O of Integers,  Up: Integer Functions
   
   Random Number Functions
   =======================
   
      The random number functions of GMP come in two groups; older function
   that rely on a global state, and newer functions that accept a state
   parameter that is read and modified.  Please see the *Note Random
   Number Functions:: for more information on how to use and not to use
   random number functions.
   
    - Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
        unsigned long int N) Generate a uniformly distributed random
        integer in the range 0 to 2^N - 1, inclusive.
   
        The variable STATE must be initialized by calling one of the
        `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
    - Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
        mpz_t N) Generate a uniform random integer in the range 0 to N -
        1, inclusive.
   
        The variable STATE must be initialized by calling one of the
        `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
    - Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
             unsigned long int N)
        Generate a random integer with long strings of zeros and ones in
        the binary representation.  Useful for testing functions and
        algorithms, since this kind of random numbers have proven to be
        more likely to trigger corner-case bugs.  The random number will
        be in the range 0 to 2^N - 1, inclusive.
   
        The variable STATE must be initialized by calling one of the
        `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
    - Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
        Generate a random integer of at most MAX_SIZE limbs.  The generated
        random number doesn't satisfy any particular requirements of
        randomness.  Negative random numbers are generated when MAX_SIZE
        is negative.
   
        This function is obsolete.  Use `mpz_urandomb' or `mpz_urandomm'
        instead.
   
    - Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
        Generate a random integer of at most MAX_SIZE limbs, with long
        strings of zeros and ones in the binary representation.  Useful
        for testing functions and algorithms, since this kind of random
        numbers have proven to be more likely to trigger corner-case bugs.
        Negative random numbers are generated when MAX_SIZE is negative.
   
        This function is obsolete.  Use `mpz_rrandomb' instead.
   
   
   File: gmp.info,  Node: Miscellaneous Integer Functions,  Prev: Integer Random Numbers,  Up: Integer Functions
   
   Miscellaneous Functions
   =======================
   
    - Function: int mpz_fits_ulong_p (mpz_t OP)
    - Function: int mpz_fits_slong_p (mpz_t OP)
    - Function: int mpz_fits_uint_p (mpz_t OP)
    - Function: int mpz_fits_sint_p (mpz_t OP)
    - Function: int mpz_fits_ushort_p (mpz_t OP)
    - Function: int mpz_fits_sshort_p (mpz_t OP)
        Return non-zero iff the value of OP fits in an `unsigned long int',
        `signed long int', `unsigned int', `signed int', `unsigned short
        int', or `signed short int', respectively.  Otherwise, return zero.
   
    - Macro: int mpz_odd_p (mpz_t OP)
    - Macro: int mpz_even_p (mpz_t OP)
        Determine whether OP is odd or even, respectively.  Return
        non-zero if yes, zero if no.  These macros evaluate their
        arguments more than once.
   
    - Function: size_t mpz_size (mpz_t OP)
        Return the size of OP measured in number of limbs.  If OP is zero,
        the returned value will be zero.
   
    - Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
        Return the size of OP measured in number of digits in base BASE.
        The base may vary from 2 to 36.  The returned value will be exact
        or 1 too big.  If BASE is a power of 2, the returned value will
        always be exact.
   
        This function is useful in order to allocate the right amount of
        space before converting OP to a string.  The right amount of
        allocation is normally two more than the value returned by
        `mpz_sizeinbase' (one extra for a minus sign and one for the
        terminating '\0').
   
   
   File: gmp.info,  Node: Rational Number Functions,  Next: Floating-point Functions,  Prev: Integer Functions,  Up: Top
   
   Rational Number Functions
   *************************
   
      This chapter describes the GMP functions for performing arithmetic
   on rational numbers.  These functions start with the prefix `mpq_'.
   
      Rational numbers are stored in objects of type `mpq_t'.
   
      All rational arithmetic functions assume operands have a canonical
   form, and canonicalize their result.  The canonical from means that the
   denominator and the numerator have no common factors, and that the
   denominator is positive.  Zero has the unique representation 0/1.
   
      Pure assignment functions do not canonicalize the assigned variable.
   It is the responsibility of the user to canonicalize the assigned
   variable before any arithmetic operations are performed on that
   variable.  *Note that this is an incompatible change from version 1 of
   the library.*
   
    - Function: void mpq_canonicalize (mpq_t OP)
        Remove any factors that are common to the numerator and
        denominator of OP, and make the denominator positive.
   
   * Menu:
   
   * Initializing Rationals::
   * Rational Arithmetic::
   * Comparing Rationals::
   * Applying Integer Functions::
   * I/O of Rationals::
   * Miscellaneous Rational Functions::
   
   
   File: gmp.info,  Node: Initializing Rationals,  Next: Rational Arithmetic,  Prev: Rational Number Functions,  Up: Rational Number Functions
   
   Initialization and Assignment Functions
   =======================================
   
    - Function: void mpq_init (mpq_t DEST_RATIONAL)
        Initialize DEST_RATIONAL and set it to 0/1.  Each variable should
        normally only be initialized once, or at least cleared out (using
        the function `mpq_clear') between each initialization.
   
    - Function: void mpq_clear (mpq_t RATIONAL_NUMBER)
        Free the space occupied by RATIONAL_NUMBER.  Make sure to call this
        function for all `mpq_t' variables when you are done with them.
   
    - Function: void mpq_set (mpq_t ROP, mpq_t OP)
    - Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
        Assign ROP from OP.
   
    - Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
             unsigned long int OP2)
    - Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
             long int OP2)
        Set the value of ROP to OP1/OP2.  Note that if OP1 and OP2 have
        common factors, ROP has to be passed to `mpq_canonicalize' before
        any operations are performed on ROP.
   
    - Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
        Swap the values ROP1 and ROP2 efficiently.
   
   
   File: gmp.info,  Node: Rational Arithmetic,  Next: Comparing Rationals,  Prev: Initializing Rationals,  Up: Rational Number Functions
   
   Arithmetic Functions
   ====================
   
    - Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
        Set SUM to ADDEND1 + ADDEND2.
   
    - Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
             SUBTRAHEND)
        Set DIFFERENCE to MINUEND - SUBTRAHEND.
   
    - Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
             MULTIPLICAND)
        Set PRODUCT to MULTIPLIER times MULTIPLICAND.
   
    - Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
             DIVISOR)
        Set QUOTIENT to DIVIDEND/DIVISOR.
   
    - Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
        Set NEGATED_OPERAND to -OPERAND.
   
    - Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
        Set INVERTED_NUMBER to 1/NUMBER.  If the new denominator is zero,
        this routine will divide by zero.
   
   
   File: gmp.info,  Node: Comparing Rationals,  Next: Applying Integer Functions,  Prev: Rational Arithmetic,  Up: Rational Number Functions
   
   Comparison Functions
   ====================
   
    - Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
        Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero
        if OP1 = OP2, and a negative value if OP1 < OP2.
   
