Package arithmetic

source code

Low-level arithmetics support.

Arithmetic package provides:

low-level number types FractionTuple, Float, Complex

base class Infinity for extended numbers

various low-level number theory functions: gcd, multinomial_coefficients, etc.

Submodules

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sympycore.arithmetic.evalf: Provides low-level evalf function.
sympycore.arithmetic.infinity: Provides base class for extended numbers: Infinity.
sympycore.arithmetic.methods: Generated arithmetic methods for numbers.
sympycore.arithmetic.mpmath
- sympycore.arithmetic.mpmath.apps
  - sympycore.arithmetic.mpmath.apps.extrafun: Numerical implementations of special functions (gamma, ...)
  - sympycore.arithmetic.mpmath.apps.interval: Defines mpi class for interval arithmetic.
  - sympycore.arithmetic.mpmath.apps.pidigits: Calculate digits of pi using AGM iteration.
  - sympycore.arithmetic.mpmath.apps.quad: Provides integration using quadratures.
  - sympycore.arithmetic.mpmath.apps.taylordemo: Interval arithmetic demo: estimating error of numerical Taylor series.
- sympycore.arithmetic.mpmath.lib
  - sympycore.arithmetic.mpmath.lib.complexop: Provides various complex valued functions.
  - sympycore.arithmetic.mpmath.lib.constants: Mathematical constants
  - sympycore.arithmetic.mpmath.lib.floatop: This module implements fundamental operations for floating-point arithmetic (bit fiddling, normalization, arithmetic, ...).
  - sympycore.arithmetic.mpmath.lib.functions: Provides real valued functons.
  - sympycore.arithmetic.mpmath.lib.numerals: Provides varios conversation functions for numerals.
  - sympycore.arithmetic.mpmath.lib.squareroot: This module provides functions for computing approximate square roots (in fixed-point or floating-point form) of positive real numbers.
  - sympycore.arithmetic.mpmath.lib.util
- sympycore.arithmetic.mpmath.mptypes: This module defines the mpf, mpc classes, and standard functions for operating with them.
sympycore.arithmetic.number_theory: Provides algorithms from number theory.
sympycore.arithmetic.numbers: Provides low-level numbers support.

Classes

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Complex
Represents complex number.

Float
Represents floating-point numbers.

FractionTuple
Represents a fraction.

Infinity
Base class for directional infinities.

Functions

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gcd(*args)
Calculate the greatest common divisor (GCD) of the arguments.

source code
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lcm(*args)
Calculate the least common multiple (LCM) of the arguments.

source code
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multinomial_coefficients(m, n, _tuple=<type 'tuple'>, _zip=<built-in function zip>)
Return a dictionary containing pairs {(k1,k2,..,km) : C_kn} where C_kn are multinomial coefficients such that n=k1+k2+..+km. source code
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Function Details

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multinomial_coefficients(m, n, _tuple=<type 'tuple'>, _zip=<built-in function zip>)

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Return a dictionary containing pairs {(k1,k2,..,km) : C_kn} where C_kn are multinomial coefficients such that n=k1+k2+..+km.

For example:

>>> print multinomial_coefficients(2,5)
{(3, 2): 10, (1, 4): 5, (2, 3): 10, (5, 0): 1, (0, 5): 1, (4, 1): 5}

The algorithm is based on the following result:

Consider a polynomial and it's m-th exponent:
P(x) = sum_{i=0}^m p_i x^k
P(x)^n = sum_{k=0}^{m n} a(n,k) x^k
The coefficients a(n,k) can be computed using the J.C.P. Miller Pure Recurrence [see D.E.Knuth, Seminumerical Algorithms, The art of Computer Programming v.2, Addison Wesley, Reading, 1981;]:
a(n,k) = 1/(k p_0) sum_{i=1}^m p_i ((n+1)i-k) a(n,k-i),
where a(n,0) = p_0^n.

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