Contexts
========
High-level code in mpmath is implemented as methods on a "context object". The context implements arithmetic, type conversions and other fundamental operations. The context also holds settings such as precision, and stores cache data. A few different contexts (with a mostly compatible interface) are provided so that the high-level algorithms can be used with different implementations of the underlying arithmetic, allowing different features and speed-accuracy tradeoffs. Currently, mpmath provides the following contexts:
* Arbitrary-precision arithmetic (``mp``)
* A faster Cython-based version of ``mp`` (used by default in Sage, and currently only available there)
* Arbitrary-precision interval arithmetic (``iv``)
* Double-precision arithmetic using Python's builtin ``float`` and ``complex`` types (``fp``)
Most global functions in the global mpmath namespace are actually methods of the ``mp``
context. This fact is usually transparent to the user, but sometimes shows up in the
form of an initial parameter called "ctx" visible in the help for the function::
>>> import mpmath
>>> help(mpmath.fsum) # doctest:+SKIP
Help on method fsum in module mpmath.ctx_mp_python:
fsum(ctx, terms, absolute=False, squared=False) method of mpmath.ctx_mp.MPContext instance
Calculates a sum containing a finite number of terms (for infinite
series, see :func:`~mpmath.nsum`). The terms will be converted to
...
The following operations are equivalent::
>>> mpmath.mp.dps = 15; mpmath.mp.pretty = False
>>> mpmath.fsum([1,2,3])
mpf('6.0')
>>> mpmath.mp.fsum([1,2,3])
mpf('6.0')
The corresponding operation using the ``fp`` context::
>>> mpmath.fp.fsum([1,2,3])
6.0
Common interface
----------------
``ctx.mpf`` creates a real number::
>>> from mpmath import mp, fp
>>> mp.mpf(3)
mpf('3.0')
>>> fp.mpf(3)
3.0
``ctx.mpc`` creates a complex number::
>>> mp.mpc(2,3)
mpc(real='2.0', imag='3.0')
>>> fp.mpc(2,3)
(2+3j)
``ctx.matrix`` creates a matrix::
>>> mp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
['0.0', '1.0']])
>>> _[0,0]
mpf('1.0')
>>> fp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
['0.0', '1.0']])
>>> _[0,0]
1.0
``ctx.prec`` holds the current precision (in bits)::
>>> mp.prec
53
>>> fp.prec
53
``ctx.dps`` holds the current precision (in digits)::
>>> mp.dps
15
>>> fp.dps
15
``ctx.pretty`` controls whether objects should be pretty-printed automatically by :func:`repr`. Pretty-printing for ``mp`` numbers is disabled by default so that they can clearly be distinguished from Python numbers and so that ``eval(repr(x)) == x`` works::
>>> mp.mpf(3)
mpf('3.0')
>>> mpf = mp.mpf
>>> eval(repr(mp.mpf(3)))
mpf('3.0')
>>> mp.pretty = True
>>> mp.mpf(3)
3.0
>>> fp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
['0.0', '1.0']])
>>> fp.pretty = True
>>> fp.matrix([[1,0],[0,1]])
[1.0 0.0]
[0.0 1.0]
>>> fp.pretty = False
>>> mp.pretty = False
Arbitrary-precision floating-point (``mp``)
---------------------------------------------
The ``mp`` context is what most users probably want to use most of the time, as it supports the most functions, is most well-tested, and is implemented with a high level of optimization. Nearly all examples in this documentation use ``mp`` functions.
See :doc:`basics` for a description of basic usage.
Arbitrary-precision interval arithmetic (``iv``)
------------------------------------------------
The ``iv.mpf`` type represents a closed interval `[a,b]`; that is, the set `\{x : a \le x \le b\}`, where `a` and `b` are arbitrary-precision floating-point values, possibly `\pm \infty`. The ``iv.mpc`` type represents a rectangular complex interval `[a,b] + [c,d]i`; that is, the set `\{z = x+iy : a \le x \le b \land c \le y \le d\}`.