        To determine if two rationals are equal, `mpq_equal' is faster than
        `mpq_cmp'.
   
    - Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
             long int DEN2)
        Compare OP1 and NUM2/DEN2.  Return a positive value if OP1 >
        NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
        NUM2/DEN2.
   
        This routine allows that NUM2 and DEN2 have common factors.
   
        This function is actually implemented as a macro.  It evaluates its
        arguments multiple times.
   
    - Macro: int mpq_sgn (mpq_t OP)
        Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
   
        This function is actually implemented as a macro.  It evaluates its
        arguments multiple times.
   
    - Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
        Return non-zero if OP1 and OP2 are equal, zero if they are
        non-equal.  Although `mpq_cmp' can be used for the same purpose,
        this function is much faster.
   
   
   File: gmp.info,  Node: Applying Integer Functions,  Next: I/O of Rationals,  Prev: Comparing Rationals,  Up: Rational Number Functions
   
   Applying Integer Functions to Rationals
   =======================================
   
      The set of `mpq' functions is quite small.  In particular, there are
   few functions for either input or output.  But there are two macros
   that allow us to apply any `mpz' function on the numerator or
   denominator of a rational number.  If these macros are used to assign
   to the rational number, `mpq_canonicalize' normally need to be called
   afterwards.
   
    - Macro: mpz_t mpq_numref (mpq_t OP)
    - Macro: mpz_t mpq_denref (mpq_t OP)
        Return a reference to the numerator and denominator of OP,
        respectively.  The `mpz' functions can be used on the result of
        these macros.
   
   
   File: gmp.info,  Node: I/O of Rationals,  Next: Miscellaneous Rational Functions,  Prev: Applying Integer Functions,  Up: Rational Number Functions
   
   Input and Output Functions
   ==========================
   
      Functions that perform input from a stdio stream, and functions that
   output to a stdio stream.  Passing a `NULL' pointer for a STREAM
   argument to any of these functions will make them read from `stdin' and
   write to `stdout', respectively.
   
      When using any of these functions, it is a good idea to include
   `stdio.h' before `gmp.h', since that will allow `gmp.h' to define
   prototypes for these functions.
   
    - Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
        Output OP on stdio stream STREAM, as a string of digits in base
        BASE.  The base may vary from 2 to 36.  Output is in the form
        `num/den' or if the denominator is 1 then just `num'.
   
        Return the number of bytes written, or if an error occurred,
        return 0.
   
   
   File: gmp.info,  Node: Miscellaneous Rational Functions,  Prev: I/O of Rationals,  Up: Rational Number Functions
   
   Miscellaneous Functions
   =======================
   
    - Function: double mpq_get_d (mpq_t OP)
        Convert OP to a double.
   
    - Function: double mpq_set_d (mpq_t ROP, double D)
        Set ROP to the value of d, without rounding.
   
      These functions assign between either the numerator or denominator
   of a rational, and an integer.  Instead of using these functions, it is
   preferable to use the more general mechanisms `mpq_numref' and
   `mpq_denref', together with `mpz_set'.
   
    - Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
        Copy NUMERATOR to the numerator of RATIONAL.  When this risks to
        make the numerator and denominator of RATIONAL have common
        factors, you have to pass RATIONAL to `mpq_canonicalize' before
        any operations are performed on RATIONAL.
   
        This function is equivalent to `mpz_set (mpq_numref (RATIONAL),
        NUMERATOR)'.
   
    - Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
        Copy DENOMINATOR to the denominator of RATIONAL.  When this risks
        to make the numerator and denominator of RATIONAL have common
        factors, or if the denominator might be negative, you have to pass
        RATIONAL to `mpq_canonicalize' before any operations are performed
        on RATIONAL.
   
        *In version 1 of the library, negative denominators were handled by
        copying the sign to the numerator.  That is no longer done.*
   
        This function is equivalent to `mpz_set (mpq_denref (RATIONAL),
        DENOMINATORS)'.
   
    - Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
        Copy the numerator of RATIONAL to the integer NUMERATOR, to
        prepare for integer operations on the numerator.
   
        This function is equivalent to `mpz_set (NUMERATOR, mpq_numref
        (RATIONAL))'.
   
    - Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
        Copy the denominator of RATIONAL to the integer DENOMINATOR, to
        prepare for integer operations on the denominator.
   
        This function is equivalent to `mpz_set (DENOMINATOR, mpq_denref
        (RATIONAL))'.
   
   
 File: gmp.info,  Node: Floating-point Functions,  Next: Low-level Functions,  Prev: Rational Number Functions,  Up: Top  File: gmp.info,  Node: Floating-point Functions,  Next: Low-level Functions,  Prev: Rational Number Functions,  Up: Top
   
 Floating-point Functions  Floating-point Functions
 ************************  ************************
   
    This is a description of the *preliminary* interface for     This chapter describes the GMP functions for performing floating
 floating-point arithmetic in GNU MP 2.  point arithmetic.  These functions start with the prefix `mpf_'.
   
    The floating-point functions expect arguments of type `mpf_t'.     GMP floating point numbers are stored in objects of type `mpf_t'.
   
    The MP floating-point functions have an interface that is similar to     The GMP floating-point functions have an interface that is similar
 the MP integer functions.  The function prefix for floating-point  to the GMP integer functions.  The function prefix for floating-point
 operations is `mpf_'.  operations is `mpf_'.
   
    There is one significant characteristic of floating-point numbers     There is one significant characteristic of floating-point numbers
 that has motivated a difference between this function class and other  that has motivated a difference between this function class and other
 MP function classes: the inherent inexactness of floating point  GMP function classes: the inherent inexactness of floating point
 arithmetic.  The user has to specify the precision of each variable.  A  arithmetic.  The user has to specify the precision of each variable.  A
 computation that assigns a variable will take place with the precision  computation that assigns a variable will take place with the precision
 of the assigned variable; the precision of variables used as input is  of the assigned variable; the precision of variables used as input is
Line 51  ignored.
Line 647  ignored.
    The precision of a calculation is defined as follows: Compute the     The precision of a calculation is defined as follows: Compute the
 requested operation exactly (with "infinite precision"), and truncate  requested operation exactly (with "infinite precision"), and truncate
 the result to the destination variable precision.  Even if the user has  the result to the destination variable precision.  Even if the user has
 asked for a very high precision, MP will not calculate with superfluous  asked for a very high precision, GMP will not calculate with
 digits.  For example, if two low-precision numbers of nearly equal  superfluous digits.  For example, if two low-precision numbers of
 magnitude are added, the precision of the result will be limited to  nearly equal magnitude are added, the precision of the result will be
 what is required to represent the result accurately.  limited to what is required to represent the result accurately.
   
