gmp guide Nomenclature and Types ====================== In this manual, "integer" usually means a multiple precision integer, as defined by the GMP library. The C data type for such integers is `mpz_t'. Here are some examples of how to declare such integers: mpz_t sum; struct foo { mpz_t x, y; }; mpz_t vec[20]; "Rational number" means a multiple precision fraction. The C data type for these fractions is `mpq_t'. For example: mpq_t quotient; "Floating point number" or "Float" for short, is an arbitrary precision mantissa with a limited precision exponent. The C data type for such objects is `mpf_t'. For example: mpf_t fp; The floating point functions accept and return exponents in the C type `mp_exp_t'. Currently this is usually a `long', but on some systems it's an `int' for efficiency. A "limb" means the part of a multi-precision number that fits in a single machine word. (We chose this word because a limb of the human body is analogous to a digit, only larger, and containing several digits.) Normally a limb is 32 or 64 bits. The C data type for a limb is `mp_limb_t'. Counts of limbs are represented in the C type `mp_size_t'. Currently this is normally a `long', but on some systems it's an `int' for efficiency. "Random state" means an algorithm selection and current state data. The C data type for such objects is `gmp_randstate_t'. For example: gmp_randstate_t rstate; Also, in general `unsigned long' is used for bit counts and ranges, and `size_t' is used for byte or character counts. Function Classes ================ There are six classes of functions in the GMP library: 1. Functions for signed integer arithmetic, with names beginning with `mpz_'. The associated type is `mpz_t'. There are about 150 functions in this class. (*note Integer Functions::) 2. Functions for rational number arithmetic, with names beginning with `mpq_'. The associated type is `mpq_t'. There are about 40 functions in this class, but the integer functions can be used for arithmetic on the numerator and denominator separately. (*note Rational Number Functions::) 3. Functions for floating-point arithmetic, with names beginning with `mpf_'. The associated type is `mpf_t'. There are about 60 functions is this class. (*note Floating-point Functions::) 4. Functions compatible with Berkeley MP, such as `itom', `madd', and `mult'. The associated type is `MINT'. (*note BSD Compatible Functions::) 5. Fast low-level functions that operate on natural numbers. These are used by the functions in the preceding groups, and you can also call them directly from very time-critical user programs. These functions' names begin with `mpn_'. The associated type is array of `mp_limb_t'. There are about 30 (hard-to-use) functions in this class. (*note Low-level Functions::) 6. Miscellaneous functions. Functions for setting up custom allocation and functions for generating random numbers. (*note Custom Allocation::, and *note Random Number Functions::) Variable Conventions ==================== GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last. GMP lets you use the same variable for both input and output in one call. For example, the main function for integer multiplication, `mpz_mul', can be used to square `x' and put the result back in `x' with mpz_mul (x, x, x); Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details. A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example, void foo (void) { mpz_t n; int i; mpz_init (n); for (i = 1; i < 100; i++) { mpz_mul (n, ...); mpz_fdiv_q (n, ...); ... } mpz_clear (n); } Parameter Conventions ===================== When a GMP variable is used as a function parameter, it's effectively a call-by-reference, meaning if the function stores a value there it will change the original in the caller. Parameters which are input-only can be designated `const' to provoke a compiler error or warning on attempting to modify them. When a function is going to return a GMP result, it should designate a parameter that it sets, like the library functions do. More than one value can be returned by having more than one output parameter, again like the library functions. A `return' of an `mpz_t' etc doesn't return the object, only a pointer, and this is almost certainly not what's wanted. Here's an example accepting an `mpz_t' parameter, doing a calculation, and storing the result to the indicated parameter. void foo (mpz_t result, const mpz_t param, unsigned long n) { unsigned long i; mpz_mul_ui (result, param, n); for (i = 1; i < n; i++) mpz_add_ui (result, result, i*7); } int main (void) { mpz_t r, n; mpz_init (r); mpz_init_set_str (n, "123456", 0); foo (r, n, 20L); gmp_printf ("%Zd\n", r); return 0; } `foo' works even if the mainline passes the same variable for `param' and `result', just like the library functions. But sometimes it's tricky to make that work, and an application might not want to bother supporting that sort of thing. For interest, the GMP types `mpz_t' etc are implemented as one-element arrays of certain structures. This is why declaring a variable creates an object with the fields GMP needs, but then using it as a parameter passes a pointer to the object. Note that the actual fields in each `mpz_t' etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases. Memory Management ================= The GMP types like `mpz_t' are small, containing only a couple of sizes, and pointers to allocated data. Once a variable is initialized, GMP takes care of all space allocation. Additional space is allocated whenever a variable doesn't have enough. `mpz_t' and `mpq_t' variables never reduce their allocated space. Normally this is the best policy, since it avoids frequent reallocation. Applications that need to return memory to the heap at some particular point can use `mpz_realloc2', or clear variables no longer needed. `mpf_t' variables, in the current implementation, use a fixed amount of space, determined by the chosen precision and allocated at initialization, so their size doesn't change. All memory is allocated using `malloc' and friends by default, but this can be changed, see *Note Custom Allocation::. Temporary memory on the stack is also used (via `alloca'), but this can be changed at build-time if desired, see *Note Build Options::. Reentrancy ========== GMP is reentrant and thread-safe, with some exceptions: * If configured with `--enable-alloca=malloc-notreentrant' (or with `--enable-alloca=notreentrant' when `alloca' is not available), then naturally GMP is not reentrant. * `mpf_set_default_prec' and `mpf_init' use a global variable for the selected precision. `mpf_init2' can be used instead, and in the C++ interface an explicit precision to the `mpf_class' constructor. * `mpz_random' and the other old random number functions use a global random state and are hence not reentrant. The newer random number functions that accept a `gmp_randstate_t' parameter can be used instead. * `gmp_randinit' (obsolete) returns an error indication through a global variable, which is not thread safe. Applications are advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp' instead. * `mp_set_memory_functions' uses global variables to store the selected memory allocation functions. * If the memory allocation functions set by a call to `mp_set_memory_functions' (or `malloc' and friends by default) are not reentrant, then GMP will not be reentrant either. * If the standard I/O functions such as `fwrite' are not reentrant then the GMP I/O functions using them will not be reentrant either. * It's safe for two threads to read from the same GMP variable simultaneously, but it's not safe for one to read while the another might be writing, nor for two threads to write simultaneously. It's not safe for two threads to generate a random number from the same `gmp_randstate_t' simultaneously, since this involves an update of that variable. Useful Macros and Constants =========================== - Global Constant: const int mp_bits_per_limb The number of bits per limb. - Macro: __GNU_MP_VERSION - Macro: __GNU_MP_VERSION_MINOR - Macro: __GNU_MP_VERSION_PATCHLEVEL The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively. - Global Constant: const char * const gmp_version The GMP version number, as a null-terminated string, in the form "i.j" or "i.j.k". This release is "4.2.1". Demonstration programs ====================== The `demos' subdirectory has some sample programs using GMP. These aren't built or installed, but there's a `Makefile' with rules for them. For instance, make pexpr ./pexpr 68^975+10 The following programs are provided * `pexpr' is an expression evaluator, the program used on the GMP web page. * The `calc' subdirectory has a similar but simpler evaluator using `lex' and `yacc'. * The `expr' subdirectory is yet another expression evaluator, a library designed for ease of use within a C program. See `demos/expr/README' for more information. * `factorize' is a Pollard-Rho factorization program. * `isprime' is a command-line interface to the `mpz_probab_prime_p' function. * `primes' counts or lists primes in an interval, using a sieve. * `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic class numbers. * The `perl' subdirectory is a comprehensive perl interface to GMP. See `demos/perl/INSTALL' for more information. Documentation is in POD format in `demos/perl/GMP.pm'. As an aside, consideration has been given at various times to some sort of expression evaluation within the main GMP library. Going beyond something minimal quickly leads to matters like user-defined functions, looping, fixnums for control variables, etc, which are considered outside the scope of GMP (much closer to language interpreters or compilers, *Note Language Bindings::.) Something simple for program input convenience may yet be a possibility, a combination of the `expr' demo and the `pexpr' tree back-end perhaps. But for now the above evaluators are offered as illustrations. Efficiency ========== Small Operands On small operands, the time for function call overheads and memory allocation can be significant in comparison to actual calculation. This is unavoidable in a general purpose variable precision library, although GMP attempts to be as efficient as it can on both large and small operands. Static Linking On some CPUs, in particular the x86s, the static `libgmp.a' should be used for maximum speed, since the PIC code in the shared `libgmp.so' will have a small overhead on each function call and global data address. For many programs this will be insignificant, but for long calculations there's a gain to be had. Initializing and Clearing Avoid excessive initializing and clearing of variables, since this can be quite time consuming, especially in comparison to otherwise fast operations like addition. A language interpreter might want to keep a free list or stack of initialized variables ready for use. It should be possible to integrate something like that with a garbage collector too. Reallocations An `mpz_t' or `mpq_t' variable used to hold successively increasing values will have its memory repeatedly `realloc'ed, which could be quite slow or could fragment memory, depending on the C library. If an application can estimate the final size then `mpz_init2' or `mpz_realloc2' can be called to allocate the necessary space from the beginning (*note Initializing Integers::). It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2' is too small, since all functions will do a further reallocation if necessary. Badly overestimating memory required will waste space though. `2exp' Functions It's up to an application to call functions like `mpz_mul_2exp' when appropriate. General purpose functions like `mpz_mul' make no attempt to identify powers of two or other special forms, because such inputs will usually be very rare and testing every time would be wasteful. `ui' and `si' Functions The `ui' functions and the small number of `si' functions exist for convenience and should be used where applicable. But if for example an `mpz_t' contains a value that fits in an `unsigned long' there's no need extract it and call a `ui' function, just use the regular `mpz' function. In-Place Operations `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and `mpf_neg' are fast when used for in-place operations like `mpz_abs(x,x)', since in the current implementation only a single field of `x' needs changing. On suitable compilers (GCC for instance) this is inlined too. `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit from an in-place operation like `mpz_add_ui(x,x,y)', since usually only one or two limbs of `x' will need to be changed. The same applies to the full precision `mpz_add' etc if `y' is small. If `y' is big then cache locality may be helped, but that's all. `mpz_mul' is currently the opposite, a separate destination is slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is only one limb, make a temporary copy of `x' before forming the result. Normally that copying will only be a tiny fraction of the time for the multiply, so this is not a particularly important consideration. `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no attempt to recognise a copy of something to itself, so a call like `mpz_set(x,x)' will be wasteful. Naturally that would never be written deliberately, but if it might arise from two pointers to the same object then a test to avoid it might be desirable. if (x != y) mpz_set (x, y); Note that it's never worth introducing extra `mpz_set' calls just to get in-place operations. If a result should go to a particular variable then just direct it there and let GMP take care of data movement. Divisibility Testing (Small Integers) `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best functions for testing whether an `mpz_t' is divisible by an individual small integer. They use an algorithm which is faster than `mpz_tdiv_ui', but which gives no useful information about the actual remainder, only whether it's zero (or a particular value). However when testing divisibility by several small integers, it's best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo 23*29*31 = 20677 and then test that. The division functions like `mpz_tdiv_q_ui' which give a quotient as well as a remainder are generally a little slower than the remainder-only functions like `mpz_tdiv_ui'. If the quotient is only rarely wanted then it's probably best to just take a remainder and then go back and calculate the quotient if and when it's wanted (`mpz_divexact_ui' can be used if the remainder is zero). Rational Arithmetic The `mpq' functions operate on `mpq_t' values with no common factors in the numerator and denominator. Common factors are checked-for and cast out as necessary. In general, cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations. However, applications that know something about the factorization of the values they're working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it's enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end. The `mpq_numref' and `mpq_denref' macros give access to the numerator and denominator to do things outside the scope of the supplied `mpq' functions. *Note Applying Integer Functions::. The canonical form for rationals allows mixed-type `mpq_t' and integer additions or subtractions to be done directly with multiples of the denominator. This will be somewhat faster than `mpq_add'. For example, /* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z); Number Sequences Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are designed for calculating isolated values. If a range of values is wanted it's probably best to call to get a starting point and iterate from there. Text Input/Output Hexadecimal or octal are suggested for input or output in text form. Power-of-2 bases like these can be converted much more efficiently than other bases, like decimal. For big numbers there's usually nothing of particular interest to be seen in the digits, so the base doesn't matter much. Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers. Integer Functions ***************** This chapter describes the GMP functions for performing integer arithmetic. These functions start with the prefix `mpz_'. GMP integers are stored in objects of type `mpz_t'. * Menu: * Initializing Integers:: * Assigning Integers:: * Simultaneous Integer Init & Assign:: * Converting Integers:: * Integer Arithmetic:: * Integer Division:: * Integer Exponentiation:: * Integer Roots:: * Number Theoretic Functions:: * Integer Comparisons:: * Integer Logic and Bit Fiddling:: * I/O of Integers:: * Integer Random Numbers:: * Integer Import and Export:: * Miscellaneous Integer Functions:: * Integer Special Functions:: Initialization Functions ======================== The functions for integer arithmetic assume that all integer objects are initialized. You do that by calling the function `mpz_init'. For example, { mpz_t integ; mpz_init (integ); ... mpz_add (integ, ...); ... mpz_sub (integ, ...); /* Unless the program is about to exit, do ... */ mpz_clear (integ); } As you can see, you can store new values any number of times, once an object is initialized. - Function: void mpz_init (mpz_t INTEGER) Initialize INTEGER, and set its value to 0. - Function: void mpz_init2 (mpz_t INTEGER, unsigned long N) Initialize INTEGER, with space for N bits, and set its value to 0. N is only the initial space, INTEGER will grow automatically in the normal way, if necessary, for subsequent values stored. `mpz_init2' makes it possible to avoid such reallocations if a maximum size is known in advance. - Function: void mpz_clear (mpz_t INTEGER) Free the space occupied by INTEGER. Call this function for all `mpz_t' variables when you are done with them. - Function: void mpz_realloc2 (mpz_t INTEGER, unsigned long N) Change the space allocated for INTEGER to N bits. The value in INTEGER is preserved if it fits, or is set to 0 if not. This function can be used to increase the space for a variable in order to avoid repeated automatic reallocations, or to decrease it to give memory back to the heap. Assignment Functions ==================== These functions assign new values to already initialized integers (*note Initializing Integers::). - Function: void mpz_set (mpz_t ROP, mpz_t OP) - Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP) - Function: void mpz_set_si (mpz_t ROP, signed long int OP) - Function: void mpz_set_d (mpz_t ROP, double OP) - Function: void mpz_set_q (mpz_t ROP, mpq_t OP) - Function: void mpz_set_f (mpz_t ROP, mpf_t OP) Set the value of ROP from OP. `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an integer. - Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE) Set the value of ROP from STR, a null-terminated C string in base BASE. White space is allowed in the string, and is simply ignored. The BASE may vary from 2 to 62, or if BASE is 0, then the leading characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B' for binary, `0' for octal, or decimal otherwise. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. This function returns 0 if the entire string is a valid number in base BASE. Otherwise it returns -1. - Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2) Swap the values ROP1 and ROP2 efficiently. Combined Initialization and Assignment Functions ================================================ For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store the value there. These functions' names have the form `mpz_init_set...' Here is an example of using one: { mpz_t pie; mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10); ... mpz_sub (pie, ...); ... mpz_clear (pie); } Once the integer has been initialized by any of the `mpz_init_set...' functions, it can be used as the source or destination operand for the ordinary integer functions. Don't use an initialize-and-set function on a variable already initialized! - Function: void mpz_init_set (mpz_t ROP, mpz_t OP) - Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP) - Function: void mpz_init_set_si (mpz_t ROP, signed long int OP) - Function: void mpz_init_set_d (mpz_t ROP, double OP) Initialize ROP with limb space and set the initial numeric value from OP. - Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE) Initialize ROP and set its value like `mpz_set_str' (see its documentation above for details). If the string is a correct base BASE number, the function returns 0; if an error occurs it returns -1. ROP is initialized even if an error occurs. (I.e., you have to call `mpz_clear' for it.) Conversion Functions ==================== This section describes functions for converting GMP integers to standard C types. Functions for converting _to_ GMP integers are described in *Note Assigning Integers:: and *Note I/O of Integers::. - Function: unsigned long int mpz_get_ui (mpz_t OP) Return the value of OP as an `unsigned long'. If OP is too big to fit an `unsigned long' then just the least significant bits that do fit are returned. The sign of OP is ignored, only the absolute value is used. - Function: signed long int mpz_get_si (mpz_t OP) If OP fits into a `signed long int' return the value of OP. Otherwise return the least significant part of OP, with the same sign as OP. If OP is too big to fit in a `signed long int', the returned result is probably not very useful. To find out if the value will fit, use the function `mpz_fits_slong_p'. - Function: double mpz_get_d (mpz_t OP) Convert OP to a `double', truncating if necessary (ie. rounding towards zero). If the exponent from the conversion is too big, the result is system dependent. An infinity is returned where available. A hardware overflow trap may or may not occur. - Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP) Convert OP to a `double', truncating if necessary (ie. rounding towards zero), and returning the exponent separately. The return value is in the range 0.5<=abs(D)<1 and the exponent is stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP is zero, the return is 0.0 and 0 is stored to `*EXP'. This is similar to the standard C `frexp' function (*note Normalization Functions: (libc)Normalization Functions.). - Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP) Convert OP to a string of digits in base BASE. The base may vary from 2 to 36. