# Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem:

• Real numbers, all are known constants - $a,c,e,h,i,k,l,m,n,o$,
• Positive real numbers, variables - $b,d$,
• Functions - $f,g,j$.

Given is equation

$a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

where $f(d)=g(d)/\sqrt{2\cdot\pi\cdot h\cdot d}$ where $log(g)=i/d$.

Instructions say that solving first equation for $d$ from a given $b$ and thus finding $f(b)$ (Essence of which confuses me.) requires Newton-Raphson method. Then they say that in similar cases people often use analytic formula simulating $f(b)$, normalized to unity at $b=1$:

$j(b)=(k\cdot (b/l)^m)\cdot (1+(b/l)^n)^o$.

Then they say to use approximation $f(b)=f_0\cdot j(b)$ and that I will find $f_0=f(1)$ (Which is second source of confusion for me.) by Newton-Raphson iteration from the first equation.

What I basically need to get is $f_0$, and I do not understand how and where does $f(b)$ come into play and how do I get to a point where I can solve the problem with Newton-Raphson method.

I have tried to look for solutions around, but to me it seems to be a per-case problem so I have decided to ask a question. Not having had any mathematical instruction in English may be a contributing factor to the issue.

I should sleep more. The problem is a non-problem, all I had to do is plug what I've got into a single equation.

$log(g)=i/d$ -> $g=10^{(i/d)}$

$f(d)=g(d)/\sqrt{2\cdot\pi \cdot h\cdot d}$ -> $f(d)=10^{(i/d)}/\sqrt{2\cdot\pi\cdot h\cdot d}$

Which, plugged into the main equation, gives us:

$a\cdot b^{-2}=c\cdot d^4+e\cdot 10^{(i/d)}/\sqrt{2\cdot\pi\cdot h\cdot d}$

Which yields us function solvable by Newton-Raphson (if we transform it to $=0$):

$f(d)=c\cdot d^4+e\cdot 10^{(i/d)}/\sqrt{2\cdot\pi\cdot h\cdot d} - a\cdot b^{-2}$