AUTHORS:
Bases: object
Bases: object
Returns the numerical integral of the function on the interval from a to b and an error bound.
INPUT:
a, b – The interval of integration, specified as two numbers or as a tuple/list with the first element the lower bound and the second element the upper bound. Use +Infinity and -Infinity for plus or minus infinity.
algorithm – valid choices are:
max_points – sets the maximum number of sample points
params – used to pass parameters to your function
eps_abs, eps_rel – absolute and relative error tolerances
rule – This controls the Gauss-Kronrod rule used in the adaptive integration:
Higher key values are more accurate for smooth functions but lower key values deal better with discontinuities.
OUTPUT:
A tuple whose first component is the answer and whose second component is an error estimate.
REMARK:
There is also a method nintegral on symbolic expressions that implements numerical integration using Maxima. It is potentially very useful for symbolic expressions.
EXAMPLES:
To integrate the function \(x^2\) from 0 to 1, we do
sage: numerical_integral(x^2, 0, 1, max_points=100)
(0.3333333333333333, 3.700743415417188e-15)
To integrate the function \(\sin(x)^3 + \sin(x)\) we do
sage: numerical_integral(sin(x)^3 + sin(x), 0, pi)
(3.333333333333333, 3.700743415417188e-14)
The input can be any callable:
sage: numerical_integral(lambda x: sin(x)^3 + sin(x), 0, pi)
(3.333333333333333, 3.700743415417188e-14)
We check this with a symbolic integration:
sage: (sin(x)^3+sin(x)).integral(x,0,pi)
10/3
If we want to change the error tolerances and gauss rule used:
sage: f = x^2
sage: numerical_integral(f, 0, 1, max_points=200, eps_abs=1e-7, eps_rel=1e-7, rule=4)
(0.3333333333333333, 3.700743415417188e-15)
For a Python function with parameters:
sage: f(x,a) = 1/(a+x^2)
sage: [numerical_integral(f, 1, 2, max_points=100, params=[n]) for n in range(10)] # random output (architecture and os dependent)
[(0.49999999999998657, 5.5511151231256336e-15),
(0.32175055439664557, 3.5721487367706477e-15),
(0.24030098317249229, 2.6678768435816325e-15),
(0.19253082576711697, 2.1375215571674764e-15),
(0.16087527719832367, 1.7860743683853337e-15),
(0.13827545676349412, 1.5351659583939151e-15),
(0.12129975935702741, 1.3466978571966261e-15),
(0.10806674191683065, 1.1997818507228991e-15),
(0.09745444625548845, 1.0819617008493815e-15),
(0.088750683050217577, 9.8533051773561173e-16)]
sage: y = var('y')
sage: numerical_integral(x*y, 0, 1)
Traceback (most recent call last):
...
ValueError: The function to be integrated depends on 2 variables (x, y),
and so cannot be integrated in one dimension. Please fix additional
variables with the 'params' argument
Note the parameters are always a tuple even if they have one component.
It is possible to integrate on infinite intervals as well by using +Infinity or -Infinity in the interval argument. For example:
sage: f = exp(-x)
sage: numerical_integral(f, 0, +Infinity) # random output
(0.99999999999957279, 1.8429811298996553e-07)
Note the coercion to the real field RR, which prevents underflow:
sage: f = exp(-x**2)
sage: numerical_integral(f, -Infinity, +Infinity) # random output
(1.7724538509060035, 3.4295192165889879e-08)
One can integrate any real-valued callable function:
sage: numerical_integral(lambda x: abs(zeta(x)), [1.1,1.5]) # random output
(1.8488570602160455, 2.052643677492633e-14)
We can also numerically integrate symbolic expressions using either this function (which uses GSL) or the native integration (which uses Maxima):
sage: exp(-1/x).nintegral(x, 1, 2) # via maxima
(0.50479221787318..., 5.60431942934407...e-15, 21, 0)
sage: numerical_integral(exp(-1/x), 1, 2)
(0.50479221787318..., 5.60431942934407...e-15)
We can also integrate constant expressions:
sage: numerical_integral(2, 1, 7)
(12.0, 0.0)
If the interval of integration is a point, then the result is always zero (this makes sense within the Lebesgue theory of integration), see Trac ticket #12047:
sage: numerical_integral(log, 0, 0)
(0.0, 0.0)
sage: numerical_integral(lambda x: sqrt(x), (-2.0, -2.0) )
(0.0, 0.0)
AUTHORS:
ALGORITHM: Uses calls to the GSL (GNU Scientific Library) C library.
TESTS:
Make sure that constant Expressions, not merely uncallable arguments, can be integrated (trac #10088), at least if we can coerce them to float:
sage: f, g = x, x-1
sage: numerical_integral(f-g, -2, 2)
(4.0, 0.0)
sage: numerical_integral(SR(2.5), 5, 20)
(37.5, 0.0)
sage: numerical_integral(SR(1+3j), 2, 3)
Traceback (most recent call last):
...
TypeError: unable to simplify to float approximation