SpECTRE
v2025.01.30
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The correction \(\delta x\) to improve a solution \(x_0\). More...
#include <Tags.hpp>
Public Types | |
using | type = typename Tag::type |
using | tag = Tag |
The correction \(\delta x\) to improve a solution \(x_0\).
A linear problem \(Ax=b\) can be equivalently formulated as the problem \(A\delta x=b-A x_0\) for the correction \(\delta x\) to an initial guess \(x_0\). More importantly, we can use a correction scheme to solve a nonlinear problem \(A_\mathrm{nonlinear}(x)=b\) by repeatedly solving a linearization of it. For instance, a Newton-Raphson scheme iteratively refines an initial guess \(x_0\) by repeatedly solving the linearized problem
\begin{equation} \frac{\delta A_\mathrm{nonlinear}}{\delta x}(x_k)\delta x_k = b-A_\mathrm{nonlinear}(x_k) \end{equation}
for the correction \(\delta x_k\) and then updating the solution as \(x_{k+1}=x_k + \delta x_k\).