Nonlinear Equations Problem
Nonlinear Problem
HydrodynamicStability.NonlinearProblem — Type`NonlinearProblem`Defines the nonlinear problem.
Mathematical Specification of the Nonlinear Problem
Problem of nonlinear equation, if
\[ f(x_0) = 0,\]
function $f(x)$ has a root when $x=x_0$.
Problem type
NonlinearProblem{F, DF, tType, P, K} <: AbstractProblemFields
f: Function for the nonlinear equations.t0: Initial guess for the nonlinear equations.p: The parameters for the problem. Defaults toNullParameterskwargs: The keyword arguments.
Algorithm
Bisection method
if $f(x)$ is a continuity function in range $[a, b]$, and which satisfy $f(a)f(b)<0$, then there exist a roof between $a$ and $b$, which means there is a $x_0$ satisfing $a<x_0<b$ and $f(x_0)=0$.
Then we can check the value of $f(c)$, where $c=\frac{a+b}{2}$, and determine whether $f(a)f(c)<0$ or $f(c)f(b)<0$, untill the value of $f(c)<\mathrm{tol}$.
Newton-Raphson method
Newton-Raphson method is a root finding method for the nonlinear equations (1). First, we give a initial guess $x_0$ and we have
\[ \begin{gather} f'(x_0)=\frac{f(x_0)-0}{x_0-x_1},\\ x_1=x_0-\frac{f(x_0)}{f'(x_0)}, \end{gather}\]
where $x_1$ is the approximate root we are looking for. Repeat the progress of (3) untill the solution converges, we have the iteration equation that
\[ x_0 = \mathrm{intial\ guess},\\ x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)},\quad i=0,1,2,\cdots\]