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of storage large enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'. The two extra bytes are for a possible minus sign, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given STR. Arithmetic Functions ==================== - Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 + OP2. - Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) - Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t OP2) Set ROP to OP1 - OP2. - Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2) - Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. - Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to ROP + OP1 times OP2. - Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to ROP - OP1 times OP2. - Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. This operation can also be defined as a left shift by OP2 bits. - Function: void mpz_neg (mpz_t ROP, mpz_t OP) Set ROP to -OP. - Function: void mpz_abs (mpz_t ROP, mpz_t OP) Set ROP to the absolute value of OP. Division Functions ================== Division is undefined if the divisor is zero. Passing a zero divisor to the division or modulo functions (including the modular powering functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional division by zero. This lets a program handle arithmetic exceptions in these functions the same way as for normal C `int' arithmetic. - Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D) - Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D) - Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) - Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_cdiv_ui (mpz_t N, unsigned long int D) - Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int B) - Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int B) - Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D) - Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D) - Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) - Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_fdiv_ui (mpz_t N, unsigned long int D) - Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int B) - Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int B) - Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D) - Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D) - Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D) - Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R, mpz_t N, unsigned long int D) - Function: unsigned long int mpz_tdiv_ui (mpz_t N, unsigned long int D) - Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, unsigned long int B) - Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, unsigned long int B) Divide N by D, forming a quotient Q and/or remainder R. For the `2exp' functions, D=2^B. The rounding is in three styles, each suiting different applications. * `cdiv' rounds Q up towards +infinity, and R will have the opposite sign to D. The `c' stands for "ceil". * `fdiv' rounds Q down towards -infinity, and R will have the same sign as D. The `f' stands for "floor". * `tdiv' rounds Q towards zero, and R will have the same sign as N. The `t' stands for "truncate". In all cases Q and R will satisfy N=Q*D+R, and R will satisfy 0<=abs(R)1, such that OP equals A raised to the power B. Under this definition both 0 and 1 are considered to be perfect powers. Negative values of OP are accepted, but of course can only be odd perfect powers. - 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. Under this definition both 0 and 1 are considered to be perfect squares. Number Theoretic Functions ========================== - Function: int mpz_probab_prime_p (mpz_t N, int REPS) Determine whether N is prime. Return 2 if N is definitely prime, return 1 if N is probably prime (without being certain), or return 0 if N is definitely composite. This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. REPS controls how many such tests are done, 5 to 10 is a reasonable number, more will reduce the chances of a composite being returned as "probably prime". Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime. - Function: void 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. For practical purposes it's adequate, 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 one 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) Set G to the greatest common divisor of A and B, and in addition set S and T to coefficients satisfying A*S + B*T = G. G is always positive, even if one or both of A and B are negative. If T is `NULL' then that value is not computed. - Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2) - Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2) Set ROP to the least common multiple of OP1 and OP2. ROP is always positive, irrespective of the signs of OP1 and OP2. ROP will be zero if either OP1 or OP2 is zero. - 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. If the inverse exists, the return value is non-zero and ROP will satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return value is zero and ROP is undefined. - Function: int mpz_jacobi (mpz_t A, mpz_t B) Calculate the Jacobi symbol (A/B). This is defined only for B odd. - Function: int mpz_legendre (mpz_t A, mpz_t P) Calculate the Legendre symbol (A/P). This is defined only for P an odd positive prime, and for such P it's identical to the Jacobi symbol. - Function: int mpz_kronecker (mpz_t 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) - Function: int mpz_si_kronecker (long A, mpz_t B) - Function: int mpz_ui_kronecker (unsigned long A, mpz_t B) Calculate the Jacobi symbol (A/B) with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even. When B is odd the Jacobi symbol and Kronecker symbol are identical, so `mpz_kronecker_ui' etc can be used for mixed precision Jacobi symbols too. For more information see Henri Cohen section 1.4.2 (*note References::), or any number theory textbook. 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. The return value is how many such occurrences were removed. - 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 FN, unsigned long int N) - Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long int N) `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number. `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1]. These functions are designed for calculating isolated Fibonacci numbers. When a sequence of values is wanted it's best to start with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or similar. - Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N) - Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned long int N) `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number. `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1]. These functions are designed for calculating isolated Lucas numbers. When a sequence of values is wanted it's best to start with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1] or similar. The Fibonacci numbers and Lucas numbers are related sequences, so it's never necessary to call both `mpz_fib2_ui' and `mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas can be found in *Note Lucas Numbers Algorithm::, the reverse is straightforward too. Comparison Functions ==================== - Function: int mpz_cmp (mpz_t OP1, mpz_t OP2) - Function: int mpz_cmp_d (mpz_t OP1, double OP2) - Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2) - Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, or a negative value if OP1 < OP2. `mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their arguments more than once. `mpz_cmp_d' can be called with an infinity, but results are undefined for a NaN. - Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2) - Function: int mpz_cmpabs_d (mpz_t OP1, double 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 abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a negative value if abs(OP1) < abs(OP2). `mpz_cmpabs_d' can be called with an infinity, but results are undefined for a NaN. - 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 argument multiple times. Logical and Bit Manipulation Functions ====================================== These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0. - Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 bitwise-and OP2. - Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 bitwise inclusive-or OP2. - Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2) Set ROP to OP1 bitwise 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) If OP>=0, return the population count of OP, which is the number of 1 bits in the binary representation. If OP<0, the number of 1s is infinite, and the return value is ULONG_MAX, the largest possible `unsigned long'. - Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2) If OP1 and OP2 are both >=0 or both <0, return the hamming distance between the two operands, which is the number of bit positions where OP1 and OP2 have different bit values. If one operand is >=0 and the other <0 then the number of bits different is infinite, and the return value is ULONG_MAX, the largest possible `unsigned long'. - Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int STARTING_BIT) - Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int STARTING_BIT) Scan OP, starting from bit STARTING_BIT, towards more significant bits, until the first 0 or 1 bit (respectively) is found. Return the index of the found bit. If the bit at STARTING_BIT is already what's sought, then STARTING_BIT is returned. If there's no bit found, then ULONG_MAX is returned. This will happen in `mpz_scan0' past the end of a positive number, or `mpz_scan1' past the end of a negative. - 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: void mpz_combit (mpz_t ROP, unsigned long int BIT_INDEX) Complement bit BIT_INDEX in ROP. - Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX) Test bit BIT_INDEX in OP and return 0 or 1 accordingly. 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 62, or if BASE is 0, then the leading characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B' for binary, `0' for octal, or decimal otherwise. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. 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. 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. Integer Import and Export ========================= `mpz_t' variables can be converted to and from arbitrary words of binary data with the following functions. - Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER, int SIZE, int ENDIAN, size_t NAILS, const void *OP) Set ROP from an array of word data at OP. The parameters specify the format of the data. COUNT many words are read, each SIZE bytes. ORDER can be 1 for most significant word first or -1 for least significant first. Within each word ENDIAN can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant NAILS bits of each word are skipped, this can be 0 to use the full words. There is no sign taken from the data, ROP will simply be a positive integer. An application can handle any sign itself, and apply it for instance with `mpz_neg'. There are no data alignment restrictions on OP, any address is allowed. Here's an example converting an array of `unsigned long' data, most significant element first, and host byte order within each value. unsigned long a[20]; mpz_t z; mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a); This example assumes the full `sizeof' bytes are used for data in the given type, which is usually true, and certainly true for `unsigned long' everywhere we know of. However on Cray vector systems it may be noted that `short' and `int' are always stored in 8 bytes (and with `sizeof' indicating that) but use only 32 or 46 bits. The NAILS feature can account for this, by passing for instance `8*sizeof(int)-INT_BIT'. - Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER, int SIZE, int ENDIAN, size_t NAILS, mpz_t OP) Fill ROP with word data from OP. The parameters specify the format of the data produced. Each word will be SIZE bytes and ORDER can be 1 for most significant word first or -1 for least significant first. Within each word ENDIAN can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant NAILS bits of each word are unused and set to zero, this can be 0 to produce full words. The number of words produced is written to `*COUNTP', or COUNTP can be `NULL' to discard the count. ROP must have enough space for the data, or if ROP is `NULL' then a result array of the necessary size is allocated using the current GMP allocation function (*note Custom Allocation::). In either case the return value is the destination used, either ROP or the allocated block. If OP is non-zero then the most significant word produced will be non-zero. If OP is zero then the count returned will be zero and nothing written to ROP. If ROP is `NULL' in this case, no block is allocated, just `NULL' is returned. The sign of OP is ignored, just the absolute value is exported. An application can use `mpz_sgn' to get the sign and handle it as desired. (*note Integer Comparisons::) There are no data alignment restrictions on ROP, any address is allowed. When an application is allocating space itself the required size can be determined with a calculation like the following. Since `mpz_sizeinbase' always returns at least 1, `count' here will be at least one, which avoids any portability problems with `malloc(0)', though