Provided by: python-gmpy2-common_2.0.2-1ubuntu2_all 
NAME
gmpy2 - gmpy2 Documentation
Contents:
INTRODUCTION TO GMPY2
gmpy2 is a C-coded Python extension module that supports multiple-precision arithmetic.
gmpy2 is the successor to the original gmpy module. The gmpy module only supported the GMP
multiple-precision library. gmpy2 adds support for the MPFR (correctly rounded real
floating-point arithmetic) and MPC (correctly rounded complex floating-point arithmetic)
libraries. gmpy2 also updates the API and naming conventions to be more consistent and
support the additional functionality.
The following libraries are supported:
· GMP for integer and rational arithmetic
Home page: http://gmplib.org
· MPIR is based on the GMP library but adds support for Microsoft's Visual Studio
compiler. It is used to create the Windows binaries.
Home page: http://www.mpir.org
· MPFR for correctly rounded real floating-point arithmetic
Home page: http://www.mpfr.org
· MPC for correctly rounded complex floating-point arithmetic
Home page: http://mpc.multiprecision.org
· Generalized Lucas sequences and primality tests are based on the following code:
mpz_lucas: http://sourceforge.net/projects/mpzlucas/
mpz_prp: http://sourceforge.net/projects/mpzprp/
Changes in gmpy2 2.0.2
· Rebuild the Windows binary installers due to a bug in MPIR.
· Correct test in is_extra_strong_lucas_prp(). Note: The incorrect test is not known to
cause any errors.
Changes in gmpy2 2.0.1
· Updated setup.py to work in more situations.
· Corrected exception handling in basic operations with mpfr type.
· Correct InvalidOperation exception not raised in certain circumstances.
· invert() now raises an exception if the modular inverse does not exist.
· Fixed internal exception in is_bpsw_prp() and is_strong_bpsw_prp().
· Updated is_extra_strong_lucas_prp() to latest version.
Changes in gmpy2 2.0.0
· Fix segmentation fault in _mpmath_normalize (an undocumented helper function
specifically for mpmath.)
· Improved setup.py See below for documentation on the changes.
· Fix issues when compiled without support for MPFR.
· Conversion of too large an mpz to float now raises OverflowError instead of returning
inf.
· Renamed min2()/max2() to minnum()/maxnum()
· The build and install process (i.e. setup.py) has been completely rewritten. See the
Installation section for more information.
· get_context() no longer accepts keyword arguments.
Known issues in gmpy2 2.0.0
· The test suite is still incomplete.
Changes in gmpy2 2.0.0b4
· Added __ceil__, __floor__, __trunc__, and __round__ methods to mpz and mpq types.
· Added __complex__ to mpc type.
· round(mpfr) now correctly returns an mpz type.
· If no arguments are given to mpz, mpq, mpfr, mpc, and xmpz, return 0 of the appropriate
type.
· Fix broken comparison between mpz and mpq when mpz is on the left.
· Added __sizeof__ to all types. Note: sys.getsizeof() calls __sizeof__ to get the memory
size of a gmpy2 object. The returned value reflects the size of the allocated memory
which may be larger than the actual minimum memory required by the object.
Known issues in gmpy2 2.0.0b4
· The new test suite (test/runtest.py) is incomplete and some tests fail on Python 2.x due
to formating issues.
Changes in gmpy2 2.0.0b3
· mp_version(), mpc_version(), and mpfr_version() now return normal strings on Python 2.x
instead of Unicode strings.
· Faster conversion of the standard library Fraction type to mpq.
· Improved conversion of the Decimal type to mpfr.
· Consistently return OverflowError when converting "inf".
· Fix mpz.__format__() when the format code includes "#".
· Add is_infinite() and deprecate is_inf().
· Add is_finite() and deprecate is_number().
· Fixed the various is_XXX() tests when used with mpc.
· Added caching for mpc objects.
· Faster code path for basic operation is both operands are mpfr or mpc.
· Fix mpfr + float segmentation fault.
Changes in gmpy2 2.0.0b2
· Allow xmpz slice assignment to increase length of xmpz instance by specifying a value
for stop.
· Fixed reference counting bug in several is_xxx_prp() tests.
· Added iter_bits(), iter_clear(), iter_set() methods to xmpz.
· Added powmod() for easy access to three argument pow().
· Removed addmul() and submul() which were added in 2.0.0b1 since they are slower than
just using Python code.
· Bug fix in gcd_ext when both arguments are not mpz.
· Added ieee() to create contexts for 32, 64, or 128 bit floats.
· Bug fix in context() not setting emax/emin correctly if they had been changed earlier.
· Contexts can be directly used in with statement without requiring
set_context()/local_context() sequence.
· local_context() now accepts an optional context.
Changes in gmpy2 2.0.0b1 and earlier
· Renamed functions that manipulate individual bits to bit_XXX() to align with
bit_length().
· Added caching for mpq.
· Added rootrem(), fib2(), lucas(), lucas2().
· Support changed hash function in Python 3.2.
· Added is_even(), is_odd().
· Add caching of the calculated hash value.
· Add xmpz (mutable mpz) type.
· Fix mpq formatting issue.
· Add read/write bit access using slices to xmpz.
· Add read-only bit access using slices to mpz.
· Add pack()/unpack() methods to split/join an integer into n-bit chunks.
· Add support for MPFR (casevh)
· Removed fcoform float conversion modifier.
· Add support for MPC.
· Added context manager.
· Allow building with just GMP/MPIR if MPFR not available.
· Allow building with GMP/MPIR and MPFR if MPC not available.
· Removed most instance methods in favor of gmpy2.function. The general guideline is that
properties of an instance can be done via instance methods but functions that return a
new result are done using gmpy2.function.
· Added __ceil__, __floor__, and __trunc__ methods since they are called by math.ceil(),
math.floor(), and math.trunc().
· Removed gmpy2.pow() to avoid conflicts.
· Removed gmpy2._copy and added xmpz.copy.
· Added support for __format__.
· Added as_integer_ratio, as_mantissa_exp, as_simple_fraction.
· Updated rich_compare.
· Require MPFR 3.1.0+ to get divby0 support.
· Added fsum(), degrees(), radians().
· Updated random number generation support.
· Changed license to LGPL 3+.
· Added lucasu, lucasu_mod, lucasv, and lucasv_mod. Based on code contributed by David
Cleaver.
· Added probable-prime tests. Based on code contributed by David Cleaver.
· Added to_binary()/from_binary.
· Renamed numdigits() to num_digits().
· Added keyword precision to constants.
· Added addmul() and submul().
· Added __round__(), round2(), round_away() for mpfr.
· round() is no longer a module level function.
· Renamed module functions min()/max() to min2()/max2().
· No longer conflicts with builtin min() and max()
· Removed set_debug() and related functionality.
INSTALLATION
Installing gmpy2 on Windows
Pre-compiled versions of gmpy2 are available at Downloads . Please select the installer
that corresponds to the version of Python installed on your computer. Note that either a
32 or 64-bit version of Python can be installed on a 64-bit version of Windows. If you get
an error message stating that Python could not be found in the registry, you have the
wrong version of the gmpy2 installer.
Installing gmpy2 on Unix/Linux
Requirements
gmpy2 has only been tested with the most recent versions of GMP, MPFR and MPC.
Specifically, for integer and rational support, gmpy2 requires GMP 5.0.x or later. To
support multiple-precision floating point arithmetic, MPFR 3.1.x or later is required. MPC
1.0.1 or later is required for complex arithmetic.
Short Instructions
If your system includes sufficiently recent versions of GMP, MPFR and MPC, and you have
the development libraries installed, compiling should be as simple as:
cd <gmpy2 source directory>
python setup.py install
If this fails, read on.
Detailed Instructions
If your Linux distribution does not support recent versions of GMP, MPFR and MPC, you will
need to compile your own versions. To avoid any possible conflict with existing libraries
on your system, it is recommended to use a directory not normally used by your
distribution. setup.py will automatically search the following directories for the
required libraries:
1. /opt/local
2. /opt
3. /usr/local
4. /usr
5. /sw
If you can't use one of these directories, you can use a directory located in your home
directory. The examples will use /home/case/local. If you use one of standard directories
(say /opt/local), then you won't need to specify --prefix=/home/case/local to setup.py but
you will need to specify the prefix when compiling GMP, MPFR, and MPC.
Create the desired destination directory for GMP, MPFR, and MPC.
$ mkdir /home/case/local
Download and un-tar the GMP source code. Change to the GMP source directory and compile
GMP.
$ cd /home/case/local/src/gmp-5.1.0
$ ./configure --prefix=/home/case/local
$ make
$ make check
$ make install
Download and un-tar the MPFR source code. Change to the MPFR source directory and compile
MPFR.