    The MP floating-point functions are *not* intended as a smooth     The GMP floating-point functions are _not_ intended as a smooth
 extension to the IEEE P754 arithmetic.  Specifically, the results  extension to the IEEE P754 arithmetic.  Specifically, the results
 obtained on one computer often differs from the results obtained on a  obtained on one computer often differs from the results obtained on a
 computer with a different word size.  computer with a different word size.
Line 73  computer with a different word size.
Line 669  computer with a different word size.
 * Miscellaneous Float Functions::  * Miscellaneous Float Functions::
   
   
 File: gmp.info,  Node: Initializing Floats,  Next: Assigning Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Initializing Floats,  Next: Assigning Floats,  Prev: Floating-point Functions,  Up: Floating-point Functions
   
 Initialization and Assignment Functions  Initialization Functions
 =======================================  ========================
   
  - Function: void mpf_set_default_prec (unsigned long int PREC)   - Function: void mpf_set_default_prec (unsigned long int PREC)
      Set the default precision to be *at least* PREC bits.  All       Set the default precision to be *at least* PREC bits.  All
Line 107  purpose.
Line 703  purpose.
      {       {
        mpf_t x, y;         mpf_t x, y;
        mpf_init (x);                    /* use default precision */         mpf_init (x);                    /* use default precision */
        mpf_init2 (y, 256);              /* precision *at least* 256 bits */         mpf_init2 (y, 256);              /* precision _at least_ 256 bits */
        ...         ...
        /* Unless the program is about to exit, do ... */         /* Unless the program is about to exit, do ... */
        mpf_clear (x);         mpf_clear (x);
Line 133  numbers.
Line 729  numbers.
      low-level function that does not change the allocation.  The PREC       low-level function that does not change the allocation.  The PREC
      argument must not be larger that the precision previously returned       argument must not be larger that the precision previously returned
      by `mpf_get_prec'.  It is crucial that the precision of ROP is       by `mpf_get_prec'.  It is crucial that the precision of ROP is
      ultimately reset to exactly the value returned by `mpf_get_prec'.       ultimately reset to exactly the value returned by `mpf_get_prec'
        before the first call to `mpf_set_prec_raw'.
   
   
 File: gmp.info,  Node: Assigning Floats,  Next: Simultaneous Float Init & Assign,  Prev: Initializing Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Assigning Floats,  Next: Simultaneous Float Init & Assign,  Prev: Initializing Floats,  Up: Floating-point Functions
   
 Assignment Functions  Assignment Functions
 --------------------  ====================
   
    These functions assign new values to already initialized floats     These functions assign new values to already initialized floats
 (*note Initializing Floats::.).  (*note Initializing Floats::).
   
  - Function: void mpf_set (mpf_t ROP, mpf_t OP)   - Function: void mpf_set (mpf_t ROP, mpf_t OP)
  - Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)   - Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
Line 168  Assignment Functions
Line 765  Assignment Functions
      This is so that numbers like `0.23' are not interpreted as octal.       This is so that numbers like `0.23' are not interpreted as octal.
   
      White space is allowed in the string, and is simply ignored.       White space is allowed in the string, and is simply ignored.
        [This is not really true; white-space is ignored in the beginning
        of the string and within the mantissa, but not in other places,
        such as after a minus sign or in the exponent.  We are considering
        changing the definition of this function, making it fail when
        there is any white-space in the input, since that makes a lot of
        sense.  Please tell us your opinion about this change.  Do you
        really want it to accept "3 14" as meaning 314 as it does now?]
   
      This function returns 0 if the entire string up to the '\0' is a       This function returns 0 if the entire string up to the '\0' is a
      valid number in base BASE.  Otherwise it returns -1.       valid number in base BASE.  Otherwise it returns -1.
   
    - Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
        Swap the values ROP1 and ROP2 efficiently.
   
   
 File: gmp.info,  Node: Simultaneous Float Init & Assign,  Next: Converting Floats,  Prev: Assigning Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Simultaneous Float Init & Assign,  Next: Converting Floats,  Prev: Assigning Floats,  Up: Floating-point Functions
   
 Combined Initialization and Assignment Functions  Combined Initialization and Assignment Functions
 ------------------------------------------------  ================================================
   
    For convenience, MP provides a parallel series of initialize-and-set     For convenience, GMP provides a parallel series of
 functions which initialize the output and then store the value there.  initialize-and-set functions which initialize the output and then store
 These functions' names have the form `mpf_init_set...'  the value there.  These functions' names have the form `mpf_init_set...'
   
    Once the float has been initialized by any of the `mpf_init_set...'     Once the float has been initialized by any of the `mpf_init_set...'
 functions, it can be used as the source or destination operand for the  functions, it can be used as the source or destination operand for the
Line 222  Conversion Functions
Line 829  Conversion Functions
      N_DIGITS is 0, the maximum number of digits accurately       N_DIGITS is 0, the maximum number of digits accurately
      representable by OP.       representable by OP.
   
      If STR is NULL, space for the mantissa is allocated using the       If STR is `NULL', space for the mantissa is allocated using the
      default allocation function, and a pointer to the string is       default allocation function.
      returned.  
   
      If STR is not NULL, it should point to a block of storage enough       If STR is not `NULL', it should point to a block of storage enough
      large for the mantissa, i.e., N_DIGITS + 2.  The two extra bytes       large for the mantissa, i.e., N_DIGITS + 2.  The two extra bytes
      are for a possible minus sign, and for the terminating null       are for a possible minus sign, and for the terminating null
      character.       character.
Line 236  Conversion Functions
Line 842  Conversion Functions
      If N_DIGITS is 0, the maximum number of digits meaningfully       If N_DIGITS is 0, the maximum number of digits meaningfully
      achievable from the precision of OP will be generated.  Note that       achievable from the precision of OP will be generated.  Note that
      the space requirements for STR in this case will be impossible for       the space requirements for STR in this case will be impossible for
      the user to predetermine.  Therefore, you need to pass NULL for       the user to predetermine.  Therefore, you need to pass `NULL' for
      the string argument whenever N_DIGITS is 0.       the string argument whenever N_DIGITS is 0.
   
      The generated string is a fraction, with an implicit radix point       The generated string is a fraction, with an implicit radix point
Line 244  Conversion Functions
Line 850  Conversion Functions
      number 3.1416 would be returned as "31416" in the string and 1       number 3.1416 would be returned as "31416" in the string and 1
      written at EXPPTR.       written at EXPPTR.
   
        A pointer to the result string is returned.  This pointer will
        will either equal STR, or if that is `NULL', will point to the
        allocated storage.
   
