Jacobian by Finite Differences

Estimate the Jacobian of a vector-valued function $f:{\mathbb{R}}^n\to {\mathbb{R}}^{m}f:\backslash mathbb\; R^n\backslash to\backslash mathbb\; R^m$ at a point, one column at a time, with centered finite differences.

Implement solve(f, x, h=1e-5).

Returns. Return an $m\times nm\backslash times\; n$ NumPy array whose entry $(i,j)(i,\; j)$ approximates $\mathrm{\partial }{f}_i\mathrm{/}\mathrm{\partial }{x}_j\backslash partial\; f\_i/\backslash partial\; x\_j$.

Examples

Example 1

Input:

solve(lambda z: np.array([z[0]**2, z[0]*z[1]]), [1.0, 2.0])

Output:

[[2.0, 0.0], [2.0, 1.0]]

Example 2

Input:

solve(lambda z: np.array([np.sin(z[0])]), [0.0])

Output:

[[1.0]]