3.9 Numerical Integration and Differentiation

Special functions are available for performing numerical integration and differentiation of expressions: int_dx() and diff_dx(). In each case, the ``x'' may be replaced with any valid variable name, to integrate or differentiate with respect to any given variable.

The function int_dx() takes three parameters - firstly the expression to be integrated, followed by the minimum and maximum integration limits. For example, the following would plot the integral of the function $\sin(x)$:

plot int_dt(sin(t),0,x)

The function diff_dx() takes two parameters and an optional third - firstly the expression to be differentiated, then the point at which the differential should be evaluated, and then an optional parameter, $\epsilon$. The following example would evaluate the differential of the function $\cos(x)$ with respect to $x$ at $x=1.0$:

print diff_dx(cos(x), 1.0)

Differentials are evaluated by a simple differencing algorithm, and the parameter $\epsilon$ controls the spacing with which to perform the differencing operation:

$$\left.\frac{\mathrm{d}f}{\mathrm{d}x}\right\vert _{x=x_0} \approx \frac{f(x_0+\epsilon/2) - f(x_0-\epsilon/2)}{\epsilon}$$

By default, $\epsilon = 10^{-6}$.

Advanced users may be interested to know that integration is performed using the quad function of the integrate package of the scipy numerical toolkit for Python - a general purpose integration routine.