These are functions that are used in tutorials and examples.

Simple real valued function

Definition:

$\‍[ y = f(x) = x * x * x = x^3 \‍]$

Jacobian:

$\‍[ \frac{\d f}{\d x}(x) = 3 * x * x = 3 * x^2 \‍]$

Simple vector valued function

$\‍[ y = f(x) = \left(\sum_{i = 1}^n x_i, \prod_{i = 1}^n x_i \right) \‍]$

Jacobian:

$\‍[ \frac{\d f}{\d x_j}(x) = \left( 1.0, \prod_{i = 1 \ldots n \wedge i \not = j } x_i \right) \‍]$

Simple real valued function for higher order derivatives

$\‍[ y = f(x) = 3*x^7 \eqdot \‍]$

Jacobian:

$\‍[ \frac{\d f}{\d x}(x) = 21 * x^6, \quad \‍]$

Higher order derivatives:

$\‍[ \frac{\d^2 f}{\d^2 x}(x) = 126 * x^5, \quad \frac{\d^3 f}{\d^3 x}(x) = 630 * x^4, \quad \frac{\d^4 f}{\d^4 x}(x) = 2520 * x^3, \quad \frac{\d^5 f}{\d^5 x}(x) = 7560 * x^2, \quad \frac{\d^6 f}{\d^6 x}(x) = 15120 * x \eqdot \‍]$

Linear system solve

$\‍[ x = A^{-1}b \‍]$

Forward mode:

$\‍[ \dot x = A^{-1}(\dot b - \dot Ax) \‍]$

Reverse mode:

$\‍[ \begin{aligned} s = & A^{-T}\bar x\\ \bar A \aeq & -s \cdot x^T \\ \bar b \aeq & s \\ \bar x = & 0 \eqdot \end{aligned} \‍]$

1D polynomial

$\‍[ w = f(x) = 3x^4 + 5x^3 - 3x^2 + 2x -4 \‍]$

Derivatives:

$\‍[ \frac{\d f}{\d x}(x) = 12x^3 + 15x^2 - 6x + 2 \‍]$

2D polynomial

$\‍[ w = f(x, y) = \sum_{i = 1 \ldots n, j = 1 \ldots n} A_{i,j} x^{i - 1} y^{j - 1} \‍]$

Derivatives:

$\‍[ \frac{\d f}{\d x}(x, y) = \sum_{i = 2 \ldots n, j = 1 \ldots n} (i - 1) A_{i,j} x^{i - 2} y^{j - 1} \‍]$

$\‍[ \frac{\d f}{\d y}(x, y) = \sum_{i = 1 \ldots n, j = 2 \ldots n} (j - 1) A_{i,j} x^{i - 1} y^{j - 2} \‍]$