... | ... | @@ -80,10 +80,9 @@ with $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathb |
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= \mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k + \mathbf{J}(\mathbf{x}_k) \mathbf{x}_k .
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```
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### Picard method
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The Picard method is a simple fixed method applied on the nonlinear function $`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$. Given an initial guess $`\mathbf{x}_0`$, Picard's method consists in computing the successive iterates $`\mathbf{x}_{k+1}`$ such that
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The Picard method is a simple fixed method applied on the equation $`\mathbf{F}(\mathbf{x})=0`$ when the nonlinear function is of the form $`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$. Given an initial guess $`\mathbf{x}_0`$, Picard's method consists in computing the successive iterates $`\mathbf{x}_{k+1}`$ such that
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```math
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\mathbf{A}(\mathbf{x}_{k}) \mathbf{x}_{k+1} = \mathbf{b},
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\quad k = 0, 1, 2, ...
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