In interactive code examples that follow, it will be assumed that all items in the mpmath namespace have been imported:
>>> from mpmath import *
Importing everything can be convenient, especially when using mpmath interactively, but be careful when mixing mpmath with other libraries! To avoid inadvertently overriding other functions or objects, explicitly import only the needed objects, or use the mpmath. or mp. namespaces:
from mpmath import sin, cos
sin(1), cos(1)
import mpmath
mpmath.sin(1), mpmath.cos(1)
from mpmath import mp # mp context object -- to be explained
mp.sin(1), mp.cos(1)
Mpmath provides the following numerical types:
Class Description mpf Real float mpc Complex float matrix Matrix
The following section will provide a very short introduction to the types mpf and mpc. Intervals and matrices are described further in the documentation chapters on interval arithmetic and matrices / linear algebra.
The mpf type is analogous to Python’s built-in float. It holds a real number or one of the special values inf (positive infinity), -inf (negative infinity) and nan (not-a-number, indicating an indeterminate result). You can create mpf instances from strings, integers, floats, and other mpf instances:
>>> mpf(4)
mpf('4.0')
>>> mpf(2.5)
mpf('2.5')
>>> mpf("1.25e6")
mpf('1250000.0')
>>> mpf(mpf(2))
mpf('2.0')
>>> mpf("inf")
mpf('+inf')
The mpc type represents a complex number in rectangular form as a pair of mpf instances. It can be constructed from a Python complex, a real number, or a pair of real numbers:
>>> mpc(2,3)
mpc(real='2.0', imag='3.0')
>>> mpc(complex(2,3)).imag
mpf('3.0')
You can mix mpf and mpc instances with each other and with Python numbers:
>>> mpf(3) + 2*mpf('2.5') + 1.0
mpf('9.0')
>>> mp.dps = 15 # Set precision (see below)
>>> mpc(1j)**0.5
mpc(real='0.70710678118654757', imag='0.70710678118654757')
Mpmath uses a global working precision; it does not keep track of the precision or accuracy of individual numbers. Performing an arithmetic operation or calling mpf() rounds the result to the current working precision. The working precision is controlled by a context object called mp, which has the following default state:
>>> print mp
Mpmath settings:
mp.prec = 53 [default: 53]
mp.dps = 15 [default: 15]
mp.trap_complex = False [default: False]
The term prec denotes the binary precision (measured in bits) while dps (short for decimal places) is the decimal precision. Binary and decimal precision are related roughly according to the formula prec = 3.33*dps. For example, it takes a precision of roughly 333 bits to hold an approximation of pi that is accurate to 100 decimal places (actually slightly more than 333 bits is used).
Changing either precision property of the mp object automatically updates the other; usually you just want to change the dps value:
>>> mp.dps = 100
>>> mp.dps
100
>>> mp.prec
336
When the precision has been set, all mpf operations are carried out at that precision:
>>> mp.dps = 50
>>> mpf(1) / 6
mpf('0.16666666666666666666666666666666666666666666666666656')
>>> mp.dps = 25
>>> mpf(2) ** mpf('0.5')
mpf('1.414213562373095048801688713')
The precision of complex arithmetic is also controlled by the mp object:
>>> mp.dps = 10
>>> mpc(1,2) / 3
mpc(real='0.3333333333321', imag='0.6666666666642')
There is no restriction on the magnitude of numbers. An mpf can for example hold an approximation of a large Mersenne prime:
>>> mp.dps = 15
>>> print mpf(2)**32582657 - 1
1.24575026015369e+9808357
Or why not 1 googolplex:
>>> print mpf(10) ** (10**100)
1.0e+100000000000000000000000000000000000000000000000000...
The (binary) exponent is stored exactly and is independent of the precision.
It is often useful to change the precision during only part of a calculation. A way to temporarily increase the precision and then restore it is as follows:
>>> mp.prec += 2
>>> # do_something()
>>> mp.prec -= 2
In Python 2.5, the with statement along with the mpmath functions workprec, workdps, extraprec and extradps can be used to temporarily change precision in a more safe manner:
>>> from __future__ import with_statement
>>> with workdps(20):
... print mpf(1)/7
... with extradps(10):
... print mpf(1)/7
...
0.14285714285714285714
0.142857142857142857142857142857
>>> mp.dps
15
The with statement ensures that the precision gets reset when exiting the block, even in the case that an exception is raised. (The effect of the with statement can be emulated in Python 2.4 by using a try/finally block.)
The workprec family of functions can also be used as function decorators:
>>> @workdps(6)
... def f():
... return mpf(1)/3
...
>>> f()
mpf('0.33333331346511841')
Some functions accept the prec and dps keyword arguments and this will override the global working precision. Note that this will not affect the precision at which the result is printed, so to get all digits, you must either use increase precision afterward when printing or use nstr/nprint:
>>> mp.dps = 15
>>> print exp(1)
2.71828182845905
>>> print exp(1, dps=50) # Extra digits won't be printed
2.71828182845905
>>> nprint(exp(1, dps=50), 50)
2.7182818284590452353602874713526624977572470937
Finally, instead of using the global context object mp, you can create custom contexts and work with methods of those instances instead of global functions. The working precision will be local to each context object:
>>> mp2 = mp.clone()
>>> mp.dps = 10
>>> mp2.dps = 20
>>> print mp.mpf(1) / 3
0.3333333333
>>> print mp2.mpf(1) / 3
0.33333333333333333333
Note: the ability to create multiple contexts is a new feature that is only partially implemented. Not all mpmath functions are yet available as context-local methods. In the present version, you are likely to encounter bugs if you try mixing different contexts.
Note that when creating a new mpf, the value will at most be as accurate as the input. Be careful when mixing mpmath numbers with Python floats. When working at high precision, fractional mpf values should be created from strings or integers:
>>> mp.dps = 30
>>> mpf(10.9) # bad
mpf('10.9000000000000003552713678800501')
>>> mpf('10.9') # good
mpf('10.8999999999999999999999999999997')
>>> mpf(109) / mpf(10) # also good
mpf('10.8999999999999999999999999999997')
>>> mp.dps = 15
(Binary fractions such as 0.5, 1.5, 0.75, 0.125, etc, are generally safe as input, however, since those can be represented exactly by Python floats.)
By default, the repr() of a number includes its type signature. This way eval can be used to recreate a number from its string representation:
>>> eval(repr(mpf(2.5)))
mpf('2.5')
Prettier output can be obtained by using str() or print, which hide the mpf and mpc signatures and also suppress rounding artifacts in the last few digits:
>>> mpf("3.14159")
mpf('3.1415899999999999')
>>> print mpf("3.14159")
3.14159
>>> print mpc(1j)**0.5
(0.707106781186548 + 0.707106781186548j)
Setting the mp.pretty option will use the str()-style output for repr() as well:
>>> mp.pretty = True
>>> mpf(0.6)
0.6
>>> mp.pretty = False
>>> mpf(0.6)
mpf('0.59999999999999998')
The number of digits with which numbers are printed by default is determined by the working precision. To specify the number of digits to show without changing the working precision, use mpmath.nstr() and mpmath.nprint():
>>> a = mpf(1) / 6
>>> a
mpf('0.16666666666666666')
>>> nstr(a, 8)
'0.16666667'
>>> nprint(a, 8)
0.16666667
>>> nstr(a, 50)
'0.16666666666666665741480812812369549646973609924316'