if `z' is zero no space at all is actually needed (or written). numb = 8*size - nail; count = (mpz_sizeinbase (z, 2) + numb-1) / numb; p = malloc (count * size); 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 argument more than once. - Function: size_t mpz_sizeinbase (mpz_t OP, int BASE) Return the size of OP measured in number of digits in the given BASE. BASE can vary from 2 to 36. The sign of OP is ignored, just the absolute value is used. The result will be either exact or 1 too big. If BASE is a power of 2, the result is always exact. If OP is zero the return value is always 1. This function can be used to determine the space required when 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 null-terminator. It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate the most significant 1 bit in OP, counting from 1. (Unlike the bitwise functions which start from 0, *Note Logical and Bit Manipulation Functions: Integer Logic and Bit Fiddling.) Special Functions ================= The functions in this section are for various special purposes. Most applications will not need them. - Function: void mpz_array_init (mpz_t INTEGER_ARRAY, size_t ARRAY_SIZE, mp_size_t FIXED_NUM_BITS) This is a special type of initialization. *Fixed* space of FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in INTEGER_ARRAY. There is no way to free the storage allocated by this function. Don't call `mpz_clear'! The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For example, mpz_t arr[20000]; mpz_array_init (arr[0], 20000, 512); This function is only intended for programs that create a large number of integers and need to reduce memory usage by avoiding the overheads of allocating and reallocating lots of small blocks. In normal programs this function is not recommended. The space allocated to each integer by this function will not be automatically increased, unlike the normal `mpz_init', so an application must ensure it is sufficient for any value stored. The following space requirements apply to various routines, * `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and `mpz_set_ui' need room for the value they store. * `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room for the larger of the two operands, plus an extra `mp_bits_per_limb'. * `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum of the number of bits in their operands, but each rounded up to a multiple of `mp_bits_per_limb'. * `mpz_swap' can be used between two array variables, but not between an array and a normal variable. For other functions, or if in doubt, the suggestion is to calculate in a regular `mpz_init' variable and copy the result to an array variable with `mpz_set'. - Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC) Change the space for INTEGER to NEW_ALLOC limbs. The value in INTEGER is preserved if it fits, or is set to 0 if not. The return value is not useful to applications and should be ignored. `mpz_realloc2' is the preferred way to accomplish allocation changes like this. `mpz_realloc2' and `_mpz_realloc' are the same except that `_mpz_realloc' takes its size in limbs. - Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N) Return limb number N from OP. The sign of OP is ignored, just the absolute value is used. The least significant limb is number 0. `mpz_size' can be used to find how many limbs make up OP. `mpz_getlimbn' returns zero if N is outside the range 0 to `mpz_size(OP)-1'. - 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. 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. - 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 Conversions:: * Rational Arithmetic:: * Comparing Rationals:: * Applying Integer Functions:: * I/O of Rationals:: 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: int mpq_set_str (mpq_t ROP, char *STR, int BASE) Set ROP from a null-terminated string STR in the given BASE. The string can be an integer like "41" or a fraction like "41/152". The fraction must be in canonical form (*note Rational Number Functions::), or if not then `mpq_canonicalize' must be called. The numerator and optional denominator are parsed the same as in `mpz_set_str' (*note Assigning Integers::). White space is allowed in the string, and is simply ignored. The BASE can vary from 2 to 62, or if BASE is 0 then the leading characters are used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for octal, or decimal otherwise. Note that this is done separately for the numerator and denominator, so for instance `0xEF/100' is 239/100, whereas `0xEF/0x100' is 239/256. The return value is 0 if the entire string is a valid number, or -1 if not. - Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2) Swap the values ROP1 and ROP2 efficiently. Conversion Functions ==================== - Function: double mpq_get_d (mpq_t OP) Convert OP to a `double', truncating if necessary (ie. rounding towards zero). If the exponent from the conversion is too big or too small to fit a `double' then the result is system dependent. For too big an infinity is returned when available. For too small 0.0 is normally returned. Hardware overflow, underflow and denorm traps may or may not occur. - Function: void mpq_set_d (mpq_t ROP, double OP) - Function: void mpq_set_f (mpq_t ROP, mpf_t OP) Set ROP to the value of OP. There is no rounding, this conversion is exact. - Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP) Convert OP to a string of digits in base BASE. The base may vary from 2 to 36. The string will be of the form `num/den', or if the denominator is 1 then just `num'. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of storage large enough for the result, that being mpz_sizeinbase (mpq_numref(OP), BASE) + mpz_sizeinbase (mpq_denref(OP), BASE) + 3 The three extra bytes are for a possible minus sign, possible slash, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given STR. 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_mul_2exp (mpq_t ROP, mpq_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. - Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t DIVISOR) Set QUOTIENT to DIVIDEND/DIVISOR. - Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, unsigned long int OP2) Set ROP to OP1 divided by 2 raised to OP2. - Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND) Set NEGATED_OPERAND to -OPERAND. - Function: void mpq_abs (mpq_t ROP, mpq_t OP) Set ROP to the absolute value of OP. - 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. 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) - Macro: int mpq_cmp_si (mpq_t OP1, 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. NUM2 and DEN2 are allowed to have common factors. These functions are implemented as a macros and evaluate their 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. Applying Integer Functions to Rationals ======================================= The set of `mpq' functions is quite small. In particular, there are few functions for either input or output. The following functions give direct access to the numerator and denominator of an `mpq_t'. Note that if an assignment to the numerator and/or denominator could take an `mpq_t' out of the canonical form described at the start of this chapter (*note Rational Number Functions::) then `mpq_canonicalize' must be called before any other `mpq' functions are applied to that `mpq_t'. - 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. - Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL) - Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL) - Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR) - Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR) Get or set the numerator or denominator of a rational. These functions are equivalent to calling `mpz_set' with an appropriate `mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or `mpq_denref' is recommended instead of these functions. Input and Output Functions ========================== When using any of these functions, it's a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions. Passing a `NULL' pointer for a STREAM argument to any of these functions will make them read from `stdin' and write to `stdout', respectively. - 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. - Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE) Read a string of digits from STREAM and convert them to a rational in ROP. Any initial white-space characters are read and discarded. Return the number of characters read (including white space), or 0 if a rational could not be read. The input can be a fraction like `17/63' or just an integer like `123'. Reading stops at the first character not in this form, and white space is not permitted within the string. If the input might not be in canonical form, then `mpq_canonicalize' must be called (*note Rational Number Functions::). The BASE can be between 2 and 36, or can be 0 in which case the leading characters of the string determine the base, `0x' or `0X' for hexadecimal, `0' for octal, or decimal otherwise. The leading characters are examined separately for the numerator and denominator of a fraction, so for instance `0x10/11' is 16/11, whereas `0x10/0x11' is 16/17. Floating-point Functions ************************ GMP floating point numbers are stored in objects of type `mpf_t' and functions operating on them have an `mpf_' prefix. The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time. The exponent of each float is a fixed precision, one machine word on most systems. In the current implementation the exponent is a count of limbs, so for example on a 32-bit system this means a range of roughly 2^-68719476768 to 2^68719476736, or on a 64-bit system this will be greater. Note however `mpf_get_str' can only return an exponent which fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents bigger than a `long'. Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high. All calculations are performed to the precision of the destination variable. Each function is defined to calculate with "infinite precision" followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition. The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details. The mantissa in stored in binary, as might be imagined from the fact precisions are expressed in bits. One consequence of this is that decimal fractions like 0.1 cannot be represented exactly. The same is true of plain IEEE `double' floats. This makes both highly unsuitable for calculations involving money or other values that should be exact decimal fractions. (Suitably scaled integers, or perhaps rationals, are better choices.) `mpf' functions and variables have no special notion of infinity or not-a-number, and applications must take care not to overflow the exponent or results will be unpredictable. This might change in a future release. Note that the `mpf' functions are _not_ intended as a smooth extension to IEEE P754 arithmetic. In particular results obtained on one computer often differ from the results on a computer with a different word size. * Menu: * Initializing Floats:: * Assigning Floats:: * Simultaneous Float Init & Assign:: * Converting Floats:: * Float Arithmetic:: * Float Comparison:: * I/O of Floats:: * Miscellaneous Float Functions:: Initialization Functions ======================== - Function: void mpf_set_default_prec (unsigned long int PREC) Set the default precision to be *at least* PREC bits. All subsequent calls to `mpf_init' will use this precision, but previously initialized variables are unaffected. - Function: unsigned long int mpf_get_default_prec (void) Return the default precision actually used. An `mpf_t' object must be initialized before storing the first value in it. The functions `mpf_init' and `mpf_init2' are used for that purpose. - Function: void mpf_init (mpf_t X) Initialize X to 0. Normally, a variable should be initialized once only or at