Newton Raphson method to find intersection points

Question

I'm supposed to use N-R method to find intersection point near $(4,-4)$ of $x^2+y^2=16$ and $e^{1/3x} + (1/5)y =1 .$

So, I equated these two functions, and got $f(x) = 5- (5*\exp(x/3)) + (16- x^2)^{1/2}$ and $f'(x)= (-5/3)*\exp(x/3) - (16-x^2)^{-1/2}$.

As stated in my assignment, I'm supposed to start with point $(4,-4)$ , but as you can see...this would give me $-\infty$ for $f'(x).$ So I can't even go on to the next iteration.

What am I doing wrong?

Answer 1

Way 1: use 2D Newton's method on the original system. Based on how the problem was formulated, this was probably the intent.

Way 2: Work with $u+v=16,e^{\sqrt{u}/3}-\sqrt{v}/5=1$, which is effectively working with $(x^2,y^2)$ as your variables under the assumption $x>0,y<0$. Then you can eliminate one variable without triggering any singularity.

Way 3: Do what you said, but start somewhere else and hope you get the right intersection. For instance you could start at $(\sqrt{8},-\sqrt{8})$ which is on the circle (and not on the axes) and will thus remove the singularity. (This might have actually been the point where they wanted you to start, i.e. $(4,-4)$ might have just been a mistake.)

Answer 2

As said in answers, they (more than likely) wanted you to use Newton-Raphson method with two varaibles.

However, you idea of eliminating $y$ and just solve for $x$ was good and works.

You have two equations $$e^{x/3}+\frac{y}{5}-1=0 \tag 1$$ $$x^2+y^2-16=0 \tag 2$$ Frm $(1)$, $y=5-5 e^{x/3}$. Plug it in $(2)$ to get $$x^2-50 e^{x/3}+25 e^{2 x/3}+9=0$$ So, you look for the zero of $$f(x)=x^2-50 e^{x/3}+25 e^{2 x/3}+9$$ $$f'(x)=2 x-\frac{50 }{3}e^{x/3}+\frac{50}{3} e^{2 x/3}$$ Now, Newton iterates using $x_0=4$ $$\left( \begin{array}{cc} n & x_n \\ 0 & 4.000000000 \\ 1 & 2.943256168 \\ 2 & 2.166300869 \\ 3 & 1.755176003 \\ 4 & 1.649488097 \\ 5 & 1.643297711 \\ 6 & 1.643277453 \end{array} \right)$$ which is the solution for ten significant figures.