... | ... | @@ -90,13 +90,21 @@ 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 $`||\mathbf{F}(\mathbf{x})||`$ in some norm (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.
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For the exact solution, $`\mathbf{F}(\mathbf{x})`$ is zero; i.e. $`\mathbf{A}(\mathbf{x})\mathbf{x}-\mathbf{b} = 0`$ if $`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$.
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In practice, the iterations are stopped if after $p$ iterations a sufficiently small value of the residual 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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In practice, the iterations are stopped if after $p$ iterations a sufficiently small value of the residual, in some norm is obtained, e.g.
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```math
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\frac{||\mathbf{\delta x}_p||_\infty}{||\mathbf{x}_p||_\infty} < \varepsilon,
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||\mathbf{F}(\mathbf{x}_p)|| < \varepsilon_\text{abs}
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```
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with $`\varepsilon`$ a small dimensionless number (e.g. $`10^{-6}`$).
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or
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```math
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\frac{||\mathbf{F}(\mathbf{x}_p)||}{||\mathbf{F}(\mathbf{x}_0)||} < \varepsilon_\text{rel}
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```
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where $`\varepsilon_\text{abs}`$ and $`\varepsilon_\text{rel}`$ are small numbers (e.g. $`10^{-6}`$).
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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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