   
 File: gmp.info,  Node: Float Arithmetic,  Next: Float Comparison,  Prev: Converting Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Float Arithmetic,  Next: Float Comparison,  Prev: Converting Floats,  Up: Floating-point Functions
   
Line 269  Arithmetic Functions
Line 879  Arithmetic Functions
   
    Division is undefined if the divisor is zero, and passing a zero     Division is undefined if the divisor is zero, and passing a zero
 divisor to the divide functions will make these functions intentionally  divisor to the divide functions will make these functions intentionally
 divide by zero.  This gives the user the possibility to handle  divide by zero.  This lets the user handle arithmetic exceptions in
 arithmetic exceptions in these functions in the same manner as other  these functions in the same manner as other arithmetic exceptions.
 arithmetic exceptions.  
   
  - Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)   - Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
  - Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t   - Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
Line 284  arithmetic exceptions.
Line 893  arithmetic exceptions.
  - Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)   - Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
      Set ROP to the square root of OP.       Set ROP to the square root of OP.
   
    - Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
             OP2)
        Set ROP to OP1 raised to the power OP2.
   
  - Function: void mpf_neg (mpf_t ROP, mpf_t OP)   - Function: void mpf_neg (mpf_t ROP, mpf_t OP)
      Set ROP to -OP.       Set ROP to -OP.
   
Line 331  Input and Output Functions
Line 944  Input and Output Functions
 ==========================  ==========================
   
    Functions that perform input from a stdio stream, and functions that     Functions that perform input from a stdio stream, and functions that
 output to a stdio stream.  Passing a NULL pointer for a STREAM argument  output to a stdio stream.  Passing a `NULL' pointer for a STREAM
 to any of these functions will make them read from `stdin' and write to  argument to any of these functions will make them read from `stdin' and
 `stdout', respectively.  write to `stdout', respectively.
   
    When using any of these functions, it is a good idea to include     When using any of these functions, it is a good idea to include
 `stdio.h' before `gmp.h', since that will allow `gmp.h' to define  `stdio.h' before `gmp.h', since that will allow `gmp.h' to define
Line 378  File: gmp.info,  Node: Miscellaneous Float Functions, 
Line 991  File: gmp.info,  Node: Miscellaneous Float Functions, 
 Miscellaneous Functions  Miscellaneous Functions
 =======================  =======================
   
    - Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
    - Function: void mpf_floor (mpf_t ROP, mpf_t OP)
    - Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
        Set ROP to OP rounded to an integer.  `mpf_ceil' rounds to the
        next higher integer, `mpf_floor' to the next lower, and
        `mpf_trunc' to the integer towards zero.
   
    - Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
             unsigned long int NBITS)
        Generate a uniformly distributed random float in ROP, such that 0
        <= ROP < 1, with NBITS significant bits in the mantissa.
   
        The variable STATE must be initialized by calling one of the
        `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
  - Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t   - Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
           MAX_EXP)            MAX_EXP)
      Generate a random float of at most MAX_SIZE limbs, with long       Generate a random float of at most MAX_SIZE limbs, with long
Line 387  Miscellaneous Functions
Line 1016  Miscellaneous Functions
      this kind of random numbers have proven to be more likely to       this kind of random numbers have proven to be more likely to
      trigger corner-case bugs.  Negative random numbers are generated       trigger corner-case bugs.  Negative random numbers are generated
      when MAX_SIZE is negative.       when MAX_SIZE is negative.
   
   
 File: gmp.info,  Node: Low-level Functions,  Next: BSD Compatible Functions,  Prev: Floating-point Functions,  Up: Top  
   
 Low-level Functions  
 *******************  
   
    This chapter describes low-level MP functions, used to implement the  
 high-level MP functions, but also intended for time-critical user code.  
   
    These functions start with the prefix `mpn_'.  
   
    The `mpn' functions are designed to be as fast as possible, *not* to  
 provide a coherent calling interface.  The different functions have  
 somewhat similar interfaces, but there are variations that make them  
 hard to use.  These functions do as little as possible apart from the  
 real multiple precision computation, so that no time is spent on things  
 that not all callers need.  
   
    A source operand is specified by a pointer to the least significant  
 limb and a limb count.  A destination operand is specified by just a  
 pointer.  It is the responsibility of the caller to ensure that the  
 destination has enough space for storing the result.  
   
    With this way of specifying operands, it is possible to perform  
 computations on subranges of an argument, and store the result into a  
 subrange of a destination.  
   
    A common requirement for all functions is that each source area  
 needs at least one limb.  No size argument may be zero.  
   
    The `mpn' functions is the base for the implementation of the `mpz_',  
 `mpf_', and `mpq_' functions.  
   
    This example adds the number beginning at SRC1_PTR and the number  
 beginning at SRC2_PTR and writes the sum at DEST_PTR.  All areas have  
 SIZE limbs.  
   
      cy = mpn_add_n (dest_ptr, src1_ptr, src2_ptr, size)  
   
 In the notation used here, a source operand is identified by the  
 pointer to the least significant limb, and the limb count in braces.  
 For example, {s1_ptr, s1_size}.  
   
  - Function: mp_limb_t mpn_add_n (mp_limb_t * DEST_PTR, const mp_limb_t  
           * SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)  
      Add {SRC1_PTR, SIZE} and {SRC2_PTR, SIZE}, and write the SIZE  
      least significant limbs of the result to DEST_PTR.  Return carry,  
      either 0 or 1.  
   
      This is the lowest-level function for addition.  It is the  
      preferred function for addition, since it is written in assembly  
      for most targets.  For addition of a variable to itself (i.e.,  
      SRC1_PTR equals SRC2_PTR, use `mpn_lshift' with a count of 1 for  
      optimal speed.  
   
  - Function: mp_limb_t mpn_add_1 (mp_limb_t * DEST_PTR, const mp_limb_t  
           * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)  
      Add {SRC1_PTR, SIZE} and SRC2_LIMB, and write the SIZE least  
      significant limbs of the result to DEST_PTR.  Return carry, either  
      0 or 1.  
   
  - Function: mp_limb_t mpn_add (mp_limb_t * DEST_PTR, const mp_limb_t *  
           SRC1_PTR, mp_size_t SRC1_SIZE, const mp_limb_t * SRC2_PTR,  
           mp_size_t SRC2_SIZE)  
      Add {SRC1_PTR, SRC1_SIZE} and {SRC2_PTR, SRC2_SIZE}, and write the  
      SRC1_SIZE least significant limbs of the result to DEST_PTR.  
      Return carry, either 0 or 1.  
   
      This function requires that SRC1_SIZE is greater than or equal to  
      SRC2_SIZE.  
   
  - Function: mp_limb_t mpn_sub_n (mp_limb_t * DEST_PTR, const mp_limb_t  
           * SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)  
      Subtract {SRC2_PTR, SRC2_SIZE} from {SRC1_PTR, SIZE}, and write  
      the SIZE least significant limbs of the result to DEST_PTR.  
      Return borrow, either 0 or 1.  
   
