Numerically computes the derivative of ,
. Optionally,
computes the
-th derivative,
, for any order
.
Basic examples
Derivatives of a simple function:
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> diff(lambda x: x**2 + x, 1.0)
3.0
>>> diff(lambda x: x**2 + x, 1.0, 2)
2.0
>>> diff(lambda x: x**2 + x, 1.0, 3)
0.0
The exponential function is invariant under differentiation:
>>> nprint([diff(exp, 3, n) for n in range(5)])
[20.0855, 20.0855, 20.0855, 20.0855, 20.0855]
Method
One of two differentiation algorithms can be chosen with the method keyword argument. The two options are 'step', and 'quad'. The default method is 'step'.
'step':
The derivative is computed using a finite difference approximation, with a small step h. This requires n+1 function evaluations and must be performed at (n+1) times the target precison. Accordingly, f must support fast evaluation at high precision.
'quad':
The derivative is computed using complex numerical integration. This requires a larger number of function evaluations, but the advantage is that not much extra precision is required. For high order derivatives, this method may thus be faster if f is very expensive to evaluate at high precision.
With 'quad' the result is likely to have a small imaginary component even if the derivative is actually real:
>>> diff(sqrt, 1, method='quad')
(0.5 - 9.44048454290863e-27j)
Scale
The scale option specifies the scale of variation of f. The step
size in the finite difference is taken to be approximately
eps*scale. Thus, for example if , the scale
should be set to 1/1000 and if
, the scale
should be 1000. By default, scale = 1.
(In practice, the default scale will work even for or
. Changing this parameter is a good idea if the scale
is something preposterous.)
If numerical integration is used, the radius of integration is taken to be equal to scale/2. Note that f must not have any singularities within the circle of radius scale/2 centered around x. If possible, a larger scale value is preferable because it typically makes the integration faster and more accurate.
Direction
By default, diff() uses a central difference approximation. This corresponds to direction=0. Alternatively, it can compute a left difference (direction=-1) or right difference (direction=1). This is useful for computing left- or right-sided derivatives of nonsmooth functions:
>>> diff(abs, 0, direction=0)
0.0
>>> diff(abs, 0, direction=1)
1.0
>>> diff(abs, 0, direction=-1)
-1.0
More generally, if the direction is nonzero, a right difference is computed where the step size is multiplied by sign(direction). For example, with direction=+j, the derivative from the positive imaginary direction will be computed.
This option only makes sense with method=’step’. If integration is used, it is assumed that f is analytic, implying that the derivative is the same in all directions.
Returns a generator that yields the sequence of derivatives
With method='step', diffs() uses only
function evaluations to generate the first
derivatives,
rather than the roughly
evaluations
required if one calls diff()
separate times.
With , the generator stops as soon as the
-th derivative has been generated. If the exact number of
needed derivatives is known in advance, this is further
slightly more efficient.
Examples
>>> from mpmath import *
>>> mp.dps = 15
>>> nprint(list(diffs(cos, 1, 5)))
[0.540302, -0.841471, -0.540302, 0.841471, 0.540302, -0.841471]
>>> for i, d in zip(range(6), diffs(cos, 1)): print i, d
...
0 0.54030230586814
1 -0.841470984807897
2 -0.54030230586814
3 0.841470984807897
4 0.54030230586814
5 -0.841470984807897
Calculates the Riemann-Liouville differintegral, or fractional derivative, defined by
where is a given (presumably well-behaved) function,
is the evaluation point,
is the order, and
is
the reference point of integration (
is an arbitrary
parameter selected automatically).
With , this is just the standard derivative
; with
,
the second derivative
, etc. With
, it gives
, with
it gives
, etc.
As is permitted to be any number, this operator generalizes
iterated differentiation and iterated integration to a single
operator with a continuous order parameter.
Examples
There is an exact formula for the fractional derivative of a
monomial , which may be used as a reference. For example,
the following gives a half-derivative (order 0.5):
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> x = mpf(3); p = 2; n = 0.5
>>> differint(lambda t: t**p, x, n)
7.81764019044672
>>> gamma(p+1)/gamma(p-n+1) * x**(p-n)
7.81764019044672
Another useful test function is the exponential function, whose
integration / differentiation formula easy generalizes
to arbitrary order. Here we first compute a third derivative,
and then a triply nested integral. (The reference point
is set to
to avoid nonzero endpoint terms.):
>>> differint(lambda x: exp(pi*x), -1.5, 3)
0.278538406900792
>>> exp(pi*-1.5) * pi**3
0.278538406900792
>>> differint(lambda x: exp(pi*x), 3.5, -3, -inf)
1922.50563031149
>>> exp(pi*3.5) / pi**3
1922.50563031149
However, for noninteger , the differentiation formula for the
exponential function must be modified to give the same result as the
Riemann-Liouville differintegral:
>>> x = mpf(3.5)
>>> c = pi
>>> n = 1+2*j
>>> differint(lambda x: exp(c*x), x, n)
(-123295.005390743 + 140955.117867654j)
>>> x**(-n) * exp(c)**x * (x*c)**n * gammainc(-n, 0, x*c) / gamma(-n)
(-123295.005390743 + 140955.117867654j)