qml.gradients.finite_diff_coeffs

finite_diff_coeffs(n, approx_order, strategy)

Generate the finite difference shift values and corresponding term coefficients for a given derivative order, approximation accuracy, and strategy.

Parameters

• n (int) – Positive integer specifying the order of the derivative. For example, n=1 corresponds to the first derivative, n=2 the second derivative, etc.

• approx_order (int) – Positive integer referring to the approximation order of the returned coefficients, e.g., approx_order=1 corresponds to the first-order approximation to the derivative.

• strategy (str) – One of "forward", "center", or "backward". For the "forward" strategy, the finite-difference shifts occur at the points $x_0, x_0+h, x_0+2h,\dots$, where $h$ is some small step size. The "backwards" strategy is similar, but in reverse: $x_0, x_0-h, x_0-2h, \dots$. Finally, the "center" strategy results in shifts symmetric around the unshifted point: $\dots, x_0-2h, x_0-h, x_0, x_0+h, x_0+2h,\dots$.

Returns

A (2, N) array. The first row corresponds to the coefficients, and the second row corresponds to the shifts.

Return type

array[float]

Example

>>> finite_diff_coeffs(n=1, approx_order=1, strategy="forward")
array([[-1.,  1.],
       [ 0.,  1.]])

For example, this results in the linear combination:

$$\frac{-y(x_0) + y(x_0 + h)}{h}$$

where $h$ is the finite-difference step size.

More examples:

>>> finite_diff_coeffs(n=1, approx_order=2, strategy="center")
array([[-0.5,  0.5],
       [-1. ,  1. ]])
>>> finite_diff_coeffs(n=2, approx_order=2, strategy="center")
array([[-2.,  1.,  1.],
       [ 0., -1.,  1.]])

Details

Consider a function $y(x)$. We wish to approximate the $n$-th derivative at point $x_0$, $y^{(n)}(x_0)$, by sampling the function at $N<n$ distinct points:

$$y^{(n)}(x_0) \approx \sum_{i=1}^N c_i y(x_i)$$

where $c_i$ are coefficients, and $x_i=x_0 + s_i$ are the points we sample the function at.

Consider the Taylor expansion of $y(x_i)$ around the point $x_0$:

$$\begin{split}y^{(n)}(x_0) \approx \sum_{i=1}^N c_i y(x_i) &= \sum_{i=1}^N c_i \left[ y(x_0) + y'(x_0)(x_i-x_0) + \frac{1}{2} y''(x_0)(x_i-x_0)^2 + \cdots \right]\\ & = \sum_{j=0}^m y^{(j)}(x_0) \left[\sum_{i=1}^N \frac{c_i s_i^j}{j!} + \mathcal{O}(s_i^m) \right],\end{split}$$

where $s_i = x_i-x_0$. For this approximation to be satisfied, we must therefore have

$$\begin{split}\sum_{i=1}^N s_i^j c_i = \begin{cases} j!, &j=n\\ 0, & j\neq n\end{cases}.\end{split}$$

Thus, to determine the coefficients $c_i \in \{c_1, \dots, c_N\}$ for particular shift values $s_i \in \{s_1, \dots, s_N\}$ and derivative order $n$, we must solve this linear system of equations.

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