Initial vector for Newton-Rhapson method for several variables

Question

I've been worked in a exercise about Newton's method on several variables, more specific the exercise 12, chapter 10.2 of Burden-Numerical Analysis pag. 646 https://faculty.ksu.edu.sa/sites/default/files/numerical_analysis_9th.pdf, So I've planted a nonlinear system equations \begin{eqnarray} k_{1}e^{k_{2}}+ k_{3} -10 = 0 \\ k_{1}e^{2k_{2}}+ 2k_{3} -12 = 0 \\ k_{1}e^{3k_{2}}+ 3k_{3} -15 = 0 \end{eqnarray} But I don't undertand what is the initial vector $K_{0}$. The first initial vector that I consider was $K_{0}=(12,12,24)$, but doesn't work. Do you know some idea to consider the initial vector?

Answer 1

Your equations have some issues, they should be

$$k_1 e^{1k_2}+1 k_3-10=0\\ k_1 e^{2 k_2}+2 k_3-12=0\\k_1 e^{3 k_2}+3 k_3-15=0$$

These equations have two closed-form solutions, but I think they are looking for the first one.

First solution $$ \begin{align} k_1 &= \frac{1}{12} \left(111-\sqrt{33}\right)\\k_2 &= \log \left(\sqrt{33}+15\right)-4 \log (2)\\k_3 &= 10-\frac{1}{192} \left(111-\sqrt{33}\right) \left(\sqrt{33}+15\right) \end{align}$$

Second solution $$ \begin{align} k_1 &= \frac{1}{12} \left(\sqrt{33}+111\right) \\ k_2 &= \log \left(15-\sqrt{33}\right)-4 \log (2) \\ k_3 &= 10-\frac{1}{192} \left(15-\sqrt{33}\right) \left(\sqrt{33}+111\right) \end{align}$$

Numerically these are

$$k_1= 8.77129,k_2 = 0.259695,k_3= -1.37228 \\ k_1= 9.72871,k_2=-0.547378,k_3= 4.37228$$

Initial conditions are not easy to find for this nonlinear Newton's.

Just trying some random values, I finally found two that work

$$K_0 = \begin{pmatrix} k_1(0)\\ k_2(0) \\ k_3(0) \end{pmatrix}= \begin{pmatrix} ~~10. \\~~1.2\\-14. \end{pmatrix}$$

Using this initial vector results in

$$\begin{align} ~~~~~k_1 &= 8.77128644612184 \\ ~~~~~k_2 &= 0.2596954489674532 \\ k_3 &= -1.372281323269029 \end{align}$$

To get the second result, use the initial vector

$$K_0 = \begin{pmatrix} k_1(0)\\ k_2(0) \\ k_3(0) \end{pmatrix}= \begin{pmatrix} 8.1 \\1.2\\5.3 \end{pmatrix}$$