GLM-Maxwell hyperbolic conservation law

conserved quantities:
${U} = {\left[\begin{array}{c} { D} _i\\ { B} _i\\ \phi\\ \psi\end{array}\right]}$
flux:
${F} = {\left[\begin{array}{c} {{{\chi_\phi}}} \cdot {{\phi}}\\ {\frac{1}{\mu}} {{ B} _z}\\ -{{\frac{1}{\mu}} {{ B} _y}}\\ {{{\chi_\psi}}} \cdot {{\psi}}\\ -{{\frac{1}{\epsilon}} {{ D} _z}}\\ {\frac{1}{\epsilon}} {{ D} _y}\\ {{{\chi_\phi}}} \cdot {{{ D} _x}}\\ {{{\chi_\psi}}} \cdot {{{ B} _x}}\end{array}\right]}$
Flux jacobian:
${A} = {\left[\begin{array}{cccccccc} 0& 0& 0& 0& 0& 0& {\chi_\phi}& 0\\ 0& 0& 0& 0& 0& \frac{1}{\mu}& 0& 0\\ 0& 0& 0& 0& \frac{-1}{\mu}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\chi_\psi}\\ 0& 0& \frac{-1}{\epsilon}& 0& 0& 0& 0& 0\\ 0& \frac{1}{\epsilon}& 0& 0& 0& 0& 0& 0\\ {\chi_\phi}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\chi_\psi}& 0& 0& 0& 0\end{array}\right]}$

Testing for homogeneity property. Does $\frac{\partial F}{\partial U} \cdot U = F$ ?

${\frac{\partial F}{\partial U}} = {\left[\begin{array}{c} {{{\chi_\phi}}} \cdot {{\phi}}\\ {\frac{1}{\mu}} {{ B} _z}\\ -{{\frac{1}{\mu}} {{ B} _y}}\\ {{{\chi_\psi}}} \cdot {{\psi}}\\ -{{\frac{1}{\epsilon}} {{ D} _z}}\\ {\frac{1}{\epsilon}} {{ D} _y}\\ {{{\chi_\phi}}} \cdot {{{ D} _x}}\\ {{{\chi_\psi}}} \cdot {{{ B} _x}}\end{array}\right]}$

Looks true.

char poly:
${{{{{{\chi_\phi}}^{2}}} {{{{\chi_\psi}}^{2}}}}{-{{{{{\chi_\phi}}^{2}}} {{{\lambda}^{2}}}}}{-{{{{{\chi_\psi}}^{2}}} {{{\lambda}^{2}}}}} + {{\lambda}^{4}} + {{{{\epsilon}^{2}}} {{{\lambda}^{8}}} {{{\mu}^{2}}}}{-{{{{{\chi_\phi}}^{2}}} {{{\epsilon}^{2}}} {{{\lambda}^{6}}} {{{\mu}^{2}}}}}{-{{{{{\chi_\psi}}^{2}}} {{{\epsilon}^{2}}} {{{\lambda}^{6}}} {{{\mu}^{2}}}}}{-{{{2}} {{\epsilon}} \cdot {{\mu}} \cdot {{{\lambda}^{6}}}}} + {{{{\epsilon}^{2}}} {{{{\chi_\phi}}^{2}}} {{{{\chi_\psi}}^{2}}} {{{\lambda}^{4}}} {{{\mu}^{2}}}} + {{{2}} {{\epsilon}} \cdot {{\mu}} \cdot {{{{\chi_\phi}}^{2}}} {{{\lambda}^{4}}}} + {{{2}} {{\epsilon}} \cdot {{\mu}} \cdot {{{{\chi_\psi}}^{2}}} {{{\lambda}^{4}}}}{-{{{2}} {{\epsilon}} \cdot {{\mu}} \cdot {{{{\chi_\phi}}^{2}}} {{{{\chi_\psi}}^{2}}} {{{\lambda}^{2}}}}}} = {0}$
char poly solutions:
${{{{{\lambda}^{0}}} {{{\epsilon}^{2}}} {{{\mu}^{2}}}} + {{{{\lambda}^{2}}} {{{\epsilon}^{2}}} {{{\mu}^{2}}}} + {{{{\lambda}^{4}}} {{{\epsilon}^{2}}} {{{\mu}^{2}}}} + {{{{\lambda}^{6}}} {{{\epsilon}^{2}}} {{{\mu}^{2}}}} + {{{{\lambda}^{8}}} {{{\epsilon}^{2}}} {{{\mu}^{2}}}}} = {0}$

