... | ... | @@ -90,10 +90,10 @@ The Picard method is a simple fixed point method applied on the equation $`\math |
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### Convergence
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For the exact solution, the residual $`||\mathbf{F}(\mathbf{x})||`$ (i.e. $`||\mathbf{A}(\mathbf{x})\mathbf{x}-\math{b}||`$ if $`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$) is zero. In practice, the iterations are stopped if after $p$ iterations a sufficiently small value of the residual (in some norm) is obtained. 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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For the exact solution, the residual $`||\mathbf{F}(\mathbf{x})||`$ (i.e. $`||\mathbf{A}(\mathbf{x})\mathbf{x}-\mathbf{b}||`$ if $`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$) is zero. In practice, the iterations are stopped if after $p$ iterations a sufficiently small value of the residual (in some norm) is obtained. 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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\end{equation}
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```
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with $`\varepsilon`$ a small dimensionless number (e.g. $`10^{-6}`$).
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