I've been using the Newton-Raphson Method in my Numerical Methods course for a while now, blindly solving non-linear equations and systems of equations . This makes me somehow lose motivation, as I can't manage to find a real problem in which this tool is applied .
Can anyone suggest such a problem in an engineering/physical or mathematical environment ? I'm interested in seeing a meaningful equation related to a possible problem and the motivation behind finding its roots.
Best Answer
Almost all engineering calculations involve solutions of nonlinear equations and, in a single steady state simulation, these are typically solved billions of times (in a dynamic simulation, zillions of time). Most of the time, these equations are solved using Newton-Raphson method (or higher order equivalents).
Just to give you a simple one, consider Colebrook equation $$\frac 1{\sqrt f}=-2\log_{10}\Big(\frac{\epsilon}{3.71D_h}+\frac{2.51}{Re \sqrt f}\Big)$$ in which $f$ is the friction factor, $\epsilon$ the roughness height, $D_h$ the hydraulic diameter and $Re$ the Reynolds number. This friction factor determines the pressure drop in a pipeline and I suppose that you perceive the impact of that when we speak about gas or oil pipelines.
Even if we know an explicit solution of this equation in terms of Lambert function, most of the time, this equation is solved using Newton method. But, in order to save iterations and ensure stability, we use to find proper transforms and "good" starting values. For example, changing variables $f=\frac 1{x^2}$ leads to $$\Phi(x)=x+2\log_{10}\Big(\frac{\epsilon}{3.71D_h}+\frac{2.51}{Re} x\Big)=0$$ Dozens of papers have been published proposing unexpensive ways for the generation of the initial estimate; these make the convergence of Newton method very fast to high accuracy.
More difficult is the case of Underwood equations $$\Phi(x)=\sum_{i=1}^n \frac {a_i}{b_i-x}-c=0$$ for which all the roots must be found ($n$ can be several hundreds). This equation is also solved zillions of time in chemical process simulations. Hundreds of papers addressed how to estimate the starting points, how to transform the equation to ensure proper convexity but by the end, Newton--Raphson does the job.
Simpler could look the next example : the key part of any process simulation is the thermodynamic model. In oil and gas industry, the most commonly used models are based on so-called cubic equations of state. Any single step in the calculation requires computing the real roots of a cubic polynomial. Because of limited precision, Cardano method is never used and again Newton.