least be cleared, using `mpf_clear', between initializations. The precision of X is undefined unless a default precision has already been established by a call to `mpf_set_default_prec'. - Function: void mpf_init2 (mpf_t X, unsigned long int PREC) Initialize X to 0 and set its precision to be *at least* PREC bits. Normally, a variable should be initialized once only or at least be cleared, using `mpf_clear', between initializations. - Function: void mpf_clear (mpf_t X) Free the space occupied by X. Make sure to call this function for all `mpf_t' variables when you are done with them. Here is an example on how to initialize floating-point variables: { mpf_t x, y; mpf_init (x); /* use default precision */ mpf_init2 (y, 256); /* precision _at least_ 256 bits */ ... /* Unless the program is about to exit, do ... */ mpf_clear (x); mpf_clear (y); } The following three functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. - Function: unsigned long int mpf_get_prec (mpf_t OP) Return the current precision of OP, in bits. - Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC) Set the precision of ROP to be *at least* PREC bits. The value in ROP will be truncated to the new precision. This function requires a call to `realloc', and so should not be used in a tight loop. - Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC) Set the precision of ROP to be *at least* PREC bits, without changing the memory allocated. PREC must be no more than the allocated precision for ROP, that being the precision when ROP was initialized, or in the most recent `mpf_set_prec'. The value in ROP is unchanged, and in particular if it had a higher precision than PREC it will retain that higher precision. New values written to ROP will use the new PREC. Before calling `mpf_clear' or the full `mpf_set_prec', another `mpf_set_prec_raw' call must be made to restore ROP to its original allocated precision. Failing to do so will have unpredictable results. `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the original allocated precision. After `mpf_set_prec_raw' it reflects the PREC value set. `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable at different precisions during a calculation, perhaps to gradually increase precision in an iteration, or just to use various different precisions for different purposes during a calculation. Assignment Functions ==================== These functions assign new values to already initialized floats (*note Initializing Floats::). - 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_si (mpf_t ROP, signed long int OP) - Function: void mpf_set_d (mpf_t ROP, double OP) - Function: void mpf_set_z (mpf_t ROP, mpz_t OP) - Function: void mpf_set_q (mpf_t ROP, mpq_t OP) Set the value of ROP from OP. - Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE) Set the value of ROP from the string in STR. The string is of the form `M@N' or, if the base is 10 or less, alternatively `MeN'. `M' is the mantissa and `N' is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if BASE is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing `localeconv'. The argument BASE may be in the ranges 2 to 62, or -62 to -2. Negative values are used to specify that the exponent is in decimal. For bases up to 36, case is ignored; upper-case and lower-case letters have the same value; for bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61. Unlike the corresponding `mpz' function, the base will not be determined from the leading characters of the string if BASE is 0. This is so that numbers like `0.23' are not interpreted as octal. 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 is a valid number in base BASE. Otherwise it returns -1. - Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2) Swap ROP1 and ROP2 efficiently. Both the values and the precisions of the two variables are swapped. Combined Initialization and Assignment Functions ================================================ For convenience, GMP provides a parallel series of initialize-and-set functions which initialize the output and then store 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...' functions, it can be used as the source or destination operand for the ordinary float functions. Don't use an initialize-and-set function on a variable already initialized! - Function: void mpf_init_set (mpf_t ROP, mpf_t OP) - Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP) - Function: void mpf_init_set_si (mpf_t ROP, signed long int OP) - Function: void mpf_init_set_d (mpf_t ROP, double OP) Initialize ROP and set its value from OP. The precision of ROP will be taken from the active default precision, as set by `mpf_set_default_prec'. - Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE) Initialize ROP and set its value from the string in STR. See `mpf_set_str' above for details on the assignment operation. Note that ROP is initialized even if an error occurs. (I.e., you have to call `mpf_clear' for it.) The precision of ROP will be taken from the active default precision, as set by `mpf_set_default_prec'. Conversion Functions ==================== - Function: double mpf_get_d (mpf_t OP) Convert OP to a `double', truncating if necessary (ie. rounding towards zero). If the exponent in OP is too big or too small to fit a `double' then the result is system dependent. For too big an infinity is returned when available. For too small 0.0 is normally returned. Hardware overflow, underflow and denorm traps may or may not occur. - Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP) Convert OP to a `double', truncating if necessary (ie. rounding towards zero), and with an exponent returned separately. The return value is in the range 0.5<=abs(D)<1 and the exponent is stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP is zero, the return is 0.0 and 0 is stored to `*EXP'. This is similar to the standard C `frexp' function (*note Normalization Functions: (libc)Normalization Functions.). - Function: long mpf_get_si (mpf_t OP) - Function: unsigned long mpf_get_ui (mpf_t OP) Convert OP to a `long' or `unsigned long', truncating any fraction part. If OP is too big for the return type, the result is undefined. See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note Miscellaneous Float Functions::). - Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE, size_t N_DIGITS, mpf_t OP) Convert OP to a string of digits in base BASE. BASE can be 2 to 36. Up to N_DIGITS digits will be generated. Trailing zeros are not returned. No more digits than can be accurately represented by OP are ever generated. If N_DIGITS is 0 then that accurate maximum number of digits are generated. If STR is `NULL', the result string is allocated using the current allocation function (*note Custom Allocation::). The block will be `strlen(str)+1' bytes, that being exactly enough for the string and null-terminator. If STR is not `NULL', it should point to a block of N_DIGITS + 2 bytes, that being enough for the mantissa, a possible minus sign, and a null-terminator. When N_DIGITS is 0 to get all significant digits, an application won't be able to know the space required, and STR should be `NULL' in that case. The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. The applicable exponent is written through the EXPPTR pointer. For example, the number 3.1416 would be returned as string "31416" and exponent 1. When OP is zero, an empty string is produced and the exponent returned is 0. A pointer to the result string is returned, being either the allocated block or the given STR. Arithmetic Functions ==================== - Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 + OP2. - Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t OP2) - Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 - OP2. - Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2) - Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 times OP2. Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions. - 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 OP2) - Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1/OP2. - Function: void mpf_sqrt (mpf_t ROP, mpf_t OP) - Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int 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) Set ROP to -OP. - Function: void mpf_abs (mpf_t ROP, mpf_t OP) Set ROP to the absolute value of OP. - Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 times 2 raised to OP2. - Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long int OP2) Set ROP to OP1 divided by 2 raised to OP2. Comparison Functions ==================== - Function: int mpf_cmp (mpf_t OP1, mpf_t OP2) - Function: int mpf_cmp_d (mpf_t OP1, double OP2) - Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2) - Function: int mpf_cmp_si (mpf_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. `mpf_cmp_d' can be called with an infinity, but results are undefined for a NaN. - Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3) Return non-zero if the first OP3 bits of OP1 and OP2 are equal, zero otherwise. I.e., test if OP1 and OP2 are approximately equal. Caution: Currently only whole limbs are compared, and only in an exact fashion. In the future values like 1000 and 0111 may be considered the same to 3 bits (on the basis that their difference is that small). - Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2) Compute the relative difference between OP1 and OP2 and store the result in ROP. This is abs(OP1-OP2)/OP1. - Macro: int mpf_sgn (mpf_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. 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 mpf_out_str (FILE *STREAM, int BASE, size_t N_DIGITS, mpf_t OP) Print OP to STREAM, as a string of digits. Return the number of bytes written, or if an error occurred, return 0. The mantissa is prefixed with an `0.' and is in the given BASE, which may vary from 2 to 36. An exponent then printed, separated by an `e', or if BASE is greater than 10 then by an `@'. The exponent is always in decimal. The decimal point follows the current locale, on systems providing `localeconv'. Up to N_DIGITS will be printed from the mantissa, except that no more digits than are accurately representable by OP will be printed. N_DIGITS can be 0 to select that accurate maximum. - Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE) Read a string in base BASE from STREAM, and put the read float in ROP. The string is of the form `M@N' or, if the base is 10 or less, alternatively `MeN'. `M' is the mantissa and `N' is the exponent. The mantissa is always in the specified base. The exponent is either in the specified base or, if BASE is negative, in decimal. The decimal point expected is taken from the current locale, on systems providing `localeconv'. The argument BASE may be in the ranges 2 to 36, or -36 to -2. Negative values are used to specify that the exponent is in decimal. Unlike the corresponding `mpz' function, the base will not be determined from the leading characters of the string if BASE is 0. This is so that numbers like `0.23' are not interpreted as octal. Return the number of bytes read, or if an error occurred, return 0. 