Applying Newton's Method to Sound Level Equation

I am trying to apply Newton's root-finding method to the following equation that measures sound level (in decibels) at a distance r meters from a source: L= L_0 - 20log10(r) - beta*r

Since we can't solve this directly, how can I find the derivative of the equation with the new function below? I'd like to print the root too. Am I able to use a function on MatLab to do both of these?

syms r

f(r) = L0 - 20 .* log10(r) - beta .* r - L;

% Given

L0 = 80;

beta = 0.00115;

L = 20;

tol = 0.000001;

This is Newton's Method from one of my previous assignments :

% Function file

function [c] = newtonsMethod(f,der,x0,tol)

% Inputs: f = function; der = derivative of function; x0 = initial guess; tol = error tolerance

err = 3*tol;

c=x0;

while err>tol

% Newton's method

c=c-f(c)/der(c);

err=abs(f(c));

end

end

Answer 1

[John D'Errico]

Since we cannot solve it directly.... Are you absolutely positive of that?

L0 = 80;
beta = 32.2; 
L = 20;

syms r
solve(L0 - 20 .* log10(r) - beta .* r - L == 0,r)
ans =
(100*lambertw(0, 1610*log(10)))/(161*log(10))

I suppose, if you say no solution exists, I could just believe you. But then why is MATLAB wrong, in claiming a solution exists? ;-) Admittedly, the solution uses the lambertw function.

help lambertw

My guess is, you are still supposed to use Newton's method. Do you mean you cannot differentiate?

diff(L0 - 20 .* log10(r) - beta .* r - L,r)
ans =
- 20/(r*log(10)) - 161/5

I'm a bit confused where the problem lies.

[emma]

Thank you for the function info! I meant that L= L_0 - 20log10(r) - beta*r cannot be solved directly, which is why we have to solve F(unknown) = RHS - LHS. Hence, solving for L0 - 20 .* log10(r) - beta .* r - L.