... | ... | @@ -38,15 +38,16 @@ the successive iterates |
|
|
```
|
|
|
where
|
|
|
```math
|
|
|
\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}(\mathbf{x})_1}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_1}{\partial x_N} \\
|
|
|
\vdots & \ddots & \vdots \\
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_N}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_N}{\partial x_N}
|
|
|
\end{array} \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}(\mathbf{x})_1}{\partial x_1} & \cdots &
|
|
|
\frac{\partial\mathbf{F}(\mathbf{x})_1}{\partial x_N} \\
|
|
|
\vdots & \ddots & \vdots \\
|
|
|
\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. $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial\mathbf{F}(\mathbf{x})_i}
|
|
|
{\partial\mathbf{x}_j}`$. In practice the Jacobian matrix $`\mathbf{J}(\mathbf{x}_k)`$ is not
|
... | ... | @@ -57,8 +58,8 @@ terms of the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_k)`$: |
|
|
```
|
|
|
Equivalently, one can solve
|
|
|
```math
|
|
|
\mathbf{J}(\mathbf{x}_k) \mathbf{x}_{k+1} = -\mathbf{F}(\mathbf{x}_k) +
|
|
|
\mathbf{J}(\mathbf{x}_k) \mathbf{x}_k ,
|
|
|
\mathbf{J}(\mathbf{x}_k) \mathbf{x}_{k+1}
|
|
|
= -\mathbf{F}(\mathbf{x}_k) + \mathbf{J}(\mathbf{x}_k) \mathbf{x}_k ,
|
|
|
```
|
|
|
in terms of the original unknown $`\mathbf{x}_{k+1}`$. A relaxation factor
|
|
|
$`\gamma_{k+1}`$ can be introduced at each iteration, leading to a modified new
|
... | ... | @@ -69,8 +70,8 @@ When the nonlinear function $`\mathbf{F}(\mathbf{x})`$ has the particular form |
|
|
$`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$,
|
|
|
the Newton-Raphson iteration becomes
|
|
|
```math
|
|
|
\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_k ) =
|
|
|
\mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k
|
|
|
\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_k )
|
|
|
= \mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k
|
|
|
```
|
|
|
with $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathbf{x})_i}{\partial\mathbf{x}_j}`$.
|
|
|
|
... | ... | |