$ cd /home/case/local/src/mpfr-3.1.1
$ ./configure --prefix=/home/case/local --with-gmp=/home/case/local
$ make
$ make check
$ make install
Download and un-tar the MPC source code. Change to the MPC source directory and compile
MPC.
$ cd /home/case/local/src/mpc-1.0.1
$ ./configure --prefix=/home/case/local --with-gmp=/home/case/local --with-mpfr=/home/case/local
$ make
$ make check
$ make install
Compile gmpy2 and specify the location of GMP, MPFR and MPC. The location of the GMP,
MPFR, and MPC libraries is embedded into the gmpy2 library so the new versions of GMP,
MPFR, and MPC do not need to be installed the system library directories. The prefix
directory is added to the beginning of the directories that are checked so it will be
found first.
$ python setup.py install --prefix=/home/case/local
If you get a "permission denied" error message, you may need to use:
$ python setup.py build --prefix=/home/case/local
$ sudo python setup.py install --prefix=/home/case/local
Options for setup.py
--force
Ignore the timestamps on all files and recompile. Normally, the results of a
previous compile are cached. To force gmpy2 to recognize external changes (updated
version of GMP, etc.), you will need to use this option.
--mpir Force the use of MPIR instead of GMP. GMP is the default library on non-Windows
operating systems.
--gmp Force the use of GMP instead of MPIR. MPIR is the default library on Windows
operating systems.
--prefix=<...>
Specify the directory prefix where GMP/MPIR, MPFR, and MPC are located. For
example, --prefix=/opt/local instructs setup.py to search /opt/local/include for
header files and /opt/local/lib for libraries.
--nompfr
Disables support for MPFR and MPC. This option is intended for testing purposes and
is not offically supported.
--nompc
Disables support MPC. This option is intended for testing purposes and is not
officially supported.
OVERVIEW OF GMPY2
Tutorial
The mpz type is compatible with Python's built-in int/long type but is significanly faster
for large values. The cutover point for performance varies, but can be as low as 20 to 40
digits. A variety of additional integer functions are provided.
>>> import gmpy2
>>> from gmpy2 import mpz,mpq,mpfr,mpc
>>> mpz(99) * 43
mpz(4257)
>>> pow(mpz(99), 37, 59)
mpz(18)
>>> gmpy2.isqrt(99)
mpz(9)
>>> gmpy2.isqrt_rem(99)
(mpz(9), mpz(18))
>>> gmpy2.gcd(123,27)
mpz(3)
>>> gmpy2.lcm(123,27)
mpz(1107)
The mpq type is compatible with the fractions.Fraction type included with Python.
>>> mpq(3,7)/7
mpq(3,49)
>>> mpq(45,3) * mpq(11,8)
mpq(165,8)
The most significant new features in gmpy2 are support for correctly rounded arbitrary
precision real and complex arithmetic based on the MPFR and MPC libraries. Floating point
contexts are used to control exceptional conditions. For example, division by zero can
either return an Infinity or raise an exception.
>>> mpfr(1)/7
mpfr('0.14285714285714285')
>>> gmpy2.get_context().precision=200
>>> mpfr(1)/7
mpfr('0.1428571428571428571428571428571428571428571428571428571428571',200)
>>> gmpy2.get_context()
context(precision=200, real_prec=Default, imag_prec=Default,
round=RoundToNearest, real_round=Default, imag_round=Default,
emax=1073741823, emin=-1073741823,
subnormalize=False,
trap_underflow=False, underflow=False,
trap_overflow=False, overflow=False,
trap_inexact=False, inexact=True,
trap_invalid=False, invalid=False,
trap_erange=False, erange=False,
trap_divzero=False, divzero=False,
trap_expbound=False,
allow_complex=False)
>>> mpfr(1)/0
mpfr('inf')
>>> gmpy2.get_context().trap_divzero=True
>>> mpfr(1)/0
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
gmpy2.DivisionByZeroError: 'mpfr' division by zero in division
>>> gmpy2.get_context()
context(precision=200, real_prec=Default, imag_prec=Default,
round=RoundToNearest, real_round=Default, imag_round=Default,
emax=1073741823, emin=-1073741823,
subnormalize=False,
trap_underflow=False, underflow=False,
trap_overflow=False, overflow=False,
trap_inexact=False, inexact=True,
trap_invalid=False, invalid=False,
trap_erange=False, erange=False,
trap_divzero=True, divzero=True,
trap_expbound=False,
allow_complex=False)
>>> gmpy2.sqrt(mpfr(-2))
mpfr('nan')
>>> gmpy2.get_context().allow_complex=True
>>> gmpy2.get_context().precision=53
>>> gmpy2.sqrt(mpfr(-2))
mpc('0.0+1.4142135623730951j')
>>>
>>> gmpy2.set_context(gmpy2.context())
>>> with gmpy2.local_context() as ctx:
... print(gmpy2.const_pi())
... ctx.precision+=20
... print(gmpy2.const_pi())
... ctx.precision+=20
... print(gmpy2.const_pi())
...
3.1415926535897931
3.1415926535897932384628
3.1415926535897932384626433831
>>> print(gmpy2.const_pi())
3.1415926535897931
>>>
Miscellaneous gmpy2 Functions
from_binary(...)
from_binary(bytes) returns a gmpy2 object from a byte sequence created by
to_binary().
get_cache(...)
get_cache() returns the current cache size (number of objects) and the maximum size
per object (number of limbs).
gmpy2 maintains an internal list of freed mpz, xmpz, mpq, mpfr, and mpc objects for
reuse. The cache significantly improves performance but also increases the memory
footprint.
license(...)
license() returns the gmpy2 license information.
mp_limbsize(...)
mp_limbsize() returns the number of bits per limb used by the GMP or MPIR libarary.
mp_version(...)
mp_version() returns the version of the GMP or MPIR library.
mpc_version(...)
mpc_version() returns the version of the MPC library.
mpfr_version(...)
mpfr_version() returns the version of the MPFR library.
random_state(...)
random_state([seed]) returns a new object containing state information for the
random number generator. An optional integer argument can be specified as the seed
value. Only the Mersenne Twister random number generator is supported.
set_cache(...)
set_cache(number, size) updates the maximum number of freed objects of each type
that are cached and the maximum size (in limbs) of each object. The maximum number
of objects of each type that can be cached is 1000. The maximum size of an object
is 16384. The maximum size of an object is approximately 64K on 32-bit systems and
128K on 64-bit systems.
NOTE:
The caching options are global to gmpy2. Changes are not thread-safe. A change
in one thread will impact all threads.
to_binary(...)
to_binary(x) returns a byte sequence from a gmpy2 object. All object types are
supported.
version(...)
version() returns the version of gmpy2.
MULTIPLE-PRECISION INTEGERS
The gmpy2 mpz type supports arbitrary precision integers. It should be a drop-in
replacement for Python's long type. Depending on the platform and the specific operation,
an mpz will be faster than Python's long once the precision exceeds 20 to 50 digits. All
the special integer functions in GMP are supported.
Examples
>>> import gmpy2
>>> from gmpy2 import mpz
>>> mpz('123') + 1
mpz(124)
>>> 10 - mpz(1)
mpz(9)
>>> gmpy2.is_prime(17)
True
NOTE:
The use of from gmpy2 import * is not recommended. The names in gmpy2 have been chosen
to avoid conflict with Python's builtin names but gmpy2 does use names that may
conflict with other modules or variable names.
mpz Methods
bit_clear(...)
x.bit_clear(n) returns a copy of x with bit n set to 0.
bit_flip(...)
x.bit_flip(n) returns a copy of x with bit n inverted.
bit_length(...)
x.bit_length() returns the number of significant bits in the radix-2 representation
of x. For compatibility with Python, mpz(0).bit_length() returns 0.
bit_scan0(...)
x.bit_scan0(n) returns the index of the first 0-bit of x with index >= n. If there
are no more 0-bits in x at or above index n (which can only happen for x < 0,
assuming an infinitely long 2's complement format), then None is returned. n must
be >= 0.
bit_scan1(...)
x.bit_scan1(n) returns the index of the first 1-bit of x with index >= n. If there
are no more 1-bits in x at or above index n (which can only happen for x >= 0,
assuming an infinitely long 2's complement format), then None is returned. n must
be >= 0.
bit_set(...)
x.bit_set(n) returns a copy of x with bit n set to 0.
bit_test(...)
x.bit_test(n) returns True if bit n of x is set, and False if it is not set.
denominator(...)
x.denominator() returns mpz(1).
digits(...)
x.digits([base=10]) returns a string representing x in radix base.
numerator(...)
x.numerator() returns a copy of x.
num_digits(...)
