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1.4.2 Gaussian quadrature (integrate.gauss_quadtol)
SciPy Reference Guide, Release 0.8.dev
A few functions are also provided in order to perform simple Gaussian quadrature over a fixed interval. The first
is fixed_quad which performs fixed-order Gaussian quadrature. The second function is quadrature which
performs Gaussian quadrature of multiple orders until the difference in the integral estimate is beneath some tolerance
supplied by the user. These functions both use the module special.orthogonal which can calculate the roots
and quadrature weights of a large variety of orthogonal polynomials (the polynomials themselves are available as
special functions returning instances of the polynomial class — e.g. special.legendre).
1.4.3 Integrating using samples
There are three functions for computing integrals given only samples: trapz , simps, and romb . The first two
functions use Newton-Coates formulas of order 1 and 2 respectively to perform integration. These two functions
can handle, non-equally-spaced samples. The trapezoidal rule approximates the function as a straight line between
adjacent points, while Simpson’s rule approximates the function between three adjacent points as a parabola.
If the samples are equally-spaced and the number of samples available is 2 k + 1 for some integer k, then Romberg
integration can be used to obtain high-precision estimates of the integral using the available samples. Romberg integration
uses the trapezoid rule at step-sizes related by a power of two and then performs Richardson extrapolation
on these estimates to approximate the integral with a higher-degree of accuracy. (A different interface to Romberg
integration useful when the function can be provided is also available as romberg).
1.4.4 Ordinary differential equations (odeint)
Integrating a set of ordinary differential equations (ODEs) given initial conditions is another useful example. The
function odeint is available in SciPy for integrating a first-order vector differential equation:
dy
dt
= f (y, t) ,
given initial conditions y (0) = y0, where y is a length N vector and f is a mapping from R N to R N . A higher-order
ordinary differential equation can always be reduced to a differential equation of this type by introducing intermediate
derivatives into the y vector.
For example suppose it is desired to find the solution to the following second-order differential equation:
d2w − zw(z) = 0
dz2 �
1
dw
with initial conditions w (0) = 3√
32 2 and �
Γ( dz = −
z=0 3) 1
3√
1 3Γ(
. It is known that the solution to this differential
3)
equation with these boundary conditions is the Airy function
w = Ai (z) ,
which gives a means to check the integrator using special.airy.
First, convert this ODE into standard form by setting y = � dw
dz , w� and t = z. Thus, the differential equation becomes
In other words,
dy
dt =
�
ty1
y0
�
=
� 0 t
1 0
� � y0
y1
f (y, t) = A (t) y.
�
=
� 0 t
1 0
1.4. Integration (scipy.integrate) 13
�
y.