      This is the lowest-level function for subtraction.  It is the  
      preferred function for subtraction, since it is written in  
      assembly for most targets.  
   
  - Function: mp_limb_t mpn_sub_1 (mp_limb_t * DEST_PTR, const mp_limb_t  
           * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)  
      Subtract SRC2_LIMB from {SRC1_PTR, SIZE}, and write the SIZE least  
      significant limbs of the result to DEST_PTR.  Return borrow,  
      either 0 or 1.  
   
  - Function: mp_limb_t mpn_sub (mp_limb_t * DEST_PTR, const mp_limb_t *  
           SRC1_PTR, mp_size_t SRC1_SIZE, const mp_limb_t * SRC2_PTR,  
           mp_size_t SRC2_SIZE)  
      Subtract {SRC2_PTR, SRC2_SIZE} from {SRC1_PTR, SRC1_SIZE}, and  
      write the SRC1_SIZE least significant limbs of the result to  
      DEST_PTR.  Return borrow, either 0 or 1.  
   
      This function requires that SRC1_SIZE is greater than or equal to  
      SRC2_SIZE.  
   
  - Function: void mpn_mul_n (mp_limb_t * DEST_PTR, const mp_limb_t *  
           SRC1_PTR, const mp_limb_t * SRC2_PTR, mp_size_t SIZE)  
      Multiply {SRC1_PTR, SIZE} and {SRC2_PTR, SIZE}, and write the  
      *entire* result to DEST_PTR.  
   
      The destination has to have space for 2SIZE limbs, even if the  
      significant result might be one limb smaller.  
   
  - Function: mp_limb_t mpn_mul_1 (mp_limb_t * DEST_PTR, const mp_limb_t  
           * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)  
      Multiply {SRC1_PTR, SIZE} and SRC2_LIMB, and write the SIZE least  
      significant limbs of the product to DEST_PTR.  Return the most  
      significant limb of the product.  
   
      This is a low-level function that is a building block for general  
      multiplication as well as other operations in MP.  It is written  
      in assembly for most targets.  
   
      Don't call this function if SRC2_LIMB is a power of 2; use  
      `mpn_lshift' with a count equal to the logarithm of SRC2_LIMB  
      instead, for optimal speed.  
   
  - Function: mp_limb_t mpn_addmul_1 (mp_limb_t * DEST_PTR, const  
           mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)  
      Multiply {SRC1_PTR, SIZE} and SRC2_LIMB, and add the SIZE least  
      significant limbs of the product to {DEST_PTR, SIZE} and write the  
      result to DEST_PTR DEST_PTR.  Return the most significant limb of  
      the product, plus carry-out from the addition.  
   
      This is a low-level function that is a building block for general  
      multiplication as well as other operations in MP.  It is written  
      in assembly for most targets.  
   
  - Function: mp_limb_t mpn_submul_1 (mp_limb_t * DEST_PTR, const  
           mp_limb_t * SRC1_PTR, mp_size_t SIZE, mp_limb_t SRC2_LIMB)  
      Multiply {SRC1_PTR, SIZE} and SRC2_LIMB, and subtract the SIZE  
      least significant limbs of the product from {DEST_PTR, SIZE} and  
      write the result to DEST_PTR.  Return the most significant limb of  
      the product, minus borrow-out from the subtraction.  
   
      This is a low-level function that is a building block for general  
      multiplication and division as well as other operations in MP.  It  
      is written in assembly for most targets.  
   
  - Function: mp_limb_t mpn_mul (mp_limb_t * DEST_PTR, const mp_limb_t *  
           SRC1_PTR, mp_size_t SRC1_SIZE, const mp_limb_t * SRC2_PTR,  
           mp_size_t SRC2_SIZE)  
      Multiply {SRC1_PTR, SRC1_SIZE} and {SRC2_PTR, SRC2_SIZE}, and  
      write the result to DEST_PTR.  Return the most significant limb of  
      the result.  
   
      The destination has to have space for SRC1_SIZE + SRC1_SIZE limbs,  
      even if the result might be one limb smaller.  
   
      This function requires that SRC1_SIZE is greater than or equal to  
      SRC2_SIZE.  The destination must be distinct from either input  
      operands.  
   
  - Function: mp_size_t mpn_divrem (mp_limb_t * R1P, mp_size_t XSIZE,  
           mp_limb_t * RS2P, mp_size_t RS2SIZE, const mp_limb_t * S3P,  
           mp_size_t S3SIZE)  
      Divide {RS2P, RS2SIZE} by {S3P, S3SIZE}, and write the quotient at  
      R1P, with the exception of the most significant limb, which is  
      returned.  The remainder replaces the dividend at RS2P.  
   
      In addition to an integer quotient, XSIZE fraction limbs are  
      developed, and stored after the integral limbs.  For most usages,  
      XSIZE will be zero.  
   
      It is required that RS2SIZE is greater than or equal to S3SIZE.  
      It is required that the most significant bit of the divisor is set.  
   
      If the quotient is not needed, pass RS2P + S3SIZE as R1P.  Aside  
      from that special case, no overlap between arguments is permitted.  
   
      Return the most significant limb of the quotient, either 0 or 1.  
   
      The area at R1P needs to be RS2SIZE - S3SIZE + XSIZE limbs large.  
   
  - Function: mp_limb_t mpn_divrem_1 (mp_limb_t * R1P, mp_size_t XSIZE,  
           mp_limb_t * S2P, mp_size_t S2SIZE, mp_limb_t S3LIMB)  
      Divide {S2P, S2SIZE} by S3LIMB, and write the quotient at R1P.  
      Return the remainder.  
   
      In addition to an integer quotient, XSIZE fraction limbs are  
      developed, and stored after the integral limbs.  For most usages,  
      XSIZE will be zero.  
   
      The areas at R1P and S2P have to be identical or completely  
      separate, not partially overlapping.  
   
  - Function: mp_size_t mpn_divmod (mp_limb_t * R1P, mp_limb_t * RS2P,  
           mp_size_t RS2SIZE, const mp_limb_t * S3P, mp_size_t S3SIZE)  
      *This interface is obsolete.  It will disappear from future  
      releases.  Use `mpn_divrem' in its stead.*  
   
  - Function: mp_limb_t mpn_divmod_1 (mp_limb_t * R1P, mp_limb_t * S2P,  
           mp_size_t S2SIZE, mp_limb_t S3LIMB)  
      *This interface is obsolete.  It will disappear from future  
      releases.  Use `mpn_divrem_1' in its stead.*  
   
  - Function: mp_limb_t mpn_mod_1 (mp_limb_t * S1P, mp_size_t S1SIZE,  
           mp_limb_t S2LIMB)  
      Divide {S1P, S1SIZE} by S2LIMB, and return the remainder.  
   