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: int mpf_integer_p (mpf_t OP) Return non-zero if OP is an integer. - Function: int mpf_fits_ulong_p (mpf_t OP) - Function: int mpf_fits_slong_p (mpf_t OP) - Function: int mpf_fits_uint_p (mpf_t OP) - Function: int mpf_fits_sint_p (mpf_t OP) - Function: int mpf_fits_ushort_p (mpf_t OP) - Function: int mpf_fits_sshort_p (mpf_t OP) Return non-zero if OP would fit in the respective C data type, when truncated to an integer. - 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 EXP) Generate a random float of at most MAX_SIZE limbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval -EXP to EXP (in limbs). This function is useful for testing functions and algorithms, since these 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. Low-level Functions ******************* This chapter describes low-level GMP functions, used to implement the high-level GMP 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. Unless otherwise stated, in-place operations are allowed where source and destination are the same, but not where they only partly overlap. The `mpn' functions are the base for the implementation of the `mpz_', `mpf_', and `mpq_' functions. This example adds the number beginning at S1P and the number beginning at S2P and writes the sum at DESTP. All areas have N limbs. cy = mpn_add_n (destp, s1p, s2p, n) It should be noted that the `mpn' functions make no attempt to identify high or low zero limbs on their operands, or other special forms. On random data such cases will be unlikely and it'd be wasteful for every function to check every time. An application knowing something about its data can take steps to trim or perhaps split its calculations. In the notation used below, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {S1P, S1N}. - Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Add {S1P, N} and {S2P, N}, and write the N least significant limbs of the result to RP. 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 CPUs. For addition of a variable to itself (i.e., S1P equals S2P, use `mpn_lshift' with a count of 1 for optimal speed. - Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Add {S1P, N} and S2LIMB, and write the N least significant limbs of the result to RP. Return carry, either 0 or 1. - Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N) Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant limbs of the result to RP. Return carry, either 0 or 1. This function requires that S1N is greater than or equal to S2N. - Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Subtract {S2P, N} from {S1P, N}, and write the N least significant limbs of the result to RP. 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 CPUs. - Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Subtract S2LIMB from {S1P, N}, and write the N least significant limbs of the result to RP. Return borrow, either 0 or 1. - Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N) Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least significant limbs of the result to RP. Return borrow, either 0 or 1. This function requires that S1N is greater than or equal to S2N. - Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to RP. The destination has to have space for 2*N limbs, even if the product's most significant limb is zero. No overlap is permitted between the destination and either source. - Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Multiply {S1P, N} by S2LIMB, and write the N least significant limbs of the product to RP. Return the most significant limb of the product. {S1P, N} and {RP, N} are allowed to overlap provided RP <= S1P. This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs. Don't call this function if S2LIMB is a power of 2; use `mpn_lshift' with a count equal to the logarithm of S2LIMB instead, for optimal speed. - Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Multiply {S1P, N} and S2LIMB, and add the N least significant limbs of the product to {RP, N} and write the result to RP. 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 GMP. It is written in assembly for most CPUs. - Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t N, mp_limb_t S2LIMB) Multiply {S1P, N} and S2LIMB, and subtract the N least significant limbs of the product from {RP, N} and write the result to RP. 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 GMP. It is written in assembly for most CPUs. - Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N) Multiply {S1P, S1N} and {S2P, S2N}, and write the result to RP. Return the most significant limb of the result. The destination has to have space for S1N + S2N limbs, even if the result might be one limb smaller. This function requires that S1N is greater than or equal to S2N. The destination must be distinct from both input operands. - Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP, mp_size_t DN) Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1} and the remainder at {RP, DN}. The quotient is rounded towards 0. No overlap is permitted between arguments. NN must be greater than or equal to DN. The most significant limb of DP must be non-zero. The QXN operand must be zero. - Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN, mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N) [This function is obsolete. Please call `mpn_tdiv_qr' instead for best performance.] Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P, with the exception of the most significant limb, which is returned. The remainder replaces the dividend at RS2P; it will be S3N limbs long (i.e., as many limbs as the divisor). In addition to an integer quotient, QXN fraction limbs are developed, and stored after the integral limbs. For most usages, QXN will be zero. It is required that RS2N is greater than or equal to S3N. It is required that the most significant bit of the divisor is set. If the quotient is not needed, pass RS2P + S3N 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 RS2N - S3N + QXN limbs large. - Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN, mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB) - Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB) Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P. Return the remainder. The integer quotient is written to {R1P+QXN, S2N} and in addition QXN fraction limbs are developed and written to {R1P, QXN}. Either or both S2N and QXN can be zero. For most usages, QXN will be zero. `mpn_divmod_1' exists for upward source compatibility and is simply a macro calling `mpn_divrem_1' with a QXN of 0. The areas at R1P and S2P have to be identical or completely separate, not partially overlapping. - Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N) [This function is obsolete. Please call `mpn_tdiv_qr' instead for best performance.] - Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP, mp_size_t N) - Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP, mp_size_t N, mp_limb_t CARRY) Divide {SP, N} by 3, expecting it to divide exactly, and writing the result to {RP, N}. If 3 divides exactly, the return value is zero and the result is the quotient. If not, the return value is non-zero and the result won't be anything useful. `mpn_divexact_by3c' takes an initial carry parameter, which can be the return value from a previous call, so a large calculation can be done piece by piece from low to high. `mpn_divexact_by3' is simply a macro calling `mpn_divexact_by3c' with a 0 carry parameter. These routines use a multiply-by-inverse and will be faster than `mpn_divrem_1' on CPUs with fast multiplication but slow division. The source a, result q, size n, initial carry i, and return value c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return c is always 0, 1 or 2, and the initial carry i must also be 0, 1 or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3. When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b == 1 mod 3 (when `mp_bits_per_limb' is even, which is always so currently). - Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N, mp_limb_t S2LIMB) Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be zero. - Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P, mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, unsigned long int D) This function puts the low floor(D/mp_bits_per_limb) limbs of Q = {S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod `mp_bits_per_limb' bits of Q. {S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed at S1P. Since the low floor(D/mp_bits_per_limb) limbs of this difference are zero, it is possible to overwrite the low limbs at S1P with this difference, provided RP <= S1P. This function requires that S1N * mp_bits_per_limb >= D, and that {S2P, S2N} is odd. *This interface is preliminary. It might change incompatibly in future revisions.* - Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP, mp_size_t N, unsigned int COUNT) Shift {SP, N} left by COUNT bits, and write the result to {RP, N}. The bits shifted out at the left are returned in the least significant COUNT bits of the return value (the rest of the return value is zero). COUNT must be in the range 1 to mp_bits_per_limb-1. The regions {SP, N} and {RP, N} may overlap, provided RP >= SP. This function is written in assembly for most CPUs. - Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP, mp_size_t N, unsigned int COUNT) Shift {SP, N} right by COUNT bits, and write the result to {RP, N}. The bits shifted out at the right are returned in the most significant COUNT bits of the return value (the rest of the return value is zero). COUNT must be in the range 1 to mp_bits_per_limb-1. The regions {SP, N} and {RP, N} may overlap, provided RP <= SP. This function is written in assembly for most CPUs. - Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Compare {S1P, N} and {S2P, N} and return a positive value if S1 > S2, 0 if they are equal, or a negative value if S1 < S2. - Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N) Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and {S2P, S2N}. The result can be up to S2N limbs, the return value is the actual number produced. Both source operands are destroyed. {S1P, S1N} must have at least as many bits as {S2P, S2N}. {S2P, S2N} must be odd. Both operands must have non-zero most significant limbs. No overlap is permitted between {S1P, S1N} and {S2P, S2N}. - Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N, mp_limb_t S2LIMB) Return the greatest common divisor of {S1P, S1N} and S2LIMB. Both operands must be non-zero. - Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P, mp_size_t *R2N, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N) Calculate the greatest common divisor of {S1P, S1N} and {S2P, S2N}. Store the gcd at {R1P, RETVAL} and the first cofactor at {R2P, *R2N}, with *R2N negative if the cofactor is negative. R1P and R2P should each have room for S1N+1 limbs, but the return value and value stored through R2N indicate the actual number produced. {S1P, S1N} >= {S2P, S2N} is required, and both must be non-zero. The regions {S1P, S1N+1} and {S2P, S2N+1} are destroyed (i.e. the operands plus an extra limb past the end of each). The cofactor R1 will satisfy R2*S1 + K*S2 = R1. The second cofactor K is not calculated but can easily be obtained from (R1 - R2*S1) / S2. - Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P, const mp_limb_t *SP, mp_size_t N) Compute the square root of {SP, N} and put the result at {R1P, ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space for N limbs, but the return value indicates how many are produced. The most significant limb of {SP, N} must be non-zero. The areas {R1P, ceil(N/2)} and {SP, N} must be completely separate. The areas {R2P, N} and {SP, N} must be either identical or completely separate. If the remainder is not wanted then R2P can be `NULL', and in this case the return value is zero or non-zero according to whether the remainder would have been zero or non-zero. A return value of zero indicates a perfect square. See also `mpz_perfect_square_p'. - Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE, mp_limb_t *S1P, mp_size_t S1N) Convert {S1P, S1N} to a raw unsigned char array at STR in base BASE, and return the number of characters produced. There may be leading zeros in the string. 