A Jacobian From the Outside

In this chapter you built Jacobians with pencil and paper: differentiate every output with respect to every input, and lay the results out so that

J_{ij} = \frac{\partial f_i}{\partial x_j}

Row $i$ belongs to output $i$ ; column $j$ belongs to input $j$ .

Sometimes pencil and paper aren't an option. The function might be a black box you can only call — a simulator, say — or you've worked a Jacobian out by hand and want an independent check that you didn't slip. Then you measure it instead.

To measure column $j$ , leave every input alone except $x_j$ . Nudge $x_j$ up by a small amount $h$ and call $f$ ; nudge it down by $h$ and call $f$ again. How far each output moved, divided by the distance $2h$ between the two calls, estimates that output's partial derivative:

J_{ij} \approx \frac{f_i(\ldots,\ x_j + h,\ \ldots) \;-\; f_i(\ldots,\ x_j - h,\ \ldots)}{2h}

Task: write numerical_jacobian(f, x, h).

f is a Python function. It takes a list of $n$ numbers and returns a list of $m$ numbers — and $m$ doesn't have to equal $n$ .
x is the point, a list of $n$ numbers. Leave it exactly as it was: the caller still needs it.
h is the nudge size, a positive number.
Return the $m \times n$ estimate as a list of $m$ rows, each holding $n$ numbers, with every entry rounded to 4 decimal places.

Use exactly the formula above with the h you are given. Several tests use a fairly large h on purpose, so the estimate can differ slightly from the exact derivative — return the estimate, not the exact value.

Example. The chapter's worked example, $f(x, y) = (x + y^2,\ xy)$ at the point $(2, 1)$ :

That agrees with the hand-derived Jacobian $\begin{bmatrix} 1 & 2y \\ y & x \end{bmatrix}$ at $(2, 1)$ .

A Jacobian From the Outside

Examples

Hints