Numerical Integration¶
AUTHORS:
- Josh Kantor (2007-02): first version
- William Stein (2007-02): rewrite of docs, conventions, etc.
- Robert Bradshaw (2008-08): fast float integration
- Jeroen Demeyer (2011-11-23): trac ticket #12047: return 0 when the integration interval is a point; reformat documentation and add to the reference manual.
-
class
sage.gsl.integration.
PyFunctionWrapper
¶ Bases:
object
-
class
sage.gsl.integration.
compiled_integrand
¶ Bases:
object
-
sage.gsl.integration.
numerical_integral
(func, a, b=None, algorithm='qag', max_points=87, params=[], eps_abs=1e-06, eps_rel=1e-06, rule=6)¶ 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:- ‘qag’ – for an adaptive integration
- ‘qags’ – for an adaptive integration with (integrable) singularities
- ‘qng’ – for a non-adaptive Gauss-Kronrod (samples at a maximum of 87pts)
max_points
– sets the maximum number of sample pointsparams
– used to pass parameters to your functioneps_abs
,eps_rel
– absolute and relative error tolerancesrule
– This controls the Gauss-Kronrod rule used in the adaptive integration:- rule=1 – 15 point rule
- rule=2 – 21 point rule
- rule=3 – 31 point rule
- rule=4 – 41 point rule
- rule=5 – 51 point rule
- rule=6 – 61 point rule
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)
In the presence of integrable singularity, the default adaptive method might fail and it is advised to use
'qags'
:sage: b = 1.81759643554688 sage: F(x) = sqrt((-x + b)/((x - 1.0)*x)) sage: numerical_integral(F, 1, b) (inf, nan) sage: numerical_integral(F, 1, b, algorithm='qags') # abs tol 1e-10 (1.1817104238446596, 3.387268288079781e-07)
AUTHORS:
- Josh Kantor
- William Stein
- Robert Bradshaw
- Jeroen Demeyer
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 ticket #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