NCHessian[afunction, $\{X_1,H_1\},\ldots,\{X_k,H_k \}$ ]

Aliases: None.

Description: NCHessian[afunction, $\{X_1,H_1\},\{X_2,H_2\},\ldots,\{X_k,H_k \}$ ]
computes the Hessian of a afunction of noncommutting variables and coefficients. The Hessian recall is the second derivative. Here we are computing the noncommutative directional derivative of a noncommutative function. Using repeated calls to DirectionalD, the Hessian of afunction is computed with respect to the variables $X_1,X_2,\ldots,X_k$ and the search directions $H_1$ , $H_2$ , $\ldots$ , $H_k$. The Hessian $\mathcal{H} \Gamma$ of a function $\Gamma$ is defined by

$$\mathcal{H} \Gamma (\vec{X})[\vec{H}] := {\displaystyle\frac{d^2}{dt^2}}\Gamma(\vec{X}+t\vec{H}) \big \vert _{t = 0}$$

One can easily show that the second derivative of a hereditary symmetric noncommutative rational function $\Gamma$ with respect to one variable $X$ has the form

$${\mathcal H}\Gamma(X) [H] = sym \left[\sum\limits^k_{\ell=1} A_\ell H^T B_\ell H C_\ell \right],$$

where $A_\ell$, $B_\ell$, and $C_\ell$ are functions of $X$ determined by $\Gamma$. (An analogous expression holds for more variables.) The Hessian will always be quadratic with respect to $\vec{H}$. (A noncommutative polynomial in variables $H_1$, $H_2$, $\dots$, $H_k$, is said to be quadratic if each monomial in the polynomial expression is of order two in the variables $H_1$, $H_2$, $\dots$, $H_k$.)

Arguments: afunction is a function of the variables $X_1,X_2,\ldots,X_k$. The Hessian will be computed with respect to the search directions $H_1$ , $H_2$ , $\ldots$ , $H_k$.
For example, suppose $F(x,y) = x + x ** y + y ** x$. Then,
NCHessian $[F,\{x,h\},\{y,k\}]$ gives $2 h**k + 2 k**h$ As another example, if $G(x,y,z)=inv[y] + z ** x$, then NCHessian $[G,\{x,h\},\{y,k\},\{z,i\}]$ gives $2 i ** h + 2 inv[y] ** k ** inv[y] ** k ** inv[y]$.
The results of NCHessian can be factored into the form $v^tMv$ by calling NCMatrixofQuadratic. (see NCMatrixofQuadratic).

Comments / Limitations: None.