Interval arithmetic provides rigorous error tracking. If `f` is a mathematical function and `\hat f` is its interval arithmetic version, then the basic guarantee of interval arithmetic is that `f(v) \subseteq \hat f(v)` for any input interval `v`. Put differently, if an interval represents the known uncertainty for a fixed number, any sequence of interval operations will produce an interval that contains what would be the result of applying the same sequence of operations to the exact number. The principal drawbacks of interval arithmetic are speed (``iv`` arithmetic is typically at least two times slower than ``mp`` arithmetic) and that it sometimes provides far too pessimistic bounds.
.. note ::
The support for interval arithmetic in mpmath is still experimental, and many functions
do not yet properly support intervals. Please use this feature with caution.
Intervals can be created from single numbers (treated as zero-width intervals) or pairs of endpoint numbers. Strings are treated as exact decimal numbers. Note that a Python float like ``0.1`` generally does not represent the same number as its literal; use ``'0.1'`` instead::
>>> from mpmath import iv
>>> iv.dps = 15; iv.pretty = False
>>> iv.mpf(3)
mpi('3.0', '3.0')
>>> print(iv.mpf(3))
[3.0, 3.0]
>>> iv.pretty = True
>>> iv.mpf([2,3])
[2.0, 3.0]
>>> iv.mpf(0.1) # probably not intended
[0.10000000000000000555, 0.10000000000000000555]
>>> iv.mpf('0.1') # good, gives a containing interval
[0.099999999999999991673, 0.10000000000000000555]
>>> iv.mpf(['0.1', '0.2'])
[0.099999999999999991673, 0.2000000000000000111]
The fact that ``'0.1'`` results in an interval of nonzero width indicates that 1/10 cannot be represented using binary floating-point numbers at this precision level (in fact, it cannot be represented exactly at any precision).
Intervals may be infinite or half-infinite::
>>> print(1 / iv.mpf([2, 'inf']))
[0.0, 0.5]
The equality testing operators ``==`` and ``!=`` check whether their operands are identical as intervals; that is, have the same endpoints. The ordering operators ``< <= > >=`` permit inequality testing using triple-valued logic: a guaranteed inequality returns ``True`` or ``False`` while an indeterminate inequality returns ``None``::
>>> iv.mpf([1,2]) == iv.mpf([1,2])
True
>>> iv.mpf([1,2]) != iv.mpf([1,2])
False
>>> iv.mpf([1,2]) <= 2
True
>>> iv.mpf([1,2]) > 0
True
>>> iv.mpf([1,2]) < 1
False
>>> iv.mpf([1,2]) < 2 # returns None
>>> iv.mpf([2,2]) < 2
False
>>> iv.mpf([1,2]) <= iv.mpf([2,3])
True
>>> iv.mpf([1,2]) < iv.mpf([2,3]) # returns None
>>> iv.mpf([1,2]) < iv.mpf([-1,0])
False
The ``in`` operator tests whether a number or interval is contained in another interval::
>>> iv.mpf([0,2]) in iv.mpf([0,10])
True
>>> 3 in iv.mpf(['-inf', 0])
False
Intervals have the properties ``.a``, ``.b`` (endpoints), ``.mid``, and ``.delta`` (width)::
>>> x = iv.mpf([2, 5])
>>> x.a
[2.0, 2.0]
>>> x.b
[5.0, 5.0]
>>> x.mid
[3.5, 3.5]
>>> x.delta
[3.0, 3.0]
Some transcendental functions are supported::
>>> iv.dps = 15
>>> mp.dps = 15
>>> iv.mpf([0.5,1.5]) ** iv.mpf([0.5, 1.5])
[0.35355339059327373086, 1.837117307087383633]
>>> iv.exp(0)
[1.0, 1.0]
>>> iv.exp(['-inf','inf'])
[0.0, +inf]
>>>
>>> iv.exp(['-inf',0])
[0.0, 1.0]
>>> iv.exp([0,'inf'])
[1.0, +inf]
>>> iv.exp([0,1])
[1.0, 2.7182818284590455349]
>>>
>>> iv.log(1)
[0.0, 0.0]
>>> iv.log([0,1])
[-inf, 0.0]
>>> iv.log([0,'inf'])
[-inf, +inf]
>>> iv.log(2)
[0.69314718055994528623, 0.69314718055994539725]
>>>
>>> iv.sin([100,'inf'])
[-1.0, 1.0]
>>> iv.cos(['-0.1','0.1'])
[0.99500416527802570954, 1.0]
Interval arithmetic is useful for proving inequalities involving irrational numbers.