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. BASE can vary from 2 to 256. The most significant limb of the input {S1P, S1N} must be non-zero. The input {S1P, S1N} is clobbered, except when BASE is a power of 2, in which case it's unchanged. The area at STR has to have space for the largest possible number represented by a S1N long limb array, plus one extra character. - Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char *STR, size_t STRSIZE, int BASE) Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP. STR[0] is the most significant byte and STR[STRSIZE-1] is the least significant. Each byte should be a value in the range 0 to BASE-1, not an ASCII character. BASE can vary from 2 to 256. The return value is the number of limbs written to RP. If the most significant input byte is non-zero then the high limb at RP will be non-zero, and only that exact number of limbs will be required there. If the most significant input byte is zero then there may be high zero limbs written to RP and included in the return value. STRSIZE must be at least 1, and no overlap is permitted between {STR,STRSIZE} and the result at RP. - 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. - 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. - Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N) - Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N) Generate a random number of length R1N and store it at R1P. The most significant limb is always non-zero. `mpn_random' generates uniformly distributed limb data, `mpn_random2' generates long strings of zeros and ones in the binary representation. `mpn_random2' is intended for testing the correctness of the `mpn' routines. - Function: unsigned long int mpn_popcount (const mp_limb_t *S1P, mp_size_t N) Count the number of set bits in {S1P, N}. - Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N) Compute the hamming distance between {S1P, N} and {S2P, N}, which is the number of bit positions where the two operands have different bit values. - Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t N) Return non-zero iff {S1P, N} is a perfect square. Nails ===== *Everything in this section is highly experimental and may disappear or be subject to incompatible changes in a future version of GMP.* Nails are an experimental feature whereby a few bits are left unused at the top of each `mp_limb_t'. This can significantly improve carry handling on some processors. All the `mpn' functions accepting limb data will expect the nail bits to be zero on entry, and will return data with the nails similarly all zero. This applies both to limb vectors and to single limb arguments. Nails can be enabled by configuring with `--enable-nails'. By default the number of bits will be chosen according to what suits the host processor, but a particular number can be selected with `--enable-nails=N'. At the mpn level, a nail build is neither source nor binary compatible with a non-nail build, strictly speaking. But programs acting on limbs only through the mpn functions are likely to work equally well with either build, and judicious use of the definitions below should make any program compatible with either build, at the source level. For the higher level routines, meaning `mpz' etc, a nail build should be fully source and binary compatible with a non-nail build. - Macro: GMP_NAIL_BITS - Macro: GMP_NUMB_BITS - Macro: GMP_LIMB_BITS `GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are not in use. `GMP_NUMB_BITS' is the number of data bits in a limb. `GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In all cases GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS - Macro: GMP_NAIL_MASK - Macro: GMP_NUMB_MASK Bit masks for the nail and number parts of a limb. `GMP_NAIL_MASK' is 0 when nails are not in use. `GMP_NAIL_MASK' is not often needed, since the nail part can be obtained with `x >> GMP_NUMB_BITS', and that means one less large constant, which can help various RISC chips. - Macro: GMP_NUMB_MAX The maximum value that can be stored in the number part of a limb. This is the same as `GMP_NUMB_MASK', but can be used for clarity when doing comparisons rather than bit-wise operations. The term "nails" comes from finger or toe nails, which are at the ends of a limb (arm or leg). "numb" is short for number, but is also how the developers felt after trying for a long time to come up with sensible names for these things. In the future (the distant future most likely) a non-zero nail might be permitted, giving non-unique representations for numbers in a limb vector. This would help vector processors since carries would only ever need to propagate one or two limbs. Random Number Functions *********************** Sequences of pseudo-random numbers in GMP are generated using a variable of type `gmp_randstate_t', which holds an algorithm selection and a current state. Such a variable must be initialized by a call to one of the `gmp_randinit' functions, and can be seeded with one of the `gmp_randseed' functions. The functions actually generating random numbers are described in *Note Integer Random Numbers::, and *Note Miscellaneous Float Functions::. The older style random number functions don't accept a `gmp_randstate_t' parameter but instead share a global variable of that type. They use a default algorithm and are currently not seeded (though perhaps that will change in the future). The new functions accepting a `gmp_randstate_t' are recommended for applications that care about randomness. * Menu: * Random State Initialization:: * Random State Seeding:: * Random State Miscellaneous:: Random State Initialization =========================== - Function: void gmp_randinit_default (gmp_randstate_t STATE) Initialize STATE with a default algorithm. This will be a compromise between speed and randomness, and is recommended for applications with no special requirements. Currently this is `gmp_randinit_mt'. - Function: int gmp_randinit_mt (gmp_randstate_t STATE) Initialize STATE for a Mersenne Twister algorithm. This algorithm is fast and has good randomness properties. - Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A, unsigned long C, unsigned long M2EXP) Initialize STATE with a linear congruential algorithm X = (A*X + C) mod 2^M2EXP. The low bits of X in this algorithm are not very random. The least significant bit will have a period no more than 2, and the second bit no more than 4, etc. For this reason only the high half of each X is actually used. When a random number of more than M2EXP/2 bits is to be generated, multiple iterations of the recurrence are used and the results concatenated. - Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE, unsigned long SIZE) Initialize STATE for a linear congruential algorithm as per `gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table, chosen so that SIZE bits (or more) of each X will be used, ie. M2EXP/2 >= SIZE. If successful the return value is non-zero. If SIZE is bigger than the table data provides then the return value is zero. The maximum SIZE currently supported is 128. - Function: int gmp_randinit_set (gmp_randstate_t ROP, gmp_randstate_t OP) Initialize ROP with a copy of the algorithm and state from OP. - Function: void gmp_randinit (gmp_randstate_t STATE, gmp_randalg_t ALG, ...) *This function is obsolete.* Initialize STATE with an algorithm selected by ALG. The only choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size' described above. A third parameter of type `unsigned long' is required, this is the SIZE for that function. `GMP_RAND_ALG_DEFAULT' or 0 are the same as `GMP_RAND_ALG_LC'. `gmp_randinit' sets bits in the global variable `gmp_errno' to indicate an error. `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is unsupported, or `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter is too big. It may be noted this error reporting is not thread safe (a good reason to use `gmp_randinit_lc_2exp_size' instead). - Function: void gmp_randclear (gmp_randstate_t STATE) Free all memory occupied by STATE. Random State Seeding ==================== - Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED) - Function: void gmp_randseed_ui (gmp_randstate_t STATE, unsigned long int SEED) Set an initial seed value into STATE. The size of a seed determines how many different sequences of random numbers that it's possible to generate. The "quality" of the seed is the randomness of a given seed compared to the previous seed used, and this affects the randomness of separate number sequences. The method for choosing a seed is critical if the generated numbers are to be used for important applications, such as generating cryptographic keys. Traditionally the system time has been used to seed, but care needs to be taken with this. If an application seeds often and the resolution of the system clock is low, then the same sequence of numbers might be repeated. Also, the system time is quite easy to guess, so if unpredictability is required then it should definitely not be the only source for the seed value. On some systems there's a special device `/dev/random' which provides random data better suited for use as a seed. Random State Miscellaneous ========================== - Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE, unsigned long N) Return a uniformly distributed random number of N bits, ie. in the range 0 to 2^N-1 inclusive. N must be less than or equal to the number of bits in an `unsigned long'. - Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE, unsigned long N) Return a uniformly distributed random number in the range 0 to N-1, inclusive. Formatted Output **************** * Menu: * Formatted Output Strings:: * Formatted Output Functions:: * C++ Formatted Output:: Format Strings ============== `gmp_printf' and friends accept format strings similar to the standard C `printf' (*note Formatted Output: (libc)Formatted Output.). A format specification is of the form % [flags] [width] [.[precision]] [type] conv GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t' respectively, `M' for `mp_limb_t', and `N' for an `mp_limb_t' array. `Z', `Q', `M' and `N' behave like integers. `Q' will print a `/' and a denominator, if needed. `F' behaves like a float. For example, mpz_t z; gmp_printf ("%s is an mpz %Zd\n", "here", z); mpq_t q; gmp_printf ("a hex rational: %#40Qx\n", q); mpf_t f; int n; gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n); mp_limb_t l; gmp_printf ("limb %Mu\n", limb); const mp_limb_t *ptr; mp_size_t size; gmp_printf ("limb array %Nx\n", ptr, size); For `N' the limbs are expected least significant first, as per the `mpn' functions (*note Low-level Functions::). A negative size can be given to print the value as a negative. All the standard C `printf' types behave the same as the C library `printf', and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to `printf' and only the GMP extensions handled directly. The flags accepted are as follows. GLIBC style ' is only for the standard C types (not the GMP types), and only if the C library supports it. 0 pad with zeros (rather than spaces) # show the base with `0x', `0X' or `0' + always show a sign (space) show a space or a `-' sign ' group digits, GLIBC style (not GMP types) The optional width and precision can be given as a number within the format string, or as a `*' to take an extra parameter of type `int', the same as the standard `printf'. The standard types accepted are as follows. `h' and `l' are portable, the rest will depend on the compiler (or include files) for the type and the C library for the output. h short hh char j intmax_t or uintmax_t l long or wchar_t ll long long L long double q quad_t or u_quad_t t ptrdiff_t z size_t The GMP types are F mpf_t, float conversions Q mpq_t, integer conversions M mp_limb_t, integer conversions N mp_limb_t array, integer conversions Z mpz_t, integer conversions The conversions accepted are as follows. `a' and `A' are always supported for `mpf_t' but depend on the C library for standard C float types. `m' and `p' depend on the C library. a A hex floats, C99 style c character d decimal integer e E scientific format float f fixed point float i same