Solved: Use the method of Steepest Descent with TOL = 0.05 to approximate the solutions

Chapter 10, Problem 10.4.1

(choose chapter or problem)

Use the method of Steepest Descent with TOL = 0.05 to approximate the solutions of the following nonlinear systems.

a. \(\begin{array}{l}
4 x_{1}^{2}-20 x_{1}+\frac{1}{4} x_{2}^{2}+8=0 \\
\frac{1}{2} x_{1} x_{2}^{2}+2 x_{1}-5 x_{2}+8=0 .
\end{array}\)

b. \(\begin{aligned}
3 x_{1}^{2}-x_{2}^{2} & =0, \\
3 x_{1} x_{2}^{2}-x_{1}^{3}-1 & =0 .
\end{aligned}\)

c. \(\begin{aligned}
\ln \left(x_{1}^{2}+x_{2}^{2}\right)-\sin \left(x_{1} x_{2}\right) & =\ln 2+\ln \pi \\
e^{x_{1}-x_{2}}+\cos \left(x_{1} x_{2}\right) & =0 .
\end{aligned}\)

d. \(\begin{aligned}
\sin \left(4 \pi x_{1} x_{2}\right)-2 x_{2}-x_{1} & =0, \\
\left(\frac{4 \pi-1}{4 \pi}\right)\left(e^{2 x_{1}}-e\right)+4 e x_{2}^{2}-2 e x_{1} & =0 .
\end{aligned}\)