Problem 1: Part 16: Newton iterations
edit
Let the nonlinear equations be 
. Recall
that the Newton method requires that we iterate using the formula
where 
is the Newton iteration number. Derive an expression for the
derivative of 
that is required in the above formula.
Let us find the derivatives term by term. For the first term
![{\displaystyle {\cfrac {d}{d\Delta \gamma }}\left[9~\mu ^{2}~(\Delta \gamma )^{2}\right]=18~\mu ^{2}~\Delta \gamma }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e28bf31c8129435b38a51779efeb46a2774303e)
For the second term
![{\displaystyle {\cfrac {d}{d\Delta \gamma }}\left[6~\mu ~{\sqrt {\cfrac {3}{2}}}~\Delta \gamma ~\mathbf {s} _{n+1}^{\text{trial}}:\mathbf {n} _{n}\right]=6~\mu ~{\sqrt {\cfrac {3}{2}}}~\mathbf {s} _{n+1}^{\text{trial}}:\mathbf {n} _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea1f4b0c8b5c2c3ebaaf3a8437aaee150b60c299)
For the fourth term
![{\displaystyle {\cfrac {d}{d\Delta \gamma }}\left[{\cfrac {3}{2}}~\mathbf {s} _{n+1}^{\text{trial}}:\mathbf {s} _{n+1}^{\text{trial}}\right]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32654d1daa8a2a3e9094a27b5f54c64e378c8cfe)
For the third term
![{\displaystyle {\cfrac {d}{d\Delta \gamma }}\left[\left\{\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right\}^{2}\left\{1-\left({\cfrac {T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}\lVert \mathbf {s} _{n}\rVert _{}-T_{0}}{T_{m}-T_{0}}}\right)\right\}^{2}\right]={\cfrac {d}{d\Delta \gamma }}[P_{n}~Q_{n}]={\cfrac {dP_{n}}{d\Delta \gamma }}~Q_{n}+{\cfrac {dQ_{n}}{d\Delta \gamma }}~P_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5665e63321fca6b0ae2ccfcd51dd98b3a965ba5)
Now,
![{\displaystyle {\begin{aligned}{\cfrac {dP_{n}}{d\Delta \gamma }}&={\cfrac {d}{d\Delta \gamma }}\left[\left\{\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right\}^{2}\right]\\&=2~\left\{\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right\}{\cfrac {d}{d\Delta \gamma }}\left[\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right]\\&=2~\left\{\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right\}\left[n~B~\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n-1}\right]{\cfrac {d}{d\Delta \gamma }}\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)\\&=2~\left\{\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right\}\left[n~B~\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n-1}\right]\left({\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)\\&=2~n~B~\left({\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n-1}\left[\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1fac77d49a405807652ba6a4983573b0ca5d0f0)
Similarly,
![{\displaystyle {\begin{aligned}{\cfrac {dQ_{n}}{d\Delta \gamma }}&={\cfrac {d}{d\Delta \gamma }}\left[\left\{1-\left({\cfrac {T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}\lVert \mathbf {s} _{n}\rVert _{}-T_{0}}{T_{m}-T_{0}}}\right)\right\}^{2}\right]\\&=2~\left\{1-\left({\cfrac {T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}\lVert \mathbf {s} _{n}\rVert _{}-T_{0}}{T_{m}-T_{0}}}\right)\right\}{\cfrac {d}{d\Delta \gamma }}\left[-\left({\cfrac {T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}\lVert \mathbf {s} _{n}\rVert _{}-T_{0}}{T_{m}-T_{0}}}\right)\right]\\&=-2~{\sqrt {\cfrac {3}{2}}}~\left\{1-\left({\cfrac {T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}~\lVert \mathbf {s} _{n}\rVert _{}-T_{0}}{T_{m}-T_{0}}}\right)\right\}~\left({\cfrac {\chi ~\lVert \mathbf {s} _{n}\rVert _{}}{\rho _{n}~C_{p}~(T_{m}-T_{0})}}\right)\\&=-{\sqrt {6}}~\left({\cfrac {\chi ~\lVert \mathbf {s} _{n}\rVert _{}}{\rho _{n}~C_{p}}}\right)~\left[{\cfrac {T_{m}-T_{n}-{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}~\lVert \mathbf {s} _{n}\rVert _{}}{(T_{m}-T_{0})^{2}}}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24ed76209db03830f7dc4461bbfb34d04ab837af)
Therefore, the full expression for the derivative is
![{\displaystyle {\begin{aligned}{\cfrac {dg(\Delta \gamma _{r})}{d\Delta \gamma }}&=18~\mu ^{2}~\Delta \gamma -6~\mu ~{\sqrt {\cfrac {3}{2}}}~\mathbf {s} _{n+1}^{\text{trial}}:\mathbf {n} _{n}\\&\qquad -2~n~B~\left({\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n-1}\left[\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right]\left[1-\left({\cfrac {T_{n}+{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}\lVert \mathbf {s} _{n}\rVert _{}-T_{0}}{T_{m}-T_{0}}}\right)\right]^{2}\\&\qquad +{\sqrt {6}}~\left({\cfrac {\chi ~\lVert \mathbf {s} _{n}\rVert _{}}{\rho _{n}~C_{p}}}\right)~\left[{\cfrac {T_{m}-T_{n}-{\sqrt {\cfrac {3}{2}}}~{\cfrac {\chi ~\Delta \gamma }{\rho _{n}~C_{p}}}~\lVert \mathbf {s} _{n}\rVert _{}}{(T_{m}-T_{0})^{2}}}\right]\left[\sigma _{0}+B\left(\alpha _{n}+\Delta \gamma ~{\cfrac {{\boldsymbol {\varepsilon }}_{n}^{p}:\mathbf {n} _{n}}{\lVert {\boldsymbol {\varepsilon }}_{n}^{p}\rVert _{}}}\right)^{n}\right]^{2}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddbd9fa4f191b142b0d01b229cacb7ab6fa8535e)
