... | ... | @@ -50,9 +50,9 @@ where |
|
|
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
|
|
|
inverted and at each iteration the following linear system is solved instead in
|
|
|
terms of the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_k)`$:
|
|
|
terms of the unknown $`\mathbf{\delta x}_{k+1} := (\mathbf{x}_{k+1} - \mathbf{x}_k)`$:
|
|
|
```math
|
|
|
\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_k ) = -\mathbf{F}(\mathbf{x}_k).
|
|
|
\mathbf{J}(\mathbf{x}_k) ( \mathbf{\delta x}_{k+1} ) = -\mathbf{F}(\mathbf{x}_k).
|
|
|
```
|
|
|
Equivalently, one can solve
|
|
|
```math
|
... | ... | @@ -62,16 +62,21 @@ Equivalently, one can solve |
|
|
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
|
|
|
(relaxed) iterate $`\tilde{\mathbf{x}}_{k+1} := \mathbf{x}_k + \gamma_{k+1}
|
|
|
(\mathbf{x}_{k+1} - \mathbf{x}_k)`$.
|
|
|
\mathbf{\delta x}_{k+1}`$.
|
|
|
|
|
|
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{J}(\mathbf{x}_k) ( \mathbf{\delta x}_{k+1} )
|
|
|
= \mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k
|
|
|
```
|
|
|
with $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathbf{x})_i}{\partial\mathbf{x}_j}`$.
|
|
|
with $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathbf{x})_i}{\partial\mathbf{x}_j}`$. Equivalently, one can solve
|
|
|
```math
|
|
|
\mathbf{J}(\mathbf{x}_k) \mathbf{x}_{k+1}
|
|
|
= \mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k + \mathbf{J}(\mathbf{x}_k) \mathbf{x}_k .
|
|
|
```
|
|
|
|
|
|
|
|
|
### Picard method
|
|
|
|
... | ... | |