x.num_digits([base=10]) returns the length of the string representing the absolute
value of x in radix base. The result is correct if base is a power of 2. For other
other bases, the result is usually correct but may be 1 too large. base can range
between 2 and 62, inclusive.
mpz Functions
add(...)
add(x, y) returns x + y. The result type depends on the input types.
bincoef(...)
bincoef(x, n) returns the binomial coefficient. n must be >= 0.
bit_clear(...)
bit_clear(x, n) returns a copy of x with bit n set to 0.
bit_flip(...)
bit_flip(x, n) returns a copy of x with bit n inverted.
bit_length(...)
bit_length(x) returns the number of significant bits in the radix-2 representation
of x. For compatibility with Python, mpz(0).bit_length() returns 0 while
mpz(0).num_digits(2) returns 1.
bit_mask(...)
bit_mask(n) returns an mpz object exactly n bits in length with all bits set.
bit_scan0(...)
bit_scan0(x, n) returns the index of the first 0-bit of x with index >= n. If there
are no more 0-bits in x at or above index n (which can only happen for x < 0,
assuming an infinitely long 2's complement format), then None is returned. n must
be >= 0.
bit_scan1(...)
bit_scan1(x, n) returns the index of the first 1-bit of x with index >= n. If there
are no more 1-bits in x at or above index n (which can only happen for x >= 0,
assuming an infinitely long 2's complement format), then None is returned. n must
be >= 0.
bit_set(...)
bit_set(x, n) returns a copy of x with bit n set to 0.
bit_test(...)
bit_test(x, n) returns True if bit n of x is set, and False if it is not set.
c_div(...)
c_div(x, y) returns the quotient of x divided by y. The quotient is rounded towards
+Inf (ceiling rounding). x and y must be integers.
c_div_2exp(...)
c_div_2exp(x, n) returns the quotient of x divided by 2**n. The quotient is rounded
towards +Inf (ceiling rounding). x must be an integer and n must be > 0.
c_divmod(...)
c_divmod(x, y) returns the quotient and remainder of x divided by y. The quotient
is rounded towards +Inf (ceiling rounding) and the remainder will have the opposite
sign of y. x and y must be integers.
c_divmod_2exp(...)
c_divmod_2exp(x ,n) returns the quotient and remainder of x divided by 2**n. The
quotient is rounded towards +Inf (ceiling rounding) and the remainder will be
negative or zero. x must be an integer and n must be > 0.
c_mod(...)
c_mod(x, y) returns the remainder of x divided by y. The remainder will have the
opposite sign of y. x and y must be integers.
c_mod_2exp(...)
c_mod_2exp(x, n) returns the remainder of x divided by 2**n. The remainder will be
negative. x must be an integer and n must be > 0.
comb(...)
comb(x, n) returns the number of combinations of x things, taking n at a time. n
must be >= 0.
digits(...)
digits(x[, base=10]) returns a string representing x in radix base.
div(...)
div(x, y) returns x / y. The result type depends on the input types.
divexact(...)
divexact(x, y) returns the quotient of x divided by y. Faster than standard
division but requires the remainder is zero!
divm(...)
divm(a, b, m) returns x such that b * x == a modulo m. Raises a ZeroDivisionError
exception if no such value x exists.
f_div(...)
f_div(x, y) returns the quotient of x divided by y. The quotient is rounded towards
-Inf (floor rounding). x and y must be integers.
f_div_2exp(...)
f_div_2exp(x, n) returns the quotient of x divided by 2**n. The quotient is rounded
towards -Inf (floor rounding). x must be an integer and n must be > 0.
f_divmod(...)
f_divmod(x, y) returns the quotient and remainder of x divided by y. The quotient
is rounded towards -Inf (floor rounding) and the remainder will have the same sign
as y. x and y must be integers.
f_divmod_2exp(...)
f_divmod_2exp(x, n) returns quotient and remainder after dividing x by 2**n. The
quotient is rounded towards -Inf (floor rounding) and the remainder will be
positive. x must be an integer and n must be > 0.
f_mod(...)
f_mod(x, y) returns the remainder of x divided by y. The remainder will have the
same sign as y. x and y must be integers.
f_mod_2exp(...)
f_mod_2exp(x, n) returns remainder of x divided by 2**n. The remainder will be
positive. x must be an integer and n must be > 0.
fac(...)
fac(n) returns the exact factorial of n. Use factorial() to get the floating-point
approximation.
fib(...)
fib(n) returns the n-th Fibonacci number.
fib2(...)
fib2(n) returns a 2-tuple with the (n-1)-th and n-th Fibonacci numbers.
gcd(...)
gcd(a, b) returns the greatest common denominator of integers a and b.
gcdext(...)
gcdext(a, b) returns a 3-element tuple (g, s, t) such that
g == gcd(a, b) and g == a * s + b * t
hamdist(...)
hamdist(x, y) returns the Hamming distance (number of bit-positions where the bits
differ) between integers x and y.
invert(...)
invert(x, m) returns y such that x * y == 1 modulo m, or 0 if no such y exists.
iroot(...)
iroot(x,n) returns a 2-element tuple (y, b) such that y is the integer n-th root of
x and b is True if the root is exact. x must be >= 0 and n must be > 0.
iroot_rem(...)
iroot_rem(x,n) returns a 2-element tuple (y, r) such that y is the integer n-th
root of x and x = y**n + r. x must be >= 0 and n must be > 0.
is_even(...)
is_even(x) returns True if x is even, False otherwise.
is_odd(...)
is_odd(x) returns True if x is odd, False otherwise.
is_power(...)
is_power(x) returns True if x is a perfect power, False otherwise.
is_prime(...)
is_prime(x[, n=25]) returns True if x is probably prime. False is returned if x is
definately composite. x is checked for small divisors and up to n Miller-Rabin
tests are performed. The actual tests performed may vary based on version of GMP or
MPIR used.
is_square(...)
is_square(x) returns True if x is a perfect square, False otherwise.
isqrt(...)
isqrt(x) returns the integer square root of an integer x. x must be >= 0.
isqrt_rem(...)
isqrt_rem(x) returns a 2-tuple (s, t) such that s = isqrt(x) and t = x - s * s. x
must be >= 0.
jacobi(...)
jacobi(x, y) returns the Jacobi symbol (x | y). y must be odd and > 0.
kronecker(...)
kronecker(x, y) returns the Kronecker-Jacobi symbol (x | y).
lcm(...)
lcm(a, b) returns the lowest common multiple of integers a and b.
legendre(...)
legendre(x, y) returns the Legendre symbol (x | y). y is assumed to be an odd
prime.
lucas(...)
lucas(n) returns the n-th Lucas number.
lucas2(...)
lucas2(n) returns a 2-tuple with the (n-1)-th and n-th Lucas numbers.
mpz(...)
mpz() returns a new mpz object set to 0.
mpz(n) returns a new mpz object from a numeric value n. If n is not an integer, it
will be truncated to an integer.
mpz(s[, base=0]) returns a new mpz object from a string s made of digits in the
given base. If base = 0, thn binary, octal, or hex Python strings are recognized by
leading 0b, 0o, or 0x characters. Otherwise the string is assumed to be decimal.
Values for base can range between 2 and 62.
mpz_random(...)
mpz_random(random_state, n) returns a uniformly distributed random integer between
0 and n-1. The parameter random_state must be created by random_state() first.
mpz_rrandomb(...)
mpz_rrandomb(random_state, b) returns a random integer between 0 and 2**b - 1 with
long sequences of zeros and one in its binary representation. The parameter
random_state must be created by random_state() first.
mpz_urandomb(...)
mpz_urandomb(random_state, b) returns a uniformly distributed random integer
between 0 and 2**b - 1. The parameter random_state must be created by
random_state() first.
mul(...)
mul(x, y) returns x * y. The result type depends on the input types.
next_prime(...)
next_prime(x) returns the next probable prime number > x.
num_digits(...)
num_digits(x[, base=10]) returns the length of the string representing the absolute
value of x in radix base. The result is correct if base is a power of 2. For other
other bases, the result is usually correct but may be 1 too large. base can range
between 2 and 62, inclusive.
popcount(...)
popcount(x) returns the number of bits with value 1 in x. If x < 0, the number of
bits with value 1 is infinite so -1 is returned in that case.
powmod(...)
powmod(x, y, m) returns (x ** y) mod m. The exponenent y can be negative, and the
correct result will be returned if the inverse of x mod m exists. Otherwise, a
ValueError is raised.
remove(...)
remove(x, f) will remove the factor f from x as many times as possible and return a
2-tuple (y, m) where y = x // (f ** m). f does not divide y. m is the multiplicity
of the factor f in x. f must be > 1.
sub(...)
sub(x, y) returns x - y. The result type depends on the input types.
t_div(...)
t_div(x, y) returns the quotient of x divided by y. The quotient is rounded towards
zero (truncation). x and y must be integers.
t_div_2exp(...)
t_div_2exp(x, n) returns the quotient of x divided by 2**n. The quotient is rounded
towards zero (truncation). n must be > 0.
t_divmod(...)
t_divmod(x, y) returns the quotient and remainder of x divided by y. The quotient
is rounded towards zero (truncation) and the remainder will have the same sign as
x. x and y must be integers.
t_divmod_2exp(...)
t_divmod_2exp(x, n) returns the quotient and remainder of x divided by 2**n. The
quotient is rounded towards zero (truncation) and the remainder will have the same
sign as x. x must be an integer and n must be > 0.
t_mod(...)
t_mod(x, y) returns the remainder of x divided by y. The remainder will have the
same sign as x. x and y must be integers.
t_mod_2exp(...)
t_mod_2exp(x, n) returns the remainder of x divided by 2**n. The remainder will
have the same sign as x. x must be an integer and n must be > 0.