  - Function: mp_limb_t mpn_preinv_mod_1 (mp_limb_t * S1P, mp_size_t  
           S1SIZE, mp_limb_t S2LIMB, mp_limb_t S3LIMB)  
      *This interface is obsolete.  It will disappear from future  
      releases.  Use `mpn_mod_1' in its stead.*  
   
  - Function: mp_limb_t mpn_bdivmod (mp_limb_t * DEST_PTR, mp_limb_t *  
           S1P, mp_size_t S1SIZE, const mp_limb_t * S2P, mp_size_t  
           S2SIZE, unsigned long int D)  
      The function puts the low [D/BITS_PER_MP_LIMB] limbs of Q = {S1P,  
      S1SIZE}/{S2P, S2SIZE} mod 2^D at DEST_PTR, and returns the high D  
      mod BITS_PER_MP_LIMB bits of Q.  
   
      {S1P, S1SIZE} - Q * {S2P, S2SIZE} mod 2^(S1SIZE*BITS_PER_MP_LIMB)  
      is placed at S1P.  Since the low [D/BITS_PER_MP_LIMB] limbs of  
      this difference are zero, it is possible to overwrite the low  
      limbs at S1P with this difference, provided DEST_PTR <= S1P.  
   
      This function requires that S1SIZE * BITS_PER_MP_LIMB >= D, and  
      that {S2P, S2SIZE} is odd.  
   
      *This interface is preliminary.  It might change incompatibly in  
      future revisions.*  
   
  - Function: mp_limb_t mpn_lshift (mp_limb_t * DEST_PTR, const  
           mp_limb_t * SRC_PTR, mp_size_t SRC_SIZE, unsigned long int  
           COUNT)  
      Shift {SRC_PTR, SRC_SIZE} COUNT bits to the left, and write the  
      SRC_SIZE least significant limbs of the result to DEST_PTR.  COUNT  
      might be in the range 1 to n - 1, on an n-bit machine. The bits  
      shifted out to the left are returned.  
   
      Overlapping of the destination space and the source space is  
      allowed in this function, provided DEST_PTR >= SRC_PTR.  
   
      This function is written in assembly for most targets.  
   
  - Function: mp_limp_t mpn_rshift (mp_limb_t * DEST_PTR, const  
           mp_limb_t * SRC_PTR, mp_size_t SRC_SIZE, unsigned long int  
           COUNT)  
      Shift {SRC_PTR, SRC_SIZE} COUNT bits to the right, and write the  
      SRC_SIZE most significant limbs of the result to DEST_PTR.  COUNT  
      might be in the range 1 to n - 1, on an n-bit machine.  The bits  
      shifted out to the right are returned.  
   
      Overlapping of the destination space and the source space is  
      allowed in this function, provided DEST_PTR <= SRC_PTR.  
   
      This function is written in assembly for most targets.  
   
  - Function: int mpn_cmp (const mp_limb_t * SRC1_PTR, const mp_limb_t *  
           SRC2_PTR, mp_size_t SIZE)  
      Compare {SRC1_PTR, SIZE} and {SRC2_PTR, SIZE} and return a  
      positive value if src1 > src2, 0 of they are equal, and a negative  
      value if src1 < src2.  
   
  - Function: mp_size_t mpn_gcd (mp_limb_t * DEST_PTR, mp_limb_t *  
           SRC1_PTR, mp_size_t SRC1_SIZE, mp_limb_t * SRC2_PTR,  
           mp_size_t SRC2_SIZE)  
      Puts at DEST_PTR the greatest common divisor of {SRC1_PTR,  
      SRC1_SIZE} and {SRC2_PTR, SRC2_SIZE}; both source operands are  
      destroyed by the operation.  The size in limbs of the greatest  
      common divisor is returned.  
   
      {SRC1_PTR, SRC1_SIZE} must be odd, and {SRC2_PTR, SRC2_SIZE} must  
      have at least as many bits as {SRC1_PTR, SRC1_SIZE}.  
   
      *This interface is preliminary.  It might change incompatibly in  
      future revisions.*  
   
  - Function: mp_limb_t mpn_gcd_1 (const mp_limb_t * SRC1_PTR, mp_size_t  
           SRC1_SIZE, mp_limb_t SRC2_LIMB)  
      Return the greatest common divisor of {SRC1_PTR, SRC1_SIZE} and  
      SRC2_LIMB, where SRC2_LIMB (as well as SRC1_SIZE) must be  
      different from 0.  
   
  - Function: mp_size_t mpn_gcdext (mp_limb_t * R1P, mp_limb_t * R2P,  
           mp_limb_t * S1P, mp_size_t S1SIZE, mp_limb_t * S2P, mp_size_t  
           S2SIZE)  
      Puts at R1P the greatest common divisor of {S1P, S1SIZE} and {S2P,  
      S2SIZE}.  The first cofactor is written at R2P.  Both source  
      operands are destroyed by the operation.  The size in limbs of the  
      greatest common divisor is returned.  
   
      *This interface is preliminary.  It might change incompatibly in  
      future revisions.*  
   
  - Function: mp_size_t mpn_sqrtrem (mp_limb_t * R1P, mp_limb_t * R2P,  
           const mp_limb_t * SP, mp_size_t SIZE)  
      Compute the square root of {SP, SIZE} and put the result at R1P.  
      Write the remainder at R2P, unless R2P is NULL.  
   
      Return the size of the remainder, whether R2P was NULL or non-NULL.  
      Iff the operand was a perfect square, the return value will be 0.  
   
      The areas at R1P and SP have to be distinct.  The areas at R2P and  
      SP have to be identical or completely separate, not partially  
      overlapping.  
   
      The area at R1P needs to have space for ceil(SIZE/2) limbs.  The  
      area at R2P needs to be SIZE limbs large.  
   
      *This interface is preliminary.  It might change incompatibly in  
      future revisions.*  
   
  - Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,  
           mp_limb_t * S1P, mp_size_t S1SIZE)  
      Convert {S1P, S1SIZE} to a raw unsigned char array in base BASE.  
      The string is not in ASCII; to convert it to printable format, add  
      the ASCII codes for `0' or `A', depending on the base and range.  
      There may be leading zeros in the string.  
   
      The area at S1P is clobbered.  
   
      Return the number of characters in STR.  
   
      The area at STR has to have space for the largest possible number  
      represented by a S1SIZE long limb array, plus one extra character.  
   
  - Function: mp_size_t mpn_set_str (mp_limb_t * R1P, const char *STR,  
           size_t strsize, int BASE)  
      Convert the raw unsigned char array at STR of length STRSIZE to a  
      limb array {S1P, S1SIZE}.  The base of STR is BASE.  
   