as d g G fixed or scientific float m `strerror' string, GLIBC style n store characters written so far o octal integer p pointer s string u unsigned integer x X hex integer `o', `x' and `X' are unsigned for the standard C types, but for types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z', `Q' and `N'. `M' is a proxy for the C library `l' or `L', according to the size of `mp_limb_t'. Unsigned conversions will be usual, but a signed conversion can be used and will interpret the value as a twos complement negative. `n' can be used with any type, even the GMP types. Other types or conversions that might be accepted by the C library `printf' cannot be used through `gmp_printf', this includes for instance extensions registered with GLIBC `register_printf_function'. Also currently there's no support for POSIX `$' style numbered arguments (perhaps this will be added in the future). The precision field has it's usual meaning for integer `Z' and float `F' types, but is currently undefined for `Q' and should not be used with that. `mpf_t' conversions only ever generate as many digits as can be accurately represented by the operand, the same as `mpf_get_str' does. Zeros will be used if necessary to pad to the requested precision. This happens even for an `f' conversion of an `mpf_t' which is an integer, for instance 2^1024 in an `mpf_t' of 128 bits precision will only produce about 40 digits, then pad with zeros to the decimal point. An empty precision field like `%.Fe' or `%.Ff' can be used to specifically request just the significant digits. The decimal point character (or string) is taken from the current locale settings on systems which provide `localeconv' (*note Locales and Internationalization: (libc)Locales.). The C library will normally do the same for standard float output. The format string is only interpreted as plain `char's, multibyte characters are not recognised. Perhaps this will change in the future. Functions ========= Each of the following functions is similar to the corresponding C library function. The basic `printf' forms take a variable argument list. The `vprintf' forms take an argument pointer, see *Note Variadic Functions: (libc)Variadic Functions, or `man 3 va_start'. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. The file based functions `gmp_printf' and `gmp_fprintf' will return -1 to indicate a write error. Output is not "atomic", so partial output may be produced if a write error occurs. All the functions can return -1 if the C library `printf' variant in use returns -1, but this shouldn't normally occur. - Function: int gmp_printf (const char *FMT, ...) - Function: int gmp_vprintf (const char *FMT, va_list AP) Print to the standard output `stdout'. Return the number of characters written, or -1 if an error occurred. - Function: int gmp_fprintf (FILE *FP, const char *FMT, ...) - Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP) Print to the stream FP. Return the number of characters written, or -1 if an error occurred. - Function: int gmp_sprintf (char *BUF, const char *FMT, ...) - Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP) Form a null-terminated string in BUF. Return the number of characters written, excluding the terminating null. No overlap is permitted between the space at BUF and the string FMT. These functions are not recommended, since there's no protection against exceeding the space available at BUF. - Function: int gmp_snprintf (char *BUF, size_t SIZE, const char *FMT, ...) - Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char *FMT, va_list AP) Form a null-terminated string in BUF. No more than SIZE bytes will be written. To get the full output, SIZE must be enough for the string and null-terminator. The return value is the total number of characters which ought to have been produced, excluding the terminating null. If RETVAL >= SIZE then the actual output has been truncated to the first SIZE-1 characters, and a null appended. No overlap is permitted between the region {BUF,SIZE} and the FMT string. Notice the return value is in ISO C99 `snprintf' style. This is so even if the C library `vsnprintf' is the older GLIBC 2.0.x style. - Function: int gmp_asprintf (char **PP, const char *FMT, ...) - Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP) Form a null-terminated string in a block of memory obtained from the current memory allocation function (*note Custom Allocation::). The block will be the size of the string and null-terminator. The address of the block in stored to *PP. The return value is the number of characters produced, excluding the null-terminator. Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1 if there's no more memory available, it lets the current allocation function handle that. - Function: int gmp_obstack_printf (struct obstack *OB, const char *FMT, ...) - Function: int gmp_obstack_vprintf (struct obstack *OB, const char *FMT, va_list AP) Append to the current object in OB. The return value is the number of characters written. A null-terminator is not written. FMT cannot be within the current object in OB, since that object might move as it grows. These functions are available only when the C library provides the obstack feature, which probably means only on GNU systems, see *Note Obstacks: (libc)Obstacks. Formatted Input *************** * Menu: * Formatted Input Strings:: * Formatted Input Functions:: * C++ Formatted Input:: Formatted Input Strings ======================= `gmp_scanf' and friends accept format strings similar to the standard C `scanf' (*note Formatted Input: (libc)Formatted Input.). A format specification is of the form % [flags] [width] [type] conv GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t' respectively. `Z' and `Q' behave like integers. `Q' will read a `/' and a denominator, if present. `F' behaves like a float. GMP variables don't require an `&' when passed to `gmp_scanf', since they're already "call-by-reference". For example, /* to read say "a(5) = 1234" */ int n; mpz_t z; gmp_scanf ("a(%d) = %Zd\n", &n, z); mpq_t q1, q2; gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2); /* to read say "topleft (1.55,-2.66)" */ mpf_t x, y; char buf[32]; gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y); All the standard C `scanf' types behave the same as in the C library `scanf', and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed to `scanf' and only the GMP extensions handled directly. The flags accepted are as follows. `a' and `'' will depend on support from the C library, and `'' cannot be used with GMP types. * read but don't store a allocate a buffer (string conversions) ' grouped digits, GLIBC style (not GMP types) The standard types accepted are as follows. `h' and `l' are portable, the rest will depend on the compiler (or include files) for the type and the C library for the input. h short hh char j intmax_t or uintmax_t l long int, double or wchar_t ll long long L long double q quad_t or u_quad_t t ptrdiff_t z size_t The GMP types are F mpf_t, float conversions Q mpq_t, integer conversions Z mpz_t, integer conversions The conversions accepted are as follows. `p' and `[' will depend on support from the C library, the rest are standard. c character or characters d decimal integer e E f g G float i integer with base indicator n characters read so far o octal integer p pointer s string of non-whitespace characters u decimal integer x X hex integer [ string of characters in a set `e', `E', `f', `g' and `G' are identical, they all read either fixed point or scientific format, and either upper or lower case `e' for the exponent in scientific format. C99 style hex float format (`printf %a', *note Formatted Output Strings::) is always accepted for `mpf_t', but for the standard float types it will depend on the C library. `x' and `X' are identical, both accept both upper and lower case hexadecimal. `o', `u', `x' and `X' all read positive or negative values. For the standard C types these are described as "unsigned" conversions, but that merely affects certain overflow handling, negatives are still allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of Integers.). For GMP types there are no overflows, so `d' and `u' are identical. `Q' type reads the numerator and (optional) denominator as given. If the value might not be in canonical form then `mpq_canonicalize' must be called before using it in any calculations (*note Rational Number Functions::). `Qi' will read a base specification separately for the numerator and denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11' would be 16/17. `n' can be used with any of the types above, even the GMP types. `*' to suppress assignment is allowed, though in that case it would do nothing at all. Other conversions or types that might be accepted by the C library `scanf' cannot be used through `gmp_scanf'. Whitespace is read and discarded before a field, except for `c' and `[' conversions. For float conversions, the decimal point character (or string) expected is taken from the current locale settings on systems which provide `localeconv' (*note Locales and Internationalization: (libc)Locales.). The C library will normally do the same for standard float input. The format string is only interpreted as plain `char's, multibyte characters are not recognised. Perhaps this will change in the future. Formatted Input Functions ========================= Each of the following functions is similar to the corresponding C library function. The plain `scanf' forms take a variable argument list. The `vscanf' forms take an argument pointer, see *Note Variadic Functions: (libc)Variadic Functions, or `man 3 va_start'. It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions. No overlap is permitted between the FMT string and any of the results produced. - Function: int gmp_scanf (const char *FMT, ...) - Function: int gmp_vscanf (const char *FMT, va_list AP) Read from the standard input `stdin'. - Function: int gmp_fscanf (FILE *FP, const char *FMT, ...) - Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP) Read from the stream FP. - Function: int gmp_sscanf (const char *S, const char *FMT, ...) - Function: int gmp_vsscanf (const char *S, const char *FMT, va_list AP) Read from a null-terminated string S. The return value from each of these functions is the same as the standard C99 `scanf', namely the number of fields successfully parsed and stored. `%n' fields and fields read but suppressed by `*' don't count towards the return value. If end of input (or a file error) is reached before a character for a field or a literal, and if no previous non-suppressed fields have matched, then the return value is `EOF' instead of 0. A whitespace character in the format string is only an optional match and doesn't induce an `EOF' in this fashion. Leading whitespace read and discarded for a field don't count as characters for that field. For the GMP types, input parsing follows C99 rules, namely one character of lookahead is used and characters are read while they continue to meet the format requirements. If this doesn't provide a complete number then the function terminates, with that field not stored nor counted towards the return value. For instance with `mpf_t' an input `1.23e-XYZ' would be read up to the `X' and that character pushed back since it's not a digit. The string `1.23e-' would then be considered invalid since an `e' must be followed by at least one digit. For the standard C types, in the current implementation GMP calls the C library `scanf' functions, which might have looser rules about what constitutes a valid input. Note that `gmp_sscanf' is the same as `gmp_fscanf' and only does one character of lookahead when parsing. Although clearly it could look at its entire input, it is deliberately made identical to `gmp_fscanf', the same way C99 `sscanf' is the same as `fscanf'. end gmp guide