Naive use of ``mp`` arithmetic may result in wrong conclusions, such as the following::
>>> mp.dps = 25
>>> x = mp.exp(mp.pi*mp.sqrt(163))
>>> y = mp.mpf(640320**3+744)
>>> print(x)
262537412640768744.0000001
>>> print(y)
262537412640768744.0
>>> x > y
True
But the correct result is `e^{\pi \sqrt{163}} < 262537412640768744`, as can be
seen by increasing the precision::
>>> mp.dps = 50
>>> print(mp.exp(mp.pi*mp.sqrt(163)))
262537412640768743.99999999999925007259719818568888
With interval arithmetic, the comparison returns ``None`` until the precision
is large enough for `x-y` to have a definite sign::
>>> iv.dps = 15
>>> iv.exp(iv.pi*iv.sqrt(163)) > (640320**3+744)
>>> iv.dps = 30
>>> iv.exp(iv.pi*iv.sqrt(163)) > (640320**3+744)
>>> iv.dps = 60
>>> iv.exp(iv.pi*iv.sqrt(163)) > (640320**3+744)
False
>>> iv.dps = 15
Fast low-precision arithmetic (``fp``)
---------------------------------------------
Although mpmath is generally designed for arbitrary-precision arithmetic, many of the high-level algorithms work perfectly well with ordinary Python ``float`` and ``complex`` numbers, which use hardware double precision (on most systems, this corresponds to 53 bits of precision). Whereas the global functions (which are methods of the ``mp`` object) always convert inputs to mpmath numbers, the ``fp`` object instead converts them to ``float`` or ``complex``, and in some cases employs basic functions optimized for double precision. When large amounts of function evaluations (numerical integration, plotting, etc) are required, and when ``fp`` arithmetic provides sufficient accuracy, this can give a significant speedup over ``mp`` arithmetic.
To take advantage of this feature, simply use the ``fp`` prefix, i.e. write ``fp.func`` instead of ``func`` or ``mp.func``::
>>> u = fp.erfc(2.5)
>>> print(u) # doctest:+SKIP
0.000406952017445
>>> type(u) # doctest:+SKIP
>>> mp.dps = 15
>>> print(mp.erfc(2.5))
0.000406952017444959
>>> fp.matrix([[1,2],[3,4]]) ** 2
matrix(
[['7.0', '10.0'],
['15.0', '22.0']])
>>>
>>> type(_[0,0]) # doctest:+SKIP
>>> print(fp.quad(fp.sin, [0, fp.pi])) # numerical integration
2.0
The ``fp`` context wraps Python's ``math`` and ``cmath`` modules for elementary functions. It supports both real and complex numbers and automatically generates complex results for real inputs (``math`` raises an exception)::
>>> fp.sqrt(5) # doctest:+SKIP
2.23606797749979
>>> fp.sqrt(-5) # doctest:+SKIP
2.23606797749979j
>>> fp.sin(10) # doctest:+SKIP
-0.5440211108893698
>>> fp.power(-1, 0.25) # doctest:+SKIP
(0.7071067811865476+0.7071067811865475j)
>>> (-1) ** 0.25 # doctest:+SKIP
Traceback (most recent call last):
...
ValueError: negative number cannot be raised to a fractional power
The ``prec`` and ``dps`` attributes can be changed (for interface compatibility with the ``mp`` context) but this has no effect::
>>> fp.prec
53
>>> fp.dps
15
>>> fp.prec = 80
>>> fp.prec
53
>>> fp.dps
15
Due to intermediate rounding and cancellation errors, results computed with ``fp`` arithmetic may be much less accurate than those computed with ``mp`` using an equivalent precision (``mp.prec = 53``), since the latter often uses increased internal precision. The accuracy is highly problem-dependent: for some functions, ``fp`` almost always gives 14-15 correct digits; for others, results can be accurate to only 2-3 digits or even completely wrong. The recommended use for ``fp`` is therefore to speed up large-scale computations where accuracy can be verified in advance on a subset of the input set, or where results can be verified afterwards.