Cadabra manual: Nval

nval

Numerically evaluate expressions

\available{2.5.12} By default Cadabra does not evaluate integrals, neither symbolically nor numerically. If you want to replace an integral with its numerical value, use nval (for symbolic evaluation, consult integrate instead). This function will in future versions be extended to handle other numerical evaluations (e.g. sums) as well.

The example below shows the basic usage of nval:

ex:= \int{ \sin(x)/\sqrt{x+3} }{x, 0, 1};

$\displaystyle{}\int_{0\,}^{1} \sin{x} {\left(x +3\,\right)}^{ - \frac{1}{2}\,}\,\,{\rm d}x$

nval(_);

$\displaystyle{}0.240842\,$

Note how this has change the original expression:

ex;

$\displaystyle{}0.240842\,$

You can pass an optional dict of values for parameters in the expression. The values passed here must be numerical (so if you want to use e.g. $\pi$, use np.pi for now). In the following example we have an integral which depends on a parameter $a$,

ex:= \int{ \sin(x) }{x, 0, a};

$\displaystyle{}\int_{0\,}^{a} \sin{x}\,\,{\rm d}x$

We can evaluate this for $a=2$ by passing that value into nval,

nval(ex, {$a$: 2} );

$\displaystyle{}1.41615\,$

If you want to evaluate only a certain part of your expression numerically, first zoom in to that part before you call nval. In the example below, we initially want to evaluate the integral containing the sine, but not the one containing the cosine.

ex:= \int{\sin(x)}{x,0,\pi} + \int{\cos(x)**2}{x,0,\pi};

$\displaystyle{}\int_{0\,}^{\pi} \sin{x}\,\,{\rm d}x +\int_{0\,}^{\pi} {\left(\cos{x}\right)}^{2\,}\,\,{\rm d}x$

zoom(_, $\int{\sin{A??}}{B??}$);

$\displaystyle{}\int_{0\,}^{\pi} \sin{x}\,\,{\rm d}x + ... $

nval(_);

$\displaystyle{}2\, + ... $

unzoom(_);

$\displaystyle{}2\, +\int_{0\,}^{\pi} {\left(\cos{x}\right)}^{2\,}\,\,{\rm d}x$

If you now still want to evaluate the cosine integral too, you can call nval on the full, unzoomed expression,

nval(_);

$\displaystyle{}3.5708\,$

While the integration variable has to be real, nval function can deal with complex-valued integrands

i::ImaginaryI; ex:= A \int{ \exp{i x} }{x, 0, \pi}; nval(_);

$\displaystyle{}\text{Property ImaginaryI attached to }i.$

$\displaystyle{}A \int_{0\,}^{\pi} \exp\left(i x\right)\,\,{\rm d}x$

$\displaystyle{}A (1.1847 \times 10^{-16} + 2i)$

Under the hood, the numerical integration method used is Gauss-Kronrod, which does not (yet) deal with infinite or semi-infinite integration ranges.