... | ... | @@ -61,8 +61,9 @@ Equivalently, one can solve |
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\mathbf{J}(\mathbf{x}_{k-1}) \mathbf{x}_k
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= -\mathbf{F}(\mathbf{x}_{k-1}) + \mathbf{J}(\mathbf{x}_{k-1}) \mathbf{x}_{k-1} ,
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```
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in terms of the original unknown $`\mathbf{x}_k`$. A relaxation factor
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$`\gamma_k`$ can be introduced at each iteration, leading to a modified new
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in terms of the original unknown $`\mathbf{x}_k`$.
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The convergence of the Newton-Raphson method is by no means guaranteed: it depends on regularity of $`\mathbf{F}`$ and the choice of the initial guess $`\mathbf{x}_0`$. In case of strong non-linearity and/or unfavorable initialization conditions, divergence is not unlikely. Sufficiently close to the solution, the convergence is quadratic. In order to ensure or accelerate convergence, relaxation techniques may be applied: a relaxation factor $`\gamma_k`$ can be introduced at each iteration, leading to a modified new
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(relaxed) iterate $`\tilde{\mathbf{x}}_k := \mathbf{x}_{k-1} + \gamma_k
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\mathbf{\delta x}_k`$. The relaxation factor is usually chosen in $`]0,1]`$.
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... | ... | @@ -101,17 +102,11 @@ where $`\varepsilon_\text{abs}`$ and $`\varepsilon_\text{rel}`$ are small number |
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Another stopping criterion can be defined on the $`p^\text{th}`$ increment $`\mathbf{\delta x}_p=\mathbf{x}_p-\mathbf{x}_{p-1}`$. For example, it could be:
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```math
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\frac{||\mathbf{\delta x}_p||_\infty}{||\mathbf{x}_p||_\infty} < \varepsilon.
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\frac{||\mathbf{\delta x}_p||}{||\mathbf{x}_p||} < \varepsilon.
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```
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<!--
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For the exact solution, the residual defined by $\vec{r}(\vec{x})=\mat{A}(\vec{x}) \; \vec{x}-\vec{b}$ is zero. If after $p$ iterations, a satisfactory convergence is obtained, the iterative process is stopped. The convergence criterion could be based on some norm of the residual $\vec{r}(\vec{x}_p)$ or on the $p^\text{th}$ increment $\vec{\delta x}_p=\vec{x}_p-\vec{x}_{p-1}$. For example, it could be :
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\begin{equation}
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\frac{||\vec{\delta x}_p||_\infty}{||\vec{x}_p||_\infty} < \varepsilon,
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\end{equation}
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with $\varepsilon$ a small dimensionless number (e.g. $10^{-6}$).
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### Newton-Raphson method
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Usually, the Newton-Raphson method (NR-method) is used. In that case, a new approximation $\vec{x}_i=\vec{x}_{i-1}+\vec{\delta x}_i$ is obtained through the linearization of the residual vector $\vec{r}(\vec{x}_i)$ around the previous approximated value $\vec{x}_{i-1}$:
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... | ... | @@ -138,7 +133,7 @@ For the system written in \eqref{Sys}, with the right-hand side $\vec{b}$ assume |
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\label{SysNR}
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\end{equation}
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Each NR-iteration provides an unique increment solution $\vec{\delta x}_i$ which depends on values taken from the previous iteration step $i-1$. Nevertheless, the convergence is by no means guaranteed. In case of strong non-linearity and/or unfavorable initialization conditions, divergence is not unlikely. Sufficiently close to the solution, the convergence of the NR-method is quadratic. In order to ensure or accelerate convergence, relaxation techniques may be applied:
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Each NR-iteration provides an unique increment solution $\vec{\delta x}_i$ which depends on values taken from the previous iteration step $i-1$. Nevertheless,
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\begin{equation}
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\vec{x}_i=\vec{x}_{i-1}+\gamma_i \; \vec{\delta x}_i,
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\end{equation}
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