MULTIPLE-PRECISION INTEGERS (ADVANCED TOPICS)
The xmpz type
gmpy2 provides access to an experimental integer type called xmpz. The xmpz type is a
mutable integer type. In-place operations (+=, //=, etc.) modify the orignal object and
do not create a new object. Instances of xmpz cannot be used as dictionary keys.
>>> import gmpy2
>>> from gmpy2 import xmpz
>>> a = xmpz(123)
>>> b = a
>>> a += 1
>>> a
xmpz(124)
>>> b
xmpz(124)
The ability to change an xmpz object in-place allows for efficient and rapid bit
manipulation.
Individual bits can be set or cleared:
>>> a[10]=1
>>> a
xmpz(1148)
Slice notation is supported. The bits referenced by a slice can be either 'read from' or
'written to'. To clear a slice of bits, use a source value of 0. In 2s-complement format,
0 is represented by an arbitrary number of 0-bits. To set a slice of bits, use a source
value of ~0. The tilde operator inverts, or complements the bits in an integer. (~0 is -1
so you can also use -1.) In 2s-complement format, -1 is represented by an arbitrary number
of 1-bits.
If a value for stop is specified in a slice assignment and the actual bit-length of the
xmpz is less than stop, then the destination xmpz is logically padded with 0-bits to
length stop.
>>> a=xmpz(0)
>>> a[8:16] = ~0
>>> bin(a)
'0b1111111100000000'
>>> a[4:12] = ~a[4:12]
>>> bin(a)
'0b1111000011110000'
Bits can be reversed:
>>> bin(a)
'0b10001111100'
>>> a[::] = a[::-1]
>>> bin(a)
'0b111110001'
The iter_bits() method returns a generator that returns True or False for each bit
position. The methods iter_clear(), and iter_set() return generators that return the bit
positions that are 1 or 0. The methods support arguments start and stop that define the
beginning and ending bit positions that are used. To mimic the behavior of slices. the bit
positions checked include start but the last position checked is stop - 1.
>>> a=xmpz(117)
>>> bin(a)
'0b1110101'
>>> list(a.iter_bits())
[True, False, True, False, True, True, True]
>>> list(a.iter_clear())
[1, 3]
>>> list(a.iter_set())
[0, 2, 4, 5, 6]
>>> list(a.iter_bits(stop=12))
[True, False, True, False, True, True, True, False, False, False, False, False]
The following program uses the Sieve of Eratosthenes to generate a list of prime numbers.
from __future__ import print_function
import time
import gmpy2
def sieve(limit=1000000):
'''Returns a generator that yields the prime numbers up to limit.'''
# Increment by 1 to account for the fact that slices do not include
# the last index value but we do want to include the last value for
# calculating a list of primes.
sieve_limit = gmpy2.isqrt(limit) + 1
limit += 1
# Mark bit positions 0 and 1 as not prime.
bitmap = gmpy2.xmpz(3)
# Process 2 separately. This allows us to use p+p for the step size
# when sieving the remaining primes.
bitmap[4 : limit : 2] = -1
# Sieve the remaining primes.
for p in bitmap.iter_clear(3, sieve_limit):
bitmap[p*p : limit : p+p] = -1
return bitmap.iter_clear(2, limit)
if __name__ == "__main__":
start = time.time()
result = list(sieve())
print(time.time() - start)
print(len(result))
Advanced Number Theory Functions
The following functions are based on mpz_lucas.c and mpz_prp.c by David Cleaver.
A good reference for probable prime testing is http://www.pseudoprime.com/pseudo.html
is_bpsw_prp(...)
is_bpsw_prp(n) will return True if n is a Baillie-Pomerance-Selfridge-Wagstaff
probable prime. A BPSW probable prime passes the is_strong_prp() test with base 2
and the is_selfridge_prp() test.
is_euler_prp(...)
is_euler_prp(n,a) will return True if n is an Euler (also known as
Solovay-Strassen) probable prime to the base a.
Assuming:
gcd(n, a) == 1
n is odd
Then an Euler probable prime requires:
a**((n-1)/2) == 1 (mod n)
is_extra_strong_lucas_prp(...)
is_extra_strong_lucas_prp(n,p) will return True if n is an extra strong Lucas
probable prime with parameters (p,1).
Assuming:
n is odd
D = p*p - 4, D != 0
gcd(n, 2*D) == 1
n = s*(2**r) + Jacobi(D,n), s odd
Then an extra strong Lucas probable prime requires:
lucasu(p,1,s) == 0 (mod n)
or
lucasv(p,1,s) == +/-2 (mod n)
or
lucasv(p,1,s*(2**t)) == 0 (mod n) for some t, 0 <= t < r
is_fermat_prp(...)
is_fermat_prp(n,a) will return True if n is a Fermat probable prime to the base a.
Assuming:
gcd(n,a) == 1
Then a Fermat probable prime requires:
a**(n-1) == 1 (mod n)
is_fibonacci_prp(...)
is_fibonacci_prp(n,p,q) will return True if n is an Fibonacci probable prime with
parameters (p,q).
Assuming:
n is odd
p > 0, q = +/-1
p*p - 4*q != 0
Then a Fibonacci probable prime requires:
lucasv(p,q,n) == p (mod n).
is_lucas_prp(...)
is_lucas_prp(n,p,q) will return True if n is a Lucas probable prime with parameters
(p,q).
Assuming:
n is odd
D = p*p - 4*q, D != 0
gcd(n, 2*q*D) == 1
Then a Lucas probable prime requires:
lucasu(p,q,n - Jacobi(D,n)) == 0 (mod n)
is_selfridge_prp(...)
is_selfridge_prp(n) will return True if n is a Lucas probable prime with Selfidge
parameters (p,q). The Selfridge parameters are chosen by finding the first element
D in the sequence {5, -7, 9, -11, 13, ...} such that Jacobi(D,n) == -1. Let p=1 and
q = (1-D)/4 and then perform a Lucas probable prime test.
is_strong_bpsw_prp(...)
is_strong_bpsw_prp(n) will return True if n is a strong
Baillie-Pomerance-Selfridge-Wagstaff probable prime. A strong BPSW probable prime
passes the is_strong_prp() test with base 2 and the is_strongselfridge_prp() test.
is_strong_lucas_prp(...)
is_strong_lucas_prp(n,p,q) will return True if n is a strong Lucas probable prime
with parameters (p,q).
Assuming:
n is odd
D = p*p - 4*q, D != 0
gcd(n, 2*q*D) == 1
n = s*(2**r) + Jacobi(D,n), s odd
Then a strong Lucas probable prime requires:
lucasu(p,q,s) == 0 (mod n)
or
lucasv(p,q,s*(2**t)) == 0 (mod n) for some t, 0 <= t < r
is_strong_prp(...)
is_strong_prp(n,a) will return True if n is an strong (also known as Miller-Rabin)
probable prime to the base a.
Assuming:
gcd(n,a) == 1
n is odd
n = s*(2**r) + 1, with s odd
Then a strong probable prime requires one of the following is true:
a**s == 1 (mod n)
or
a**(s*(2**t)) == -1 (mod n) for some t, 0 <= t < r.
is_strong_selfridge_prp(...)
is_strong_selfridge_prp(n) will return True if n is a strong Lucas probable prime
with Selfidge parameters (p,q). The Selfridge parameters are chosen by finding the
first element D in the sequence {5, -7, 9, -11, 13, ...} such that Jacobi(D,n) ==
-1. Let p=1 and q = (1-D)/4 and then perform a strong Lucas probable prime test.
lucasu(...)
lucasu(p,q,k) will return the k-th element of the Lucas U sequence defined by p,q.
p*p - 4*q must not equal 0; k must be greater than or equal to 0.
lucasu_mod(...)
lucasu_mod(p,q,k,n) will return the k-th element of the Lucas U sequence defined by
p,q (mod n). p*p - 4*q must not equal 0; k must be greater than or equal to 0; n
must be greater than 0.
lucasv(...)
lucasv(p,q,k) will return the k-th element of the Lucas V sequence defined by
parameters (p,q). p*p - 4*q must not equal 0; k must be greater than or equal to 0.
lucasv_mod(...)
lucasv_mod(p,q,k,n) will return the k-th element of the Lucas V sequence defined by
parameters (p,q) (mod n). p*p - 4*q must not equal 0; k must be greater than or
equal to 0; n must be greater than 0.