Flux in the x direction:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{cccccccc} -{\frac{1}{\sqrt{\epsilon}}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}& 0\\ 0& 0& 0& -{\sqrt{\epsilon}}& 0& \sqrt{\epsilon}& 0& 0\\ 0& 0& \sqrt{\epsilon}& 0& -{\sqrt{\epsilon}}& 0& 0& 0\\ 0& -{\frac{1}{\sqrt{\epsilon}}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}\\ 0& 0& \sqrt{\mu}& 0& \sqrt{\mu}& 0& 0& 0\\ 0& 0& 0& \sqrt{\mu}& 0& \sqrt{\mu}& 0& 0\\ \frac{1}{\sqrt{\epsilon}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}& 0\\ 0& \frac{1}{\sqrt{\epsilon}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{{\chi_\phi}}& 0& 0& 0& 0& 0& 0& 0\\ 0& -{{\chi_\psi}}& 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{-1}{\sqrt{{{\mu}} \cdot {{\epsilon}}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& \frac{-1}{\sqrt{{{\mu}} \cdot {{\epsilon}}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{1}{\sqrt{{{\mu}} \cdot {{\epsilon}}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{1}{\sqrt{{{\mu}} \cdot {{\epsilon}}}}& 0& 0\\ 0& 0& 0& 0& 0& 0& {\chi_\phi}& 0\\ 0& 0& 0& 0& 0& 0& 0& {\chi_\psi}\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{{\frac{1}{2}} {\sqrt{\epsilon}}}& 0& 0& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0\\ 0& 0& 0& -{{\frac{1}{2}} {\sqrt{\epsilon}}}& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}\\ 0& 0& \frac{1}{{{2}} {{\sqrt{\epsilon}}}}& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0& 0\\ 0& -{\frac{1}{{{2}} {{\sqrt{\epsilon}}}}}& 0& 0& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ 0& 0& -{\frac{1}{{{2}} {{\sqrt{\epsilon}}}}}& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0& 0\\ 0& \frac{1}{{{2}} {{\sqrt{\epsilon}}}}& 0& 0& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ {\frac{1}{2}} {\sqrt{\epsilon}}& 0& 0& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0\\ 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}\end{array}\right]}}}$

Verify reconstruction from eigensystem:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccccc} 0& 0& 0& 0& 0& 0& {\chi_\phi}& 0\\ 0& 0& 0& 0& 0& \frac{1}{\mu}& 0& 0\\ 0& 0& 0& 0& -{\frac{1}{\mu}}& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& {\chi_\psi}\\ 0& 0& -{\frac{1}{\epsilon}}& 0& 0& 0& 0& 0\\ 0& \frac{1}{\epsilon}& 0& 0& 0& 0& 0& 0\\ {\chi_\phi}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\chi_\psi}& 0& 0& 0& 0\end{array}\right]}$