      Return the number of limbs stored in R1P.  
   
  - Function: unsigned long int mpn_scan0 (const mp_limb_t * S1P,  
           unsigned long int BIT)  
      Scan S1P from bit position BIT for the next clear bit.  
   
      It is required that there be a clear bit within the area at S1P at  
      or beyond bit position BIT, so that the function has something to  
      return.  
   
      *This interface is preliminary.  It might change incompatibly in  
      future revisions.*  
   
  - Function: unsigned long int mpn_scan1 (const mp_limb_t * S1P,  
           unsigned long int BIT)  
      Scan S1P from bit position BIT for the next set bit.  
   
      It is required that there be a set bit within the area at S1P at or  
      beyond bit position BIT, so that the function has something to  
      return.  
   
      *This interface is preliminary.  It might change incompatibly in  
      future revisions.*  
   
  - Function: void mpn_random2 (mp_limb_t * R1P, mp_size_t R1SIZE)  
      Generate a random number of length R1SIZE with long strings of  
      zeros and ones in the binary representation, and store it at R1P.  
   
      The generated random numbers are intended for testing the  
      correctness of the implementation of the `mpn' routines.  
   
  - Function: unsigned long int mpn_popcount (const mp_limb_t * S1P,  
           unsigned long int SIZE)  
      Count the number of set bits in {S1P, SIZE}.  
   
  - Function: unsigned long int mpn_hamdist (const mp_limb_t * S1P,  
           const mp_limb_t * S2P, unsigned long int SIZE)  
      Compute the hamming distance between {S1P, SIZE} and {S2P, SIZE}.  
   
  - Function: int mpn_perfect_square_p (const mp_limb_t * S1P, mp_size_t  
           SIZE)  
      Return non-zero iff {S1P, SIZE} is a perfect square.  
   
   
 File: gmp.info,  Node: BSD Compatible Functions,  Next: Custom Allocation,  Prev: Low-level Functions,  Up: Top  
   
 Berkeley MP Compatible Functions  
 ********************************  
   
    These functions are intended to be fully compatible with the  
 Berkeley MP library which is available on many BSD derived U*ix systems.  
   
    The original Berkeley MP library has a usage restriction: you cannot  
 use the same variable as both source and destination in a single  
 function call.  The compatible functions in GNU MP do not share this  
 restriction--inputs and outputs may overlap.  
   
    It is not recommended that new programs are written using these  
 functions.  Apart from the incomplete set of functions, the interface  
 for initializing `MINT' objects is more error prone, and the `pow'  
 function collides with `pow' in `libm.a'.  
   
    Include the header `mp.h' to get the definition of the necessary  
 types and functions.  If you are on a BSD derived system, make sure to  
 include GNU `mp.h' if you are going to link the GNU `libmp.a' to you  
 program.  This means that you probably need to give the -I<dir> option  
 to the compiler, where <dir> is the directory where you have GNU `mp.h'.  
   
  - Function: MINT * itom (signed short int INITIAL_VALUE)  
      Allocate an integer consisting of a `MINT' object and dynamic limb  
      space.  Initialize the integer to INITIAL_VALUE.  Return a pointer  
      to the `MINT' object.  
   
  - Function: MINT * xtom (char *INITIAL_VALUE)  
      Allocate an integer consisting of a `MINT' object and dynamic limb  
      space.  Initialize the integer from INITIAL_VALUE, a hexadecimal,  
      '\0'-terminate C string.  Return a pointer to the `MINT' object.  
   
  - Function: void move (MINT *SRC, MINT *DEST)  
      Set DEST to SRC by copying.  Both variables must be previously  
      initialized.  
   
  - Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)  
      Add SRC_1 and SRC_2 and put the sum in DESTINATION.  
   
  - Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)  
      Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.  
   
  - Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)  
      Multiply SRC_1 and SRC_2 and put the product in DESTINATION.  
   
  - Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,  
           MINT *REMAINDER)  
  - Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT  
           *QUOTIENT, signed short int *REMAINDER)  
      Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod  
      DIVISOR.  The quotient is rounded towards zero; the remainder has  
      the same sign as the dividend unless it is zero.  
   
      Some implementations of these functions work differently--or not  
      at all--for negative arguments.  
   
  - Function: void msqrt (MINT *OPERAND, MINT *ROOT, MINT *REMAINDER)  
      Set ROOT to the truncated integer part of the square root of  
      OPERAND.  Set REMAINDER to OPERAND-ROOT*ROOT, (i.e., zero if  
      OPERAND is a perfect square).  
   
      If ROOT and REMAINDER are the same variable, the results are  
      undefined.  
   
  - Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)  
      Set DEST to (BASE raised to EXP) modulo MOD.  
   
  - Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)  
      Set DEST to BASE raised to EXP.  
   
  - Function: void gcd (MINT *OPERAND1, MINT *OPERAND2, MINT *RES)  
      Set RES to the greatest common divisor of OPERAND1 and OPERAND2.  
   
  - Function: int mcmp (MINT *OPERAND1, MINT *OPERAND2)  
      Compare OPERAND1 and OPERAND2.  Return a positive value if  
      OPERAND1 > OPERAND2, zero if OPERAND1 = OPERAND2, and a negative  
      value if OPERAND1 < OPERAND2.  
   
  - Function: void min (MINT *DEST)  
      Input a decimal string from `stdin', and put the read integer in  
      DEST.  SPC and TAB are allowed in the number string, and are  
      ignored.  
   
  - Function: void mout (MINT *SRC)  
      Output SRC to `stdout', as a decimal string.  Also output a  
      newline.  
   
  - Function: char * mtox (MINT *OPERAND)  
      Convert OPERAND to a hexadecimal string, and return a pointer to  
      the string.  The returned string is allocated using the default  
      memory allocation function, `malloc' by default.  
   
  - Function: void mfree (MINT *OPERAND)  
      De-allocate, the space used by OPERAND.  *This function should  
      only be passed a value returned by `itom' or `xtom'.*  
   
   
 File: gmp.info,  Node: Custom Allocation,  Next: Contributors,  Prev: BSD Compatible Functions,  Up: Top  
   
 Custom Allocation  
 *****************  
   
    By default, the MP functions use `malloc', `realloc', and `free' for  
 memory allocation.  If `malloc' or `realloc' fails, the MP library  
 terminates execution after printing a fatal error message to standard  
 error.  
   
    For some applications, you may wish to allocate memory in other  
 ways, or you may not want to have a fatal error when there is no more  
 memory available.  To accomplish this, you can specify alternative  
 memory allocation functions.  
   