MULTIPLE-PRECISION RATIONALS
gmpy2 provides a rational type call mpq. It should be a replacement for Python's
fractions.Fraction module.
>>> import gmpy2
>>> from gmpy2 import mpq
>>> mpq(1,7)
mpq(1,7)
>>> mpq(1,7) * 11
mpq(11,7)
>>> mpq(11,7)/13
mpq(11,91)
mpq Methods
digits(...)
x.digits([base=10]) returns a Python string representing x in the given base (2 to
62, default is 10). A leading '-' is present if x < 0, but no leading '+' is
present if x >= 0.
mpq Attributes
denominator
x.denomintor returns the denominator of x.
numerator
x.numerator returns the numerator of x.
mpq Functions
add(...)
add(x, y) returns x + y. The result type depends on the input types.
div(...)
div(x, y) returns x / y. The result type depends on the input types.
f2q(...)
f2q(x[, err]) returns the best mpq approximating x to within relative error err.
Default is the precision of x. If x is not an mpfr, it is converted to an mpfr.
Uses Stern-Brocot tree to find the best approximation. An mpz is returned if the
the denominator is 1. If err < 0, then the relative error sought is 2.0 ** err.
mpq(...)
mpq() returns an mpq object set to 0/1.
mpq(n) returns an mpq object with a numeric value n. Decimal and Fraction values
are converted exactly.
mpq(n, m) returns an mpq object with a numeric value n / m.
mpq(s[, base=10]) returns an mpq object from a string s made up of digits in the
given base. s may be made up of two numbers in the same base separated by a '/'
character. If base == 10, then an embedded '.' indicates a number with a decimal
fractional part.
mul(...)
mul(x, y) returns x * y. The result type depends on the input types.
qdiv(...)
qdiv(x[, y=1]) returns x/y as mpz if possible, or as mpq if x is not exactly
divisible by y.
sub(...)
sub(x, y) returns x - y. The result type depends on the input types.
MULTIPLE-PRECISION REALS
gmpy2 replaces the mpf type from gmpy 1.x with a new mpfr type based on the MPFR library.
The new mpfr type supports correct rounding, selectable rounding modes, and many
trigonometric, exponential, and special functions. A context manager is used to control
precision, rounding modes, and the behavior of exceptions.
The default precision of an mpfr is 53 bits - the same precision as Python's float type.
If the precison is changed, then mpfr(float('1.2')) differs from mpfr('1.2'). To take
advantage of the higher precision provided by the mpfr type, always pass constants as
strings.
>>> import gmpy2
>>> from gmpy2 import mpfr
>>> mpfr('1.2')
mpfr('1.2')
>>> mpfr(float('1.2'))
mpfr('1.2')
>>> gmpy2.get_context().precision=100
>>> mpfr('1.2')
mpfr('1.2000000000000000000000000000006',100)
>>> mpfr(float('1.2'))
mpfr('1.1999999999999999555910790149937',100)
>>>
Contexts
WARNING:
Contexts and context managers are not thread-safe! Modifying the context in one thread
will impact all other threads.
A context is used to control the behavior of mpfr and mpc arithmetic. In addition to
controlling the precision, the rounding mode can be specified, minimum and maximum
exponent values can be changed, various exceptions can be raised or ignored, gradual
underflow can be enabled, and returning complex results can be enabled.
gmpy2.context() creates a new context with all options set to default.
gmpy2.set_context(ctx) will set the active context to ctx. gmpy2.get_context() will
return a reference to the active context. Note that contexts are mutable: modifying the
reference returned by get_context() will modify the active context until a new context is
enabled with set_context(). The copy() method of a context will return a copy of the
context.
The following example just modifies the precision. The remaining options will be discussed
later.
>>> gmpy2.set_context(gmpy2.context())
>>> gmpy2.get_context()
context(precision=53, real_prec=Default, imag_prec=Default,
round=RoundToNearest, real_round=Default, imag_round=Default,
emax=1073741823, emin=-1073741823,
subnormalize=False,
trap_underflow=False, underflow=False,
trap_overflow=False, overflow=False,
trap_inexact=False, inexact=False,
trap_invalid=False, invalid=False,
trap_erange=False, erange=False,
trap_divzero=False, divzero=False,
trap_expbound=False,
allow_complex=False)
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997898')
>>> gmpy2.get_context().precision=100
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997896964091736687316',100)
>>> gmpy2.get_context().precision+=20
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997896964091736687312762351',120)
>>> ctx=gmpy2.get_context()
>>> ctx.precision+=20
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997896964091736687312762354406182',140)
>>> gmpy2.set_context(gmpy2.context())
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997898')
>>> ctx.precision+=20
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997898')
>>> gmpy2.set_context(ctx)
>>> gmpy2.sqrt(5)
mpfr('2.2360679774997896964091736687312762354406183596116',160)
>>>
Context Attributes
precision
This attribute controls the precision of an mpfr result. The precision is specified
in bits, not decimal digits. The maximum precision that can be specified is
platform dependent and can be retrieved with get_max_precision().
NOTE:
Specifying a value for precision that is too close to the maximum precision will cause
the MPFR library to fail.
real_prec
This attribute controls the precision of the real part of an mpc result. If the
value is Default, then the value of the precision attribute is used.
imag_prec
This attribute controls the precision of the imaginary part of an mpc result. If
the value is Default, then the value of real_prec is used.
round There are five rounding modes availble to mpfr types:
RoundAwayZero
The result is rounded away from 0.0.
RoundDown
The result is rounded towards -Infinity.
RoundToNearest
Round to the nearest value; ties are rounded to an even value.
RoundToZero
The result is rounded towards 0.0.
RoundUp
The result is rounded towards +Infinity.
real_round
This attribute controls the rounding mode for the real part of an mpc result. If
the value is Default, then the value of the round attribute is used. Note:
RoundAwayZero is not a valid rounding mode for mpc.
imag_round
This attribute controls the rounding mode for the imaginary part of an mpc result.
If the value is Default, then the value of the real_round attribute is used. Note:
RoundAwayZero is not a valid rounding mode for mpc.
emax This attribute controls the maximum allowed exponent of an mpfr result. The
maximum exponent is platform dependent and can be retrieved with get_emax_max().
emin This attribute controls the minimum allowed exponent of an mpfr result. The
minimum exponent is platform dependent and can be retrieved with get_emin_min().
NOTE:
It is possible to change the values of emin/emax such that previous mpfr values are no
longer valid numbers but should either underflow to +/-0.0 or overflow to +/-Infinity.
To raise an exception if this occurs, see trap_expbound.
subnormalize
The usual IEEE-754 floating point representation supports gradual underflow when
the minimum exponent is reached. The MFPR library does not enable gradual underflow
by default but it can be enabled to precisely mimic the results of IEEE-754
floating point operations.
trap_underflow
If set to False, a result that is smaller than the smallest possible mpfr given the
current exponent range will be replaced by +/-0.0. If set to True, an
UnderflowResultError exception is raised.
underflow
This flag is not user controllable. It is automatically set if a result underflowed
to +/-0.0 and trap_underflow is False.
trap_overflow
If set to False, a result that is larger than the largest possible mpfr given the
current exponent range will be replaced by +/-Infinity. If set to True, an
OverflowResultError exception is raised.
overflow
This flag is not user controllable. It is automatically set if a result overflowed
to +/-Infinity and trap_overflow is False.
trap_inexact
This attribute controls whether or not an InexactResultError exception is raised if
an inexact result is returned. To check if the result is greater or less than the
exact result, check the rc attribute of the mpfr result.
inexact
This flag is not user controllable. It is automatically set if an inexact result is
returned.
trap_invalid
This attribute controls whether or not an InvalidOperationError exception is raised
if a numerical result is not defined. A special NaN (Not-A-Number) value will be
returned if an exception is not raised. The InvalidOperationError is a sub-class
of Python's ValueError.
