91Ó°ÊÓ

Use simple fixed-point iteration to locate the root of \\[ f(x)=2 \sin (\sqrt{x})-x \\] Use an initial guess of \(x_{0}=0.5\) and iterate until \(\varepsilon_{a} \leq 0.001 \%\). Verify that the process is linearly convergent as described in Box 6.1.

Use (a) fixed-point iteration and (b) the Newton-Raphson method to determine a root of \(f(x)=-x^{2}+1.8 x+2.5\) using \(x_{0}=5 .\) Perform the computation until \(\varepsilon_{a}\) is less than \(\varepsilon_{s}=0.05 \%\) Also perform an error check of your final answer.

Locate the first positive root of \\[ f(x)=\sin x+\cos \left(1+x^{2}\right)-1 \\] where \(x\) is in radians. Use four iterations of the secant method with initial guesses of \((a) x_{i-1}=1.0\) and \(x_{i}=3.0 ;(b), x_{i-1}=1.5\) and \(x_{i}=2.5,\) and \((\mathrm{c}) x_{i-1}=1.5\) and \(x_{i}=2.25\) to locate the root. (d) Use the graphical method to explain your results.

Determine the real root of \(x^{3.5}=80,\) with the modified secant method to within \(\varepsilon_{\mathrm{s}}=0.1 \%\) using an initial guess of \(x_{0}=3.5\) and \(\delta=0.01.\)

Determine the roots of the following simultaneous nonlinear equations using (a) fixed-point iteration and (b) the NewtonRaphson method: \\[ \begin{array}{l} y=-x^{2}+x+0.75 \\ y+5 x y=x^{2} \end{array} \\] Employ initial guesses of \(x=y=1.2\) and discuss the results.

A mass balance for a pollutant in a well-mixed lake can be written as \\[ V \frac{d c}{d t}=W-Q c-k V \sqrt{c} \\] Given the parameter values \(V=1 \times 10^{6} \mathrm{m}^{3}, Q=1 \times 10^{5} \mathrm{m}^{3} / \mathrm{yr}\) \(W=1 \times 10^{6} \mathrm{g} / \mathrm{yr},\) and \(k=0.25 \mathrm{m}^{0.5} / \mathrm{g}^{0.5} / \mathrm{yr},\) use the modified secant method to solve for the steady-state concentration. Employ an initial guess of \(c=4 \mathrm{g} / \mathrm{m}^{3}\) and \(\delta=0.5 .\) Perform three iterations and determine the percent relative error after the third iteration.