Flux in a normal basis:
${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{cccccccc} { {(n_1)}} _1& { {(n_2)}} _1& { {(n_3)}} _1& 0& 0& 0& 0& 0\\ { {(n_1)}} _2& { {(n_2)}} _2& { {(n_3)}} _2& 0& 0& 0& 0& 0\\ { {(n_1)}} _3& { {(n_2)}} _3& { {(n_3)}} _3& 0& 0& 0& 0& 0\\ 0& 0& 0& { {(n_1)}} _1& { {(n_2)}} _1& { {(n_3)}} _1& 0& 0\\ 0& 0& 0& { {(n_1)}} _2& { {(n_2)}} _2& { {(n_3)}} _2& 0& 0\\ 0& 0& 0& { {(n_1)}} _3& { {(n_2)}} _3& { {(n_3)}} _3& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{\frac{1}{\sqrt{\epsilon}}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}& 0\\ 0& 0& 0& -{\sqrt{\epsilon}}& 0& \sqrt{\epsilon}& 0& 0\\ 0& 0& \sqrt{\epsilon}& 0& -{\sqrt{\epsilon}}& 0& 0& 0\\ 0& -{\frac{1}{\sqrt{\epsilon}}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}\\ 0& 0& \sqrt{\mu}& 0& \sqrt{\mu}& 0& 0& 0\\ 0& 0& 0& \sqrt{\mu}& 0& \sqrt{\mu}& 0& 0\\ \frac{1}{\sqrt{\epsilon}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}& 0\\ 0& \frac{1}{\sqrt{\epsilon}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{{\frac{1}{\sqrt{|g|}}} {{\chi_\phi}}}& 0& 0& 0& 0& 0& 0& 0\\ 0& -{{\frac{1}{\sqrt{|g|}}} {{\chi_\psi}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}& 0& 0\\ 0& 0& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{\chi_\phi}}& 0\\ 0& 0& 0& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{\chi_\psi}}\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{{\frac{1}{2}} {\sqrt{\epsilon}}}& 0& 0& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0\\ 0& 0& 0& -{{\frac{1}{2}} {\sqrt{\epsilon}}}& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}\\ 0& 0& \frac{1}{{{2}} {{\sqrt{\epsilon}}}}& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0& 0\\ 0& -{\frac{1}{{{2}} {{\sqrt{\epsilon}}}}}& 0& 0& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ 0& 0& -{\frac{1}{{{2}} {{\sqrt{\epsilon}}}}}& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0& 0\\ 0& \frac{1}{{{2}} {{\sqrt{\epsilon}}}}& 0& 0& 0& \frac{1}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ {\frac{1}{2}} {\sqrt{\epsilon}}& 0& 0& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0\\ 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}\end{array}\right]}} {{\left[\begin{array}{cccccccc} { {(n_1)}} _1& { {(n_1)}} _2& { {(n_1)}} _3& 0& 0& 0& 0& 0\\ { {(n_2)}} _1& { {(n_2)}} _2& { {(n_2)}} _3& 0& 0& 0& 0& 0\\ { {(n_3)}} _1& { {(n_3)}} _2& { {(n_3)}} _3& 0& 0& 0& 0& 0\\ 0& 0& 0& { {(n_1)}} _1& { {(n_1)}} _2& { {(n_1)}} _3& 0& 0\\ 0& 0& 0& { {(n_2)}} _1& { {(n_2)}} _2& { {(n_2)}} _3& 0& 0\\ 0& 0& 0& { {(n_3)}} _1& { {(n_3)}} _2& { {(n_3)}} _3& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right]}}}$