For example, gmpy2.sqrt(-2) will normally return mpfr('nan'). However, if
allow_complex is set to True, then an mpc result will be returned.
invalid
This flag is not user controllable. It is automatically set if an invalid
(Not-A-Number) result is returned.
trap_erange
This attribute controls whether or not a RangeError exception is raised when
certain operations are performed on NaN and/or Infinity values. Setting
trap_erange to True can be used to raise an exception if comparisons are attempted
with a NaN.
>>> gmpy2.set_context(gmpy2.context())
>>> mpfr('nan') == mpfr('nan')
False
>>> gmpy2.get_context().trap_erange=True
>>> mpfr('nan') == mpfr('nan')
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
gmpy2.RangeError: comparison with NaN
>>>
erange This flag is not user controllable. It is automatically set if an erange error
occurred.
trap_divzero
This attribute controls whether or not a DivisionByZeroError exception is raised if
division by 0 occurs. The DivisionByZeroError is a sub-class of Python's
ZeroDivisionError.
divzero
This flag is not user controllable. It is automatically set if a division by zero
occurred and NaN result was returned.
trap_expbound
This attribute controls whether or not an ExponentOutOfBoundsError exception is
raised if exponents in an operand are outside the current emin/emax limits.
allow_complex
This attribute controls whether or not an mpc result can be returned if an mpfr
result would normally not be possible.
Context Methods
clear_flags()
Clear the underflow, overflow, inexact, invalid, erange, and divzero flags.
copy() Return a copy of the context.
Contexts and the with statement
Contexts can also be used in conjunction with Python's with ... statement to temporarily
change the context settings for a block of code and then restore the original settings
when the block of code exits.
gmpy2.local_context() first save the current context and then creates a new context based
on a context passed as the first argument, or the current context if no context is passed.
The new context is modified if any optional keyword arguments are given. The orginal
active context is restored when the block completes.
In the following example, the current context is saved by gmpy2.local_context() and then
the block begins with a copy of the default context and the precision set to 100. When the
block is finished, the original context is restored.
>>> with gmpy2.local_context(gmpy2.context(), precision=100) as ctx:
... print(gmpy2.sqrt(2))
... ctx.precision += 100
... print(gmpy2.sqrt(2))
...
1.4142135623730950488016887242092
1.4142135623730950488016887242096980785696718753769480731766796
>>>
A context object can also be used directly to create a context manager block. However,
instead of restoring the context to the active context when the with ... statement is
executed, the restored context is the context used before any keyword argument
modifications.
The code:
:: with gmpy2.ieee(64) as ctx:
is equivalent to:
:: gmpy2.set_context(gmpy2.ieee(64)) with gmpy2.local_context() as ctx:
Contexts that implement the standard single, double, and quadruple precision floating
point types can be created using ieee().
mpfr Methods
as_integer_ratio()
Returns a 2-tuple containing the numerator and denominator after converting the
mpfr object into the exact rational equivalent. The return 2-tuple is equivalent to
Python's as_integer_ratio() method of built-in float objects.
as_mantissa_exp()
Returns a 2-tuple containing the mantissa and exponent.
as_simple_fraction()
Returns an mpq containing the simpliest rational value that approximates the mpfr
value with an error less than 1/(2**precision).
conjugate()
Returns the complex conjugate. For mpfr objects, returns a copy of the original
object.
digits()
Returns a 3-tuple containing the mantissa, the exponent, and the number of bits of
precision. The mantissa is represented as a string in the specified base with up to
'prec' digits. If 'prec' is 0, as many digits that are available are returned. No
more digits than available given x's precision are returned. 'base' must be between
2 and 62, inclusive.
is_integer()
Returns True if the mpfr object is an integer.
mpfr Attributes
imag Returns the imaginary component. For mpfr objects, returns 0.
precision
Returns the precision of the mpfr object.
rc The result code (also known as ternary value in the MPFR documentation) is 0 if the
value of the mpfr object is exactly equal to the exact, infinite precision value.
If the result code is 1, then the value of the mpfr object is greater than the
exact value. If the result code is -1, then the value of the mpfr object is less
than the exact, infinite precision value.
real Returns the real component. For mpfr objects, returns a copy of the original
object.
mpfr Functions
acos(...)
acos(x) returns the arc-cosine of x. x is measured in radians. If
context.allow_complex is True, then an mpc result will be returned for abs(x) > 1.
acosh(...)
acosh(x) returns the inverse hyperbolic cosine of x.
add(...)
add(x, y) returns x + y. The type of the result is based on the types of the
arguments.
agm(...)
agm(x, y) returns the arithmetic-geometric mean of x and y.
ai(...)
ai(x) returns the Airy function of x.
asin(...)
asin(x) returns the arc-sine of x. x is measured in radians. If
context.allow_complex is True, then an mpc result will be returned for abs(x) > 1.
asinh(...)
asinh(x) return the inverse hyperbolic sine of x.
atan(...)
atan(x) returns the arc-tangent of x. x is measured in radians.
atan2(...)
atan2(y, x) returns the arc-tangent of (y/x).
atanh(...)
atanh(x) returns the inverse hyperbolic tangent of x. If context.allow_complex is
True, then an mpc result will be returned for abs(x) > 1.
cbrt(...)
cbrt(x) returns the cube root of x.
ceil(...)
ceil(x) returns the 'mpfr' that is the smallest integer >= x.
check_range(...)
check_range(x) return a new 'mpfr' with exponent that lies within the current range
of emin and emax.
const_catalan(...)
const_catalan([precision=0]) returns the catalan constant using the specified
precision. If no precision is specified, the default precision is used.
const_euler(...)
const_euler([precision=0]) returns the euler constant using the specified
precision. If no precision is specified, the default precision is used.
const_log2(...)
const_log2([precision=0]) returns the log2 constant using the specified precision.
If no precision is specified, the default precision is used.
const_pi(...)
const_pi([precision=0]) returns the constant pi using the specified precision. If
no precision is specified, the default precision is used.
context(...)
context() returns a new context manager controlling MPFR and MPC arithmetic.
cos(...)
cos(x) seturns the cosine of x. x is measured in radians.
cosh(...)
cosh(x) returns the hyperbolic cosine of x.
cot(...)
cot(x) returns the cotangent of x. x is measured in radians.
coth(...)
coth(x) returns the hyperbolic cotangent of x.
csc(...)
csc(x) returns the cosecant of x. x is measured in radians.
csch(...)
csch(x) returns the hyperbolic cosecant of x.
degrees(...)
degrees(x) converts an angle measurement x from radians to degrees.
digamma(...)
digamma(x) returns the digamma of x.
div(...)
div(x, y) returns x / y. The type of the result is based on the types of the
arguments.
div_2exp(...)
div_2exp(x, n) returns an 'mpfr' or 'mpc' divided by 2**n.
eint(...)
eint(x) returns the exponential integral of x.
erf(...)
erf(x) returns the error function of x.
erfc(...)
erfc(x) returns the complementary error function of x.
exp(...)
exp(x) returns e**x.
exp10(...)
exp10(x) returns 10**x.
exp2(...)
exp2(x) returns 2**x.
expm1(...)
expm1(x) returns e**x - 1. expm1() is more accurate than exp(x) - 1 when x is
small.
f2q(...)
f2q(x[,err]) returns the simplest mpq approximating x to within relative error err.
Default is the precision of x. Uses Stern-Brocot tree to find the simplist
approximation. An mpz is returned if the the denominator is 1. If err<0, error
sought is 2.0 ** err.
factorial(...)
factorial(n) returns the floating-point approximation to the factorial of n.
See fac(n) to get the exact integer result.
floor(...)
floor(x) returns the 'mpfr' that is the smallest integer <= x.
fma(...)
fma(x, y, z) returns correctly rounded result of (x * y) + z.
fmod(...)
fmod(x, y) returns x - n*y where n is the integer quotient of x/y, rounded to 0.
fms(...)
fms(x, y, z) returns correctly rounded result of (x * y) - z.
frac(...)
frac(x) returns the fractional part of x.
frexp(...)
frexp(x) returns a tuple containing the exponent and mantissa of x.
fsum(...)
fsum(iterable) returns the accurate sum of the values in the iterable.
gamma(...)
gamma(x) returns the gamma of x.
get_exp(...)
get_exp(mpfr) returns the exponent of an mpfr. Returns 0 for NaN or Infinity and
sets the erange flag and will raise an exception if trap_erange is set.
hypot(...)
hypot(y, x) returns square root of (x**2 + y**2).
ieee(...)
ieee(bitwidth) returns a context with settings for 32-bit (aka single), 64-bit (aka
double), or 128-bit (aka quadruple) precision floating point types.
inf(...)
inf(n) returns an mpfr initialized to Infinity with the same sign as n. If n is
not given, +Infinity is returned.
is_finite(...)
is_finite(x) returns True if x is an actual number (i.e. not NaN or Infinity).
is_inf(...)
is_inf(x) returns True if x is Infinity or -Infinity.
