MODELING & NUMERICAL SOLUTIONS $2$MS$3$ AUTO TECH ASSIGNMENT $2$ Roots of Non$-$linear Equations

The upward velocity of a rocket can be computed by the following formula: $v=u*ln(\frac{m_0}{m_0-qt})-gt,$ where $v$ is the upward velocity, $u$ is the velocity at which the fuel is expelled relative to the rocket, $m_0$ is the initial mass of the rocket at time $t=0,q$ is the fuel consumption rate and $g$ is the downwards acceleration due to gravity. If $u=2700\frac{m}{s},m_0=250,000Kg,q=3680k\frac{g}{s},$ and $g=9\mathrm{.}81\frac{m}{s^2},$ compute the time at which $v=2000\frac{m}{s}.$ Use single $($float$)$ precision for all your manual calculations. Single precision i$.$e$.$ with $6$ significant digits. For example: $-2345\mathrm{.}01,0\mathrm{.}220344,-0\mathrm{.}000112233,10233400$

a$)$ Bisection method $($manual calculations, starting points $10$ and $50,$ three iterations, single precision$)$

b$)$ False$-$position method $($manual calc., starting points $10$ and $50,$ three iterations, single precision$)$

c$)$ Secant method $($manual calculations, starting points $50$ and $40,$ three iterations, single precision$)$

d$)$ Newton$-$Raphson $($C$++$ program, use pseudo$-$code, single precision float, eps $=0\mathrm{.}0001,$ delta $=0\mathrm{.}01)$

e$)$ Error analysis $($compare $a,b,c$ with d respectively, express errors in percentage$)$

procedure Newton$(f,f^{\text{'}},x,$nmax,$,)$

integer $n,$nmax; $,$ real $x,fx,fp,,$

external function $f,f^{\text{'}}$

fxlarrf$(x)$

output $0,x,fx$

for $n=1$ to nmax do

$fplarr{f}^{\text{'}}(x)$

if $|fp|<mi\; id="EditorElement\_13886751"></mi>$ then

output "small derivative"

return

end if

dlarrf$\frac{x}{f}p$

xlarrx$-d$

fxlarrf$(x)$

output $n,x,fx$

if $|d|<mi\; id="EditorElement\_13886529"></mi>$ then

output "convergence"

return

end if

end for

end procedure Newton

Newton Raphson p$106,$ p$92.$

Legend:

$f$ is the input function being solved

$f^{\text{'}}$ derivative of $f,$ obtained through analytical derivation

$x$ is the starting point of the algorithm

nmax is the max number of iterations after which the algorithm is aborted

$$ is desired precision

$$ is a limit used to indicate when the derivative becomes almost horizontal.