... | ... | @@ -26,7 +26,7 @@ For a general nonlinear problem, we want to find $`\mathbf{x}`$ solution to |
|
|
```math
|
|
|
\mathbf{F}(\mathbf{x}) = 0,
|
|
|
```
|
|
|
where $`\mathbf{F}`$ is a (nonlinear) function from $`R^N`$ to $`R^N`$.
|
|
|
where $`\mathbf{F}(\mathbf{x})`$ is a (nonlinear) function from $`R^N`$ to $`R^N`$.
|
|
|
|
|
|
### Newton-Raphson method
|
|
|
|
... | ... | @@ -38,18 +38,18 @@ the successive iterates |
|
|
```
|
|
|
where
|
|
|
```math
|
|
|
\mathbf{J}=\left[ \frac{\partial\mathbf{F}}{\partial x_1} \cdots
|
|
|
\frac{\partial\mathbf{F}}{\partial x_N} \right]
|
|
|
\mathbf{J}(\mathbf{x})=\left[ \frac{\partial\mathbf{F}(\mathbf{x})}{\partial x_1} \cdots
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})}{\partial x_N} \right]
|
|
|
=\left[ \begin{array}{ccc}
|
|
|
\frac{\partial\mathbf{F}_1}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}_1}{\partial x_N} \\
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_1}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_1}{\partial x_N} \\
|
|
|
\vdots & \ddots & \vdots \\
|
|
|
\frac{\partial\mathbf{F}_N}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}_N}{\partial x_N}
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_N}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_N}{\partial x_N}
|
|
|
\end{array} \right]
|
|
|
```
|
|
|
is the Jacobian matrix, i.e. $`\mathbf{J}_{ij} = \frac{\partial\mathbf{F}_i}
|
|
|
{\partial\mathbf{x}_j}`$. In practice the Jacobian matrix $`\mathbf{J}`$ is not
|
|
|
is the Jacobian matrix, i.e. $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial\mathbf{F}(\mathbf{x})_i}
|
|
|
{\partial\mathbf{x}_j}`$. In practice the Jacobian matrix $`\mathbf{J}(\mathbf{x})`$ is not
|
|
|
inverted and at each iteration the following linear system is solved instead in
|
|
|
terms of the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_k)`$:
|
|
|
```math
|
... | ... | @@ -72,7 +72,7 @@ the Newton-Raphson iteration becomes |
|
|
\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_k ) =
|
|
|
\mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k
|
|
|
```
|
|
|
with $`\mathbf{J}_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathbf{x})_i}{\partial\mathbf{x}_j}`$.
|
|
|
with $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathbf{x})_i}{\partial\mathbf{x}_j}`$.
|
|
|
|
|
|
<!--
|
|
|
|
... | ... | @@ -551,4 +551,4 @@ Here a complete example is given for a electro-thermal problem. We have an objec |
|
|
<img src="ElectroThermal_Sim.jpg" width=50%>
|
|
|
|
|
|
---
|
|
|
*Initially written by @AsamMich.* |
|
|
*Initially written by @AsamMich.* |
|
|
\ No newline at end of file |