  - Function: void mp_set_memory_functions (  
           void *(*ALLOC_FUNC_PTR) (size_t),  
           void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),  
           void (*FREE_FUNC_PTR) (void *, size_t))  
      Replace the current allocation functions from the arguments.  If  
      an argument is NULL, the corresponding default function is  
      retained.  
   
      *Make sure to call this function in such a way that there are no  
      active MP objects that were allocated using the previously active  
      allocation function!  Usually, that means that you have to call  
      this function before any other MP function.*  
   
    The functions you supply should fit the following declarations:  
   
  - Function: void * allocate_function (size_t ALLOC_SIZE)  
      This function should return a pointer to newly allocated space  
      with at least ALLOC_SIZE storage units.  
   
  - Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,  
           size_t NEW_SIZE)  
      This function should return a pointer to newly allocated space of  
      at least NEW_SIZE storage units, after copying at least the first  
      OLD_SIZE storage units from PTR.  It should also de-allocate the  
      space at PTR.  
   
      You can assume that the space at PTR was formerly returned from  
      `allocate_function' or `reallocate_function', for a request for  
      OLD_SIZE storage units.  
   
  - Function: void deallocate_function (void *PTR, size_t SIZE)  
      De-allocate the space pointed to by PTR.  
   
      You can assume that the space at PTR was formerly returned from  
      `allocate_function' or `reallocate_function', for a request for  
      SIZE storage units.  
   
    (A "storage unit" is the unit in which the `sizeof' operator returns  
 the size of an object, normally an 8 bit byte.)  
   
   
 File: gmp.info,  Node: Contributors,  Next: References,  Prev: Custom Allocation,  Up: Top  
   
 Contributors  
 ************  
   
    I would like to thank Gunnar Sjoedin and Hans Riesel for their help  
 with mathematical problems, Richard Stallman for his help with design  
 issues and for revising the first version of this manual, Brian Beuning  
 and Doug Lea for their testing of early versions of the library.  
   
    John Amanatides of York University in Canada contributed the function  
 `mpz_probab_prime_p'.  
   
    Paul Zimmermann of Inria sparked the development of GMP 2, with his  
 comparisons between bignum packages.  
   
    Ken Weber (Kent State University, Universidade Federal do Rio Grande  
 do Sul) contributed `mpz_gcd', `mpz_divexact', `mpn_gcd', and  
 `mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2.  
   
    Per Bothner of Cygnus Support helped to set up MP to use Cygnus'  
 configure.  He has also made valuable suggestions and tested numerous  
 intermediary releases.  
   
    Joachim Hollman was involved in the design of the `mpf' interface,  
 and in the `mpz' design revisions for version 2.  
   
    Bennet Yee contributed the functions `mpz_jacobi' and `mpz_legendre'.  
   
    Andreas Schwab contributed the files `mpn/m68k/lshift.S' and  
 `mpn/m68k/rshift.S'.  
   
    The development of floating point functions of GNU MP 2, were  
 supported in part by the ESPRIT-BRA (Basic Research Activities) 6846  
 project POSSO (POlynomial System SOlving).  
   
    GNU MP 2 was finished and released by TMG Datakonsult,  
 Sodermannagatan 5, 116 23 STOCKHOLM, SWEDEN, in cooperation with the  
 IDA Center for Computing Sciences, USA.  
   
   
 File: gmp.info,  Node: References,  Prev: Contributors,  Up: Top  
   
 References  
 **********  
   
    * Donald E. Knuth, "The Art of Computer Programming", vol 2,  
      "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.  
   
    * John D. Lipson, "Elements of Algebra and Algebraic Computing", The  
      Benjamin Cummings Publishing Company Inc, 1981.  
   
    * Richard M. Stallman, "Using and Porting GCC", Free Software  
      Foundation, 1995.  
   
    * Peter L. Montgomery, "Modular Multiplication Without Trial  
      Division", in Mathematics of Computation, volume 44, number 170,  
      April 1985.  
   
    * Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant  
      Integers using Multiplication", in Proceedings of the SIGPLAN  
      PLDI'94 Conference, June 1994.  
   
    * Tudor Jebelean, "An algorithm for exact division", Journal of  
      Symbolic Computation, v. 15, 1993, pp. 169-180.  
   
    * Kenneth Weber, "The accelerated integer GCD algorithm", ACM  
      Transactions on Mathematical Software, v. 21 (March), 1995, pp.  
      111-122.  
   
   
 File: gmp.info,  Node: Concept Index,  Up: Top  
   
 Concept Index  
 *************  
   
 * Menu:  
   
 * gmp.h:                                MP Basics.  
 * mp.h:                                 BSD Compatible Functions.  
 * Arithmetic functions <1>:             Float Arithmetic.  
 * Arithmetic functions:                 Integer Arithmetic.  
 * Bit manipulation functions:           Integer Logic and Bit Fiddling.  
 * BSD MP compatible functions:          BSD Compatible Functions.  
 * Comparison functions:                 Float Comparison.  
 * Conditions for copying GNU MP:        Copying.  
 * Conversion functions <1>:             Converting Integers.  
 * Conversion functions:                 Converting Floats.  
 * Copying conditions:                   Copying.  
 * Float arithmetic functions:           Float Arithmetic.  
 * Float assignment functions:           Assigning Floats.  
 * Float comparisons functions:          Float Comparison.  
 * Float functions:                      Floating-point Functions.  
 * Float input and output functions:     I/O of Floats.  
 * Floating-point functions:             Floating-point Functions.  
 * Floating-point number:                MP Basics.  
 * I/O functions <1>:                    I/O of Floats.  
 * I/O functions:                        I/O of Integers.  
 * Initialization and assignment functions <1>: Simultaneous Float Init & Assign.  
 * Initialization and assignment functions: Simultaneous Integer Init & Assign.  
 * Input functions <1>:                  I/O of Integers.  
 * Input functions:                      I/O of Floats.  
 * Installation:                         Installing MP.  
 * Integer:                              MP Basics.  
 * Integer arithmetic functions:         Integer Arithmetic.  
 * Integer assignment functions:         Assigning Integers.  
 * Integer conversion functions:         Converting Integers.  
 * Integer functions:                    Integer Functions.  
 * Integer input and output functions:   I/O of Integers.  
 * Limb:                                 MP Basics.  
 * Logical functions:                    Integer Logic and Bit Fiddling.  
 * Low-level functions:                  Low-level Functions.  
 * Miscellaneous float functions:        Miscellaneous Float Functions.  
 * Miscellaneous integer functions:      Miscellaneous Integer Functions.  
 * Output functions <1>:                 I/O of Floats.  
 * Output functions:                     I/O of Integers.  
 * Rational number:                      MP Basics.  
 * Rational number functions:            Rational Number Functions.  
 * Reporting bugs:                       Reporting Bugs.  
 * User-defined precision:               Floating-point Functions.  
   
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