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {{{\left[\begin{array}{cccccccc} -{\frac{{ {(n_1)}} _1}{\sqrt{\epsilon}}}& 0& {{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}}& -{{{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}}}& -{{{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}}}& {{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}}& \frac{{ {(n_1)}} _1}{\sqrt{\epsilon}}& 0\\ -{\frac{{ {(n_1)}} _2}{\sqrt{\epsilon}}}& 0& {{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}}& -{{{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}}}& -{{{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}}}& {{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}}& \frac{{ {(n_1)}} _2}{\sqrt{\epsilon}}& 0\\ -{\frac{{ {(n_1)}} _3}{\sqrt{\epsilon}}}& 0& {{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}}& -{{{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}}}& -{{{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}}}& {{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}}& \frac{{ {(n_1)}} _3}{\sqrt{\epsilon}}& 0\\ 0& -{\frac{{ {(n_1)}} _1}{\sqrt{\epsilon}}}& {{{ {(n_2)}} _1}} {{\sqrt{\mu}}}& {{{ {(n_3)}} _1}} {{\sqrt{\mu}}}& {{{ {(n_2)}} _1}} {{\sqrt{\mu}}}& {{{ {(n_3)}} _1}} {{\sqrt{\mu}}}& 0& \frac{{ {(n_1)}} _1}{\sqrt{\epsilon}}\\ 0& -{\frac{{ {(n_1)}} _2}{\sqrt{\epsilon}}}& {{{ {(n_2)}} _2}} {{\sqrt{\mu}}}& {{{ {(n_3)}} _2}} {{\sqrt{\mu}}}& {{{ {(n_2)}} _2}} {{\sqrt{\mu}}}& {{{ {(n_3)}} _2}} {{\sqrt{\mu}}}& 0& \frac{{ {(n_1)}} _2}{\sqrt{\epsilon}}\\ 0& -{\frac{{ {(n_1)}} _3}{\sqrt{\epsilon}}}& {{{ {(n_2)}} _3}} {{\sqrt{\mu}}}& {{{ {(n_3)}} _3}} {{\sqrt{\mu}}}& {{{ {(n_2)}} _3}} {{\sqrt{\mu}}}& {{{ {(n_3)}} _3}} {{\sqrt{\mu}}}& 0& \frac{{ {(n_1)}} _3}{\sqrt{\epsilon}}\\ \frac{1}{\sqrt{\epsilon}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}& 0\\ 0& \frac{1}{\sqrt{\epsilon}}& 0& 0& 0& 0& 0& \frac{1}{\sqrt{\epsilon}}\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{{\frac{1}{\sqrt{|g|}}} {{\chi_\phi}}}& 0& 0& 0& 0& 0& 0& 0\\ 0& -{{\frac{1}{\sqrt{|g|}}} {{\chi_\psi}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& -{\frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}}& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{\frac{1}{\sqrt{\mu}}}{{{\sqrt{|g|}}} \cdot {{\sqrt{\epsilon}}}}& 0& 0\\ 0& 0& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{\chi_\phi}}& 0\\ 0& 0& 0& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{\chi_\psi}}\end{array}\right]}} {{\left[\begin{array}{cccccccc} -{{\frac{1}{2}} {{{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}}}& -{{\frac{1}{2}} {{{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}}}& -{{\frac{1}{2}} {{{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}}& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0\\ 0& 0& 0& -{{\frac{1}{2}} {{{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}}}& -{{\frac{1}{2}} {{{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}}}& -{{\frac{1}{2}} {{{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}}& 0& {\frac{1}{2}} {\sqrt{\epsilon}}\\ \frac{{ {(n_3)}} _1}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{ {(n_3)}} _2}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{ {(n_3)}} _3}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{ {(n_2)}} _1}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_2)}} _2}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_2)}} _3}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ -{\frac{{ {(n_2)}} _1}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{ {(n_2)}} _2}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{ {(n_2)}} _3}{{{2}} {{\sqrt{\epsilon}}}}}& \frac{{ {(n_3)}} _1}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_3)}} _2}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_3)}} _3}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ -{\frac{{ {(n_3)}} _1}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{ {(n_3)}} _2}{{{2}} {{\sqrt{\epsilon}}}}}& -{\frac{{ {(n_3)}} _3}{{{2}} {{\sqrt{\epsilon}}}}}& \frac{{ {(n_2)}} _1}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_2)}} _2}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_2)}} _3}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ \frac{{ {(n_2)}} _1}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{ {(n_2)}} _2}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{ {(n_2)}} _3}{{{2}} {{\sqrt{\epsilon}}}}& \frac{{ {(n_3)}} _1}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_3)}} _2}{{{2}} {{\sqrt{\mu}}}}& \frac{{ {(n_3)}} _3}{{{2}} {{\sqrt{\mu}}}}& 0& 0\\ {\frac{1}{2}} {{{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}}& {\frac{1}{2}} {{{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}}& {\frac{1}{2}} {{{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}& 0& 0& 0& {\frac{1}{2}} {\sqrt{\epsilon}}& 0\\ 0& 0& 0& {\frac{1}{2}} {{{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}}& {\frac{1}{2}} {{{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}}& {\frac{1}{2}} {{{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}& 0& {\frac{1}{2}} {\sqrt{\epsilon}}\end{array}\right]}}}$

${\frac{\partial { F} ^I}{\partial { U} ^J}} = {\left[\begin{array}{cccccccc} 0& 0& 0& 0& \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\phi}}} \cdot {{{ {(n_1)}} _1}}}& 0\\ 0& 0& 0& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& 0& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\phi}}} \cdot {{{ {(n_1)}} _2}}}& 0\\ 0& 0& 0& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}}}{{{\mu}} \cdot {{\sqrt{|g|}}}}& 0& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\phi}}} \cdot {{{ {(n_1)}} _3}}}& 0\\ 0& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}{-{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\psi}}} \cdot {{{ {(n_1)}} _1}}}\\ \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _1}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& \frac{{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}{-{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\psi}}} \cdot {{{ {(n_1)}} _2}}}\\ \frac{{{{{ {(n_2)}} _1}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _1}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& \frac{{{{{ {(n_2)}} _2}} {{{ {(n_3)}} _3}}}{-{{{{ {(n_2)}} _3}} {{{ {(n_3)}} _2}}}}}{{{\epsilon}} \cdot {{\sqrt{|g|}}}}& 0& 0& 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\psi}}} \cdot {{{ {(n_1)}} _3}}}\\ {\frac{1}{\sqrt{|g|}}} {{{{\chi_\phi}}} \cdot {{{ {(n_1)}} _1}}}& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\phi}}} \cdot {{{ {(n_1)}} _2}}}& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\phi}}} \cdot {{{ {(n_1)}} _3}}}& 0& 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\psi}}} \cdot {{{ {(n_1)}} _1}}}& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\psi}}} \cdot {{{ {(n_1)}} _2}}}& {\frac{1}{\sqrt{|g|}}} {{{{\chi_\psi}}} \cdot {{{ {(n_1)}} _3}}}& 0& 0\end{array}\right]}$

