Solve using python that utilizes the False$-$Position method instead of using fixed point iteration. Prepare a graphical solution of the problem and discuss whether you expect any nonzero initial guess to converge. Note that the given equation yields a complex number for the flow rate for negative values of the depth. Is it possible to modify your program to avoid root estimates that are negative?

If so$,$ does this guarantee the method will converge? $6\mathrm{.}20$ The Manning equation can be written for a rectangular open

channel as

$Q=\frac{\sqrt[\phantom{2}]{S}(BH{)}^{\frac{5}{3}}}{n(B+2H{)}^{\frac{2}{3}}}$

where $Q=$ flow $(\frac{m^3}{s}),S=$ slope $(\frac{m}{m}),B=$ channel width $(m),$

$H=$ depth $($m$),$ and $n=$ the Manning roughness coefficient. De$-$

velop a fixed$-$point iteration scheme to solve this equation for

$H$ given $Q=10,S=0\mathrm{.}0001,B=30,$ and $n=0\mathrm{.}025.$ Prove that your

scheme converges for all initial guesses greater than or equal to zero.