NOTE:
is_inf() is deprecated; please use if_infinite().
is_infinite(...)
is_infinite(x) returns True if x Infinity or -Infinity.
is_nan(...)
is_nan(x) returns True if x is NaN (Not-A-Number).
is_number(...)
is_number(x) returns True if x is an actual number (i.e. not NaN or Infinity).
NOTE:
is_number() is deprecated; please use is_finite().
is_regular(...)
is_regular(x) returns True if x is not zero, NaN, or Infinity.
is_signed(...)
is_signed(x) returns True if the sign bit of x is set.
is_unordered(...)
is_unordered(x,y) returns True if either x and/or y is NaN.
is_zero(...)
is_zero(x) returns True if x is zero.
j0(...)
j0(x) returns the Bessel function of the first kind of order 0 of x.
j1(...)
j1(x) returns the Bessel function of the first kind of order 1 of x.
jn(...)
jn(x,n) returns the Bessel function of the first kind of order n of x.
lgamma(...)
lgamma(x) returns a tuple containing the logarithm of the absolute value of
gamma(x) and the sign of gamma(x)
li2(...)
li2(x) returns the real part of dilogarithm of x.
lngamma(...)
lngamma(x) returns the logarithm of gamma(x).
log(...)
log(x) returns the natural logarithm of x.
log10(...)
log10(x) returns the base-10 logarithm of x.
log1p(...)
log1p(x) returns the natural logarithm of (1+x).
log2(...)
log2(x) returns the base-2 logarithm of x.
max2(...)
max2(x, y) returns the maximum of x and y. The result may be rounded to match the
current context. Use the builtin max() to get an exact copy of the largest object
without any rounding.
min2(...)
min2(x, y) returns the minimum of x and y. The result may be rounded to match the
current context. Use the builtin min() to get an exact copy of the smallest object
without any rounding.
modf(...)
modf(x) returns a tuple containing the integer and fractional portions of x.
mpfr(...)
mpfr() returns and mpfr object set to 0.0.
mpfr(n[, precison=0]) returns an mpfr object after converting a numeric value n. If
no precision, or a precision of 0, is specified; the precision is taken from the
current context.
mpfr(s[, precision=0[, [base=0]]) returns an mpfr object after converting a string
's' made up of digits in the given base, possibly with fractional part (with period
as a separator) and/or exponent (with exponent marker 'e' for base<=10, else '@').
If no precision, or a precision of 0, is specified; the precison is taken from the
current context. The base of the string representation must be 0 or in the interval
2 ... 62. If the base is 0, the leading digits of the string are used to identify
the base: 0b implies base=2, 0x implies base=16, otherwise base=10 is assumed.
mpfr_from_old_binary(...)
mpfr_from_old_binary(string) returns an mpfr from a GMPY 1.x binary mpf format.
Please use to_binary()/from_binary() to convert GMPY2 objects to or from a binary
format.
mpfr_grandom(...)
mpfr_grandom(random_state) returns two random numbers with gaussian distribution.
The parameter random_state must be created by random_state() first.
mpfr_random(...)
mpfr_random(random_state) returns a uniformly distributed number between [0,1]. The
parameter random_state must be created by random_state() first.
mul(...)
mul(x, y) returns x * y. The type of the result is based on the types of the
arguments.
mul_2exp(...)
mul_2exp(x, n) returns 'mpfr' or 'mpc' multiplied by 2**n.
nan(...)
nan() returns an 'mpfr' initialized to NaN (Not-A-Number).
next_above(...)
next_above(x) returns the next 'mpfr' from x toward +Infinity.
next_below(...)
next_below(x) returns the next 'mpfr' from x toward -Infinity.
radians(...)
radians(x) converts an angle measurement x from degrees to radians.
rec_sqrt(...)
rec_sqrt(x) returns the reciprocal of the square root of x.
reldiff(...)
reldiff(x, y) returns the relative difference between x and y. Result is equal to
abs(x-y)/x.
remainder(...)
remainder(x, y) returns x - n*y where n is the integer quotient of x/y, rounded to
the nearest integer and ties rounded to even.
remquo(...)
remquo(x, y) returns a tuple containing the remainder(x,y) and the low bits of the
quotient.
rint(...)
rint(x) returns x rounded to the nearest integer using the current rounding mode.
rint_ceil(...)
rint_ceil(x) returns x rounded to the nearest integer by first rounding to the next
higher or equal integer and then, if needed, using the current rounding mode.
rint_floor(...)
rint_floor(x) returns x rounded to the nearest integer by first rounding to the
next lower or equal integer and then, if needed, using the current rounding mode.
rint_round(...)
rint_round(x) returns x rounded to the nearest integer by first rounding to the
nearest integer (ties away from 0) and then, if needed, using the current rounding
mode.
rint_trunc(...)
rint_trunc(x) returns x rounded to the nearest integer by first rounding towards
zero and then, if needed, using the current rounding mode.
root(...)
root(x, n) returns n-th root of x. The result always an mpfr.
round2(...)
round2(x[, n]) returns x rounded to n bits. Uses default precision if n is not
specified. See round_away() to access the mpfr_round() function. Use the builtin
round() to round x to n decimal digits.
round_away(...)
round_away(x) returns an mpfr by rounding x the nearest integer, with ties rounded
away from 0.
sec(...)
sec(x) returns the secant of x. x is measured in radians.
sech(...)
sech(x) returns the hyperbolic secant of x.
set_exp(...)
set_exp(x, n) sets the exponent of a given mpfr to n. If n is outside the range of
valid exponents, set_exp() will set the erange flag and either return the original
value or raise an exception if trap_erange is set.
set_sign(...)
set_sign(x, bool) returns a copy of x with it's sign bit set if bool evaluates to
True.
sign(...)
sign(x) returns -1 if x < 0, 0 if x == 0, or +1 if x >0.
sin(...)
sin(x) returns the sine of x. x is measured in radians.
sin_cos(...)
sin_cos(x) returns a tuple containing the sine and cosine of x. x is measured in
radians.
sinh(...)
sinh(x) returns the hyberbolic sine of x.
sinh_cosh(...)
sinh_cosh(x) returns a tuple containing the hyperbolic sine and cosine of x.
sqrt(...)
sqrt(x) returns the square root of x. If x is integer, rational, or real, then an
mpfr will be returned. If x is complex, then an mpc will be returned. If
context.allow_complex is True, negative values of x will return an mpc.
square(...)
square(x) returns x * x. The type of the result is based on the types of the
arguments.
sub(...)
sub(x, y) returns x - y. The type of the result is based on the types of the
arguments.
tan(...)
tan(x) returns the tangent of x. x is measured in radians.
tanh(...)
tanh(x) returns the hyperbolic tangent of x.
trunc(...)
trunc(x) returns an 'mpfr' that is x truncated towards 0. Same as x.floor() if x>=0
or x.ceil() if x<0.
y0(...)
y0(x) returns the Bessel function of the second kind of order 0 of x.
y1(...)
y1(x) returns the Bessel function of the second kind of order 1 of x.
yn(...)
yn(x,n) returns the Bessel function of the second kind of order n of x.
zero(...)
zero(n) returns an mpfr inialized to 0.0 with the same sign as n. If n is not
given, +0.0 is returned.
zeta(...)
zeta(x) returns the Riemann zeta of x.
mpfr Formatting
The mpfr type supports the __format__() special method to allow custom output formatting.
__format__(...)
x.__format__(fmt) returns a Python string by formatting 'x' using the format string
'fmt'. A valid format string consists of:
optional alignment code:
'<' -> left shifted in field
'>' -> right shifted in field
'^' -> centered in field
optional leading sign code
'+' -> always display leading sign
'-' -> only display minus for negative values
' ' -> minus for negative values, space for positive values
optional width.precision
optional rounding mode:
'U' -> round toward plus infinity
'D' -> round toward minus infinity
'Y' -> round away from zero
'Z' -> round toward zero
'N' -> round to nearest
optional conversion code:
'a','A' -> hex format
'b' -> binary format
'e','E' -> scientific format
'f','F' -> fixed point format
'g','G' -> fixed or scientific format
NOTE:
The formatting codes must be specified in the order shown above.
>>> import gmpy2
>>> from gmpy2 import mpfr
>>> a=mpfr("1.23456")
>>> "{0:15.3f}".format(a)
' 1.235'
>>> "{0:15.3Uf}".format(a)
' 1.235'
>>> "{0:15.3Df}".format(a)
' 1.234'
>>> "{0:.3Df}".format(a)
'1.234'
>>> "{0:+.3Df}".format(a)
'+1.234'
MULTIPLE-PRECISION COMPLEX
gmpy2 adds a multiple-precision complex type called mpc that is based on the MPC library.