L(v) = $\left[\begin{array}{c} {{{\frac{1}{2}}} {{{v_7}}} \cdot {{\sqrt{\epsilon}}}}{-{{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}}}{-{{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}}}{-{{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}}\\ {{{\frac{1}{2}}} {{{v_8}}} \cdot {{\sqrt{\epsilon}}}}{-{{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}}}{-{{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}}}{-{{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}}\\ {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_2)}} _1}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_2)}} _2}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_2)}} _3}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_3)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_3)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_3)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}}\\ {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_3)}} _1}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_3)}} _2}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_3)}} _3}} {{\frac{1}{\sqrt{\mu}}}}}{-{{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_2)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_2)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_2)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}}}\\ {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_2)}} _1}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_2)}} _2}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_2)}} _3}} {{\frac{1}{\sqrt{\mu}}}}}{-{{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_3)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_3)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}}}{-{{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_3)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}}}\\ {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_3)}} _1}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_3)}} _2}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_3)}} _3}} {{\frac{1}{\sqrt{\mu}}}}} + {{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_2)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_2)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_2)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}}\\ {{{\frac{1}{2}}} {{{v_7}}} \cdot {{\sqrt{\epsilon}}}} + {{{\frac{1}{2}}} {{{v_1}}} \cdot {{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}} + {{{\frac{1}{2}}} {{{v_2}}} \cdot {{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}} + {{{\frac{1}{2}}} {{{v_3}}} \cdot {{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}\\ {{{\frac{1}{2}}} {{{v_8}}} \cdot {{\sqrt{\epsilon}}}} + {{{\frac{1}{2}}} {{{v_4}}} \cdot {{{ {(n_1)}} _1}} {{\sqrt{\epsilon}}}} + {{{\frac{1}{2}}} {{{v_5}}} \cdot {{{ {(n_1)}} _2}} {{\sqrt{\epsilon}}}} + {{{\frac{1}{2}}} {{{v_6}}} \cdot {{{ {(n_1)}} _3}} {{\sqrt{\epsilon}}}}\end{array}\right]$

R(v) = $\left[\begin{array}{c} {-{{{{v_1}}} \cdot {{{ {(n_1)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{{v_7}}} \cdot {{{ {(n_1)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_3}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}}}{-{{{{v_4}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}}}}{-{{{{v_5}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\epsilon}}}}} + {{{{v_6}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\epsilon}}}}\\ {-{{{{v_1}}} \cdot {{{ {(n_1)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{{v_7}}} \cdot {{{ {(n_1)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_3}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}}}{-{{{{v_4}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}}}}{-{{{{v_5}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\epsilon}}}}} + {{{{v_6}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\epsilon}}}}\\ {-{{{{v_1}}} \cdot {{{ {(n_1)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{{v_7}}} \cdot {{{ {(n_1)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_3}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}}}{-{{{{v_4}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}}}}{-{{{{v_5}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\epsilon}}}}} + {{{{v_6}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\epsilon}}}}\\ {-{{{{v_2}}} \cdot {{{ {(n_1)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{{v_8}}} \cdot {{{ {(n_1)}} _1}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_3}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\mu}}}} + {{{{v_4}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\mu}}}} + {{{{v_5}}} \cdot {{{ {(n_2)}} _1}} {{\sqrt{\mu}}}} + {{{{v_6}}} \cdot {{{ {(n_3)}} _1}} {{\sqrt{\mu}}}}\\ {-{{{{v_2}}} \cdot {{{ {(n_1)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{{v_8}}} \cdot {{{ {(n_1)}} _2}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_3}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\mu}}}} + {{{{v_4}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\mu}}}} + {{{{v_5}}} \cdot {{{ {(n_2)}} _2}} {{\sqrt{\mu}}}} + {{{{v_6}}} \cdot {{{ {(n_3)}} _2}} {{\sqrt{\mu}}}}\\ {-{{{{v_2}}} \cdot {{{ {(n_1)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}}} + {{{{v_8}}} \cdot {{{ {(n_1)}} _3}} {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_3}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\mu}}}} + {{{{v_4}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\mu}}}} + {{{{v_5}}} \cdot {{{ {(n_2)}} _3}} {{\sqrt{\mu}}}} + {{{{v_6}}} \cdot {{{ {(n_3)}} _3}} {{\sqrt{\mu}}}}\\ {{{{v_1}}} \cdot {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_7}}} \cdot {{\frac{1}{\sqrt{\epsilon}}}}}\\ {{{{v_2}}} \cdot {{\frac{1}{\sqrt{\epsilon}}}}} + {{{{v_8}}} \cdot {{\frac{1}{\sqrt{\epsilon}}}}}\end{array}\right]$

