convert/mathorner writes the polynomial poly in the name var in horner or ``nested'' form.

If there is only one indeterminate in poly then it is not necessary to specify the third argument var.

Horner form allows you to evaluate polynomials of Matrices in the most efficient manner. For a polynomial of degree n there will be n adds and n multiplications needed to evaluate the Horner form.

$p≔-56-7x^5+22x^4-55x^3-94x^2+87x$

$p≔-7x^5+22x^4-55x^3-94x^2+87x-56$

$A≔\mathrm{Matrix}\left(\left[\left[1\,-3\right]\,\left[4\,7\right]\right]\right)$

$A≔\left[\begin{array}{cc}1& \mathrm{-3}\\ 4& 7\end{array}\right]$

$\mathrm{convert}\left(p\,\mathrm{mathorner}\right)$

$-56+\left(87+\left(-94+\left(-55+\left(22-7x\right)\&*x\right)\&*x\right)\&*x\right)\&*x$

$\mathrm{subs}\left(x=A\,\right)$

$-56+\left(87+\left(-94+\left(-55+\left(22-7\left[\begin{array}{cc}1& \mathrm{-3}\\ 4& 7\end{array}\right]\right)\&*\left[\begin{array}{cc}1& \mathrm{-3}\\ 4& 7\end{array}\right]\right)\&*\left[\begin{array}{cc}1& \mathrm{-3}\\ 4& 7\end{array}\right]\right)\&*\left[\begin{array}{cc}1& \mathrm{-3}\\ 4& 7\end{array}\right]\right)\&*\left[\begin{array}{cc}1& \mathrm{-3}\\ 4& 7\end{array}\right]$

$\mathrm{eval}\left(\,\mathrm{`\&*`}=\mathrm{`.`}\right)$

$\left[\begin{array}{cc}14717& 12681\\ \mathrm{-16908}& \mathrm{-10645}\end{array}\right]$

$\mathrm{eval}\left(\,\left[x=A\,\mathrm{`\&*`}=\mathrm{`.`}\right]\right)$

$\left[\begin{array}{cc}14717& 12681\\ \mathrm{-16908}& \mathrm{-10645}\end{array}\right]$

$P≔\mathrm{eval}\left(p\,x=\mathrm{\%A}\right)$

$P≔-7A^5+22A^4-55A^3-94A^2+87A-56$

$\mathrm{eval}\left(\mathrm{convert}\left(P\,\mathrm{mathorner}\,\mathrm{\%A}\right)\,\mathrm{`\&*`}=\mathrm{`.`}\right)$

$-56+\left(87+\left(-94+\left(-55+\left(22-7A\right)·A\right)·A\right)·A\right)·A$

$\mathrm{value}\left(\right)$

$\left[\begin{array}{cc}14717& 12681\\ \mathrm{-16908}& \mathrm{-10645}\end{array}\right]$