The context manager settings for mpfr arithmetic are applied to mpc arithmetic by default.
It is possible to specifiy different precision and rounding modes for both the real and
imaginary components of an mpc.
>>> import gmpy2
>>> from gmpy2 import mpc
>>> gmpy2.sqrt(mpc("1+2j"))
mpc('1.272019649514069+0.78615137775742328j')
>>> gmpy2.get_context(real_prec=100,imag_prec=200)
context(precision=53, real_prec=100, imag_prec=200,
round=RoundToNearest, real_round=Default, imag_round=Default,
emax=1073741823, emin=-1073741823,
subnormalize=False,
trap_underflow=False, underflow=False,
trap_overflow=False, overflow=False,
trap_inexact=False, inexact=True,
trap_invalid=False, invalid=False,
trap_erange=False, erange=False,
trap_divzero=False, divzero=False,
trap_expbound=False,
allow_complex=False)
>>> gmpy2.sqrt(mpc("1+2j"))
mpc('1.2720196495140689642524224617376+0.78615137775742328606955858584295892952312205783772323766490213j',(100,200))
Exceptions are normally raised in Python when the result of a real operation is not
defined over the reals; for example, sqrt(-4) will raise an exception. The default context
in gmpy2 implements the same behavior but by setting allow_complex to True, complex
results will be returned.
>>> import gmpy2
>>> from gmpy2 import mpc
>>> gmpy2.sqrt(-4)
mpfr('nan')
>>> gmpy2.get_context(allow_complex=True)
context(precision=53, real_prec=Default, imag_prec=Default,
round=RoundToNearest, real_round=Default, imag_round=Default,
emax=1073741823, emin=-1073741823,
subnormalize=False,
trap_underflow=False, underflow=False,
trap_overflow=False, overflow=False,
trap_inexact=False, inexact=False,
trap_invalid=False, invalid=True,
trap_erange=False, erange=False,
trap_divzero=False, divzero=False,
trap_expbound=False,
allow_complex=True)
>>> gmpy2.sqrt(-4)
mpc('0.0+2.0j')
mpc Methods
conjugate()
Returns the complex conjugate.
digits()
Returns a two element tuple where each element represents the real and imaginary
components as a 3-tuple containing the mantissa, the exponent, and the number of
bits of precision. The mantissa is represented as a string in the specified base
with up to 'prec' digits. If 'prec' is 0, as many digits that are available are
returned. No more digits than available given x's precision are returned. 'base'
must be between 2 and 62, inclusive.
mpc Attributes
imag Returns the imaginary component.
precision
Returns a 2-tuple containing the the precision of the real and imaginary
components.
rc Returns a 2-tuple containing the ternary value of the real and imaginary
components. The ternary value is 0 if the value of the component is exactly equal
to the exact, infinite precision value. If the result code is 1, then the value of
the component is greater than the exact value. If the result code is -1, then the
value of the component is less than the exact, infinite precision value.
real Returns the real component.
mpc Functions
acos(...)
acos(x) returns the arc-cosine of x.
acosh(...)
acosh(x) returns the inverse hyperbolic cosine of x.
add(...)
add(x, y) returns x + y. The type of the result is based on the types of the
arguments.
asin(...)
asin(x) returns the arc-sine of x.
asinh(...)
asinh(x) return the inverse hyperbolic sine of x.
atan(...)
atan(x) returns the arc-tangent of x.
atanh(...)
atanh(x) returns the inverse hyperbolic tangent of x.
cos(...)
cos(x) seturns the cosine of x.
cosh(...)
cosh(x) returns the hyperbolic cosine of x.
div(...)
div(x, y) returns x / y. The type of the result is based on the types of the
arguments.
div_2exp(...)
div_2exp(x, n) returns an 'mpfr' or 'mpc' divided by 2**n.
exp(...)
exp(x) returns e**x.
fma(...)
fma(x, y, z) returns correctly rounded result of (x * y) + z.
fms(...)
fms(x, y, z) returns correctly rounded result of (x * y) - z.
is_inf(...)
is_inf(x) returns True if either the real or imaginary component of x is Infinity
or -Infinity.
is_nan(...)
is_nan(x) returns True if either the real or imaginary component of x is NaN
(Not-A-Number).
is_zero(...)
is_zero(x) returns True if x is zero.
log(...)
log(x) returns the natural logarithm of x.
log10(...)
log10(x) returns the base-10 logarithm of x.
mpc(...)
mpc() returns an mpc object set to 0.0+0.0j.
mpc(c[, precision=0]) returns a new 'mpc' object from an existing complex number
(either a Python complex object or another 'mpc' object). If the precision is not
specified, then the precision is taken from the current context. The rounding mode
is always taken from the current context.
mpc(r[, i=0[, precision=0]]) returns a new 'mpc' object by converting two
non-complex numbers into the real and imaginary components of an 'mpc' object. If
the precision is not specified, then the precision is taken from the current
context. The rounding mode is always taken from the current context.
mpc(s[, [precision=0[, base=10]]) returns a new 'mpc' object by converting a string
s into a complex number. If base is omitted, then a base-10 representation is
assumed otherwise a base between 2 and 36 can be specified. If the precision is not
specified, then the precision is taken from the current context. The rounding mode
is always taken from the current context.
In addition to the standard Python string representation of a complex number:
"1+2j", the string representation used by the MPC library: "(1 2)" is also
supported.
NOTE:
The precision can be specified either a single number that is used for both the
real and imaginary components, or as a 2-tuple that can specify different
precisions for the real and imaginary components.
mpc_random(...)
mpfc_random(random_state) returns a uniformly distributed number in the unit square
[0,1]x[0,1]. The parameter random_state must be created by random_state() first.
mul(...)
mul(x, y) returns x * y. The type of the result is based on the types of the
arguments.
mul_2exp(...)
mul_2exp(x, n) returns 'mpfr' or 'mpc' multiplied by 2**n.
norm(...)
norm(x) returns the norm of a complex x. The norm(x) is defined as x.real**2 +
x.imag**2. abs(x) is the square root of norm(x).
phase(...)
phase(x) returns the phase angle, also known as argument, of a complex x.
polar(...)
polar(x) returns the polar coordinate form of a complex x that is in rectangular
form.
proj(...)
proj(x) returns the projection of a complex x on to the Riemann sphere.
rect(...)
rect(x) returns the polar coordinate form of a complex x that is in rectangular
form.
sin(...)
sin(x) returns the sine of x.
sinh(...)
sinh(x) returns the hyberbolic sine of x.
sqrt(...)
sqrt(x) returns the square root of x. If x is integer, rational, or real, then an
mpfr will be returned. If x is complex, then an mpc will be returned. If
context.allow_complex is True, negative values of x will return an mpc.
square(...)
square(x) returns x * x. The type of the result is based on the types of the
arguments.
sub(...)
sub(x, y) returns x - y. The type of the result is based on the types of the
arguments.
tan(...)
tan(x) returns the tangent of x. x is measured in radians.
tanh(...)
tanh(x) returns the hyperbolic tangent of x.
mpc Formatting
The mpc type supports the __format__() special method to allow custom output formatting.
__format__(...)
x.__format__(fmt) returns a Python string by formatting 'x' using the format string
'fmt'. A valid format string consists of:
optional alignment code:
'<' -> left shifted in field
'>' -> right shifted in field
'^' -> centered in field
optional leading sign code
'+' -> always display leading sign
'-' -> only display minus for negative values
' ' -> minus for negative values, space for positive values
optional width.real_precision.imag_precision
optional rounding mode:
'U' -> round toward plus infinity
'D' -> round toward minus infinity
'Z' -> round toward zero
'N' -> round to nearest
optional output style:
'P' -> Python style, 1+2j, (default)
'M' -> MPC style, (1 2)
optional conversion code:
'a','A' -> hex format
'b' -> binary format
'e','E' -> scientific format
'f','F' -> fixed point format
'g','G' -> fixed or scientific format
NOTE:
The formatting codes must be specified in the order shown above.
>>> import gmpy2
>>> from gmpy2 import mpc
>>> a=gmpy2.sqrt(mpc("1+2j"))
>>> a
mpc('1.272019649514069+0.78615137775742328j')
>>> "{0:.4.4Mf}".format(a)
'(1.2720 0.7862)'
>>> "{0:.4.4f}".format(a)
'1.2720+0.7862j'
>>> "{0:^20.4.4U}".format(a)
' 1.2721+0.7862j '
>>> "{0:^20.4.4D}".format(a)
' 1.2720+0.7861j '
· genindex
· modindex
· search
AUTHOR
Case Van Horsen
COPYRIGHT
2012, 2013, Case Van Horsen