left transform code:

real const tmp1 = 1. / sqrt_1_eps;
real const tmp2 = 1. / 2.;
real const tmp3 = (X)->ptr[6] * tmp1;
real const tmp4 = n_l.x * tmp1;
real const tmp5 = (X)->ptr[0] * tmp4;
real const tmp6 = tmp2 * tmp5;
real const tmp7 = n_l.y * tmp1;
real const tmp8 = (X)->ptr[1] * tmp7;
real const tmp9 = tmp2 * tmp8;
real const tmp10 = n_l.z * tmp1;
real const tmp11 = (X)->ptr[2] * tmp10;
real const tmp12 = tmp2 * tmp11;
real const tmp13 = tmp2 * tmp3;
real const tmp14 = (X)->ptr[7] * tmp1;
real const tmp15 = (X)->ptr[3] * tmp4;
real const tmp16 = tmp2 * tmp15;
real const tmp17 = (X)->ptr[4] * tmp7;
real const tmp18 = tmp2 * tmp17;
real const tmp19 = (X)->ptr[5] * tmp10;
real const tmp20 = tmp2 * tmp19;
real const tmp21 = tmp2 * tmp14;
real const tmp22 = sqrt_1_eps * n3_l.x;
real const tmp23 = (X)->ptr[0] * tmp22;
real const tmp24 = sqrt_1_eps * n3_l.y;
real const tmp25 = (X)->ptr[1] * tmp24;
real const tmp26 = sqrt_1_eps * n3_l.z;
real const tmp27 = (X)->ptr[2] * tmp26;
real const tmp28 = sqrt_1_mu * n2_l.x;
real const tmp29 = (X)->ptr[3] * tmp28;
real const tmp30 = sqrt_1_mu * n2_l.y;
real const tmp31 = (X)->ptr[4] * tmp30;
real const tmp32 = sqrt_1_mu * n2_l.z;
real const tmp33 = (X)->ptr[5] * tmp32;
real const tmp34 = tmp2 * tmp31;
real const tmp35 = tmp2 * tmp33;
real const tmp36 = tmp2 * tmp29;
real const tmp37 = tmp34 + tmp35;
real const tmp38 = tmp2 * tmp27;
real const tmp39 = tmp36 + tmp37;
real const tmp40 = tmp2 * tmp25;
real const tmp41 = tmp2 * tmp23;
real const tmp42 = sqrt_1_eps * n2_l.x;
real const tmp43 = (X)->ptr[0] * tmp42;
real const tmp44 = tmp2 * tmp43;
real const tmp45 = sqrt_1_eps * n2_l.y;
real const tmp46 = (X)->ptr[1] * tmp45;
real const tmp47 = tmp2 * tmp46;
real const tmp48 = sqrt_1_eps * n2_l.z;
real const tmp49 = (X)->ptr[2] * tmp48;
real const tmp50 = tmp2 * tmp49;
real const tmp51 = sqrt_1_mu * n3_l.x;
real const tmp52 = (X)->ptr[3] * tmp51;
real const tmp53 = sqrt_1_mu * n3_l.y;
real const tmp54 = (X)->ptr[4] * tmp53;
real const tmp55 = sqrt_1_mu * n3_l.z;
real const tmp56 = (X)->ptr[5] * tmp55;
real const tmp57 = tmp2 * tmp54;
real const tmp58 = tmp2 * tmp56;
real const tmp59 = tmp2 * tmp52;
real const tmp60 = tmp57 + tmp58;
real const tmp61 = tmp59 + tmp60;
real const (Y)->ptr[0] = -tmp6 + -tmp12 + -tmp9 + tmp13;
real const (Y)->ptr[1] = -tmp16 + -tmp20 + -tmp18 + tmp21;
real const (Y)->ptr[2] = tmp38 + tmp39 + tmp40 + tmp41;
real const (Y)->ptr[3] = -tmp44 + -tmp47 + -tmp50 + tmp61;
real const (Y)->ptr[4] = -tmp41 + -tmp40 + -tmp38 + tmp39;
real const (Y)->ptr[5] = tmp44 + tmp47 + tmp50 + tmp61;
real const (Y)->ptr[6] = tmp6 + tmp9 + tmp12 + tmp13;
real const (Y)->ptr[7] = tmp16 + tmp18 + tmp20 + tmp21;

right transform code:

real const tmp1 = 1. / sqrt_1_eps;
real const tmp2 = n3_l.x * tmp1;
real const tmp3 = n2_l.x * tmp1;
real const tmp4 = sqrt_1_eps * n_l.x;
real const tmp5 = n3_l.y * tmp1;
real const tmp6 = n2_l.y * tmp1;
real const tmp7 = sqrt_1_eps * n_l.y;
real const tmp8 = n3_l.z * tmp1;
real const tmp9 = n2_l.z * tmp1;
real const tmp10 = sqrt_1_eps * n_l.z;
real const tmp11 = 1. / sqrt_1_mu;
real const tmp12 = n2_l.x * tmp11;
real const tmp13 = n3_l.x * tmp11;
real const tmp14 = n2_l.y * tmp11;
real const tmp15 = n3_l.y * tmp11;
real const tmp16 = n2_l.z * tmp11;
real const tmp17 = n3_l.z * tmp11;
real const (Y)->ptr[0] = (X)->ptr[2] * tmp2 + -(X)->ptr[3] * tmp3 + -(X)->ptr[4] * tmp2 + (X)->ptr[5] * tmp3 + (X)->ptr[6] * tmp4 + -(X)->ptr[0] * tmp4;
real const (Y)->ptr[1] = (X)->ptr[2] * tmp5 + -(X)->ptr[3] * tmp6 + -(X)->ptr[4] * tmp5 + (X)->ptr[5] * tmp6 + (X)->ptr[6] * tmp7 + -(X)->ptr[0] * tmp7;
real const (Y)->ptr[2] = (X)->ptr[2] * tmp8 + -(X)->ptr[3] * tmp9 + -(X)->ptr[4] * tmp8 + (X)->ptr[5] * tmp9 + (X)->ptr[6] * tmp10 + -(X)->ptr[0] * tmp10;
real const (Y)->ptr[3] = (X)->ptr[2] * tmp12 + (X)->ptr[3] * tmp13 + (X)->ptr[4] * tmp12 + (X)->ptr[5] * tmp13 + (X)->ptr[7] * tmp4 + -(X)->ptr[1] * tmp4;
real const (Y)->ptr[4] = (X)->ptr[2] * tmp14 + (X)->ptr[3] * tmp15 + (X)->ptr[4] * tmp14 + (X)->ptr[5] * tmp15 + (X)->ptr[7] * tmp7 + -(X)->ptr[1] * tmp7;
real const (Y)->ptr[5] = (X)->ptr[2] * tmp16 + (X)->ptr[3] * tmp17 + (X)->ptr[4] * tmp16 + (X)->ptr[5] * tmp17 + (X)->ptr[7] * tmp10 + -(X)->ptr[1] * tmp10;
real const (Y)->ptr[6] = (X)->ptr[6] * sqrt_1_eps + (X)->ptr[0] * sqrt_1_eps;
real const (Y)->ptr[7] = (X)->ptr[7] * sqrt_1_eps + (X)->ptr[1] * sqrt_1_eps;