... | ... | @@ -33,7 +33,7 @@ where $`\mathbf{F}`$ is a (nonlinear) function from $`R^N`$ to $`R^N`$. |
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Given an initial guess $`\mathbf{x}_0`$, Newton's method consists in computing
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the successive iterates
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```math
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\mathbf{x}_{k+1} = \mathbf{x}_{k} - \mathbf{J}^{-1}(\mathbf{x}_k) F(\mathbf{x}_k),
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\mathbf{x}_{k+1} = \mathbf{x}_k - \mathbf{J}^{-1}(\mathbf{x}_k) F(\mathbf{x}_k),
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\quad k = 1, 2, ...
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```
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where
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... | ... | @@ -51,20 +51,28 @@ where |
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is the Jacobian matrix, i.e. $`\mathbf{J}_{ij} = \frac{\partial\mathbf{F}_i}
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{\partial\mathbf{x}_j}`$. In practice the Jacobian matrix $`\mathbf{J}`$ is not
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inverted and at each iteration the following linear system is solved instead in
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terms of the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_{k})`$:
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terms of the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_k)`$:
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```math
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\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_{k} ) = -\mathbf{F}(\mathbf{x}_k).
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\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_k ) = -\mathbf{F}(\mathbf{x}_k).
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```
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Equivalently, one can solve
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```math
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\mathbf{J}(\mathbf{x}_k) \mathbf{x}_{k+1} = -\mathbf{F}(\mathbf{x}_k) +
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\mathbf{J}(\mathbf{x}_k) \mathbf{x}_{k} ,
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\mathbf{J}(\mathbf{x}_k) \mathbf{x}_k ,
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```
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in terms of the original unknown $`\mathbf{x}_{k+1}`$.
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in terms of the original unknown $`\mathbf{x}_{k+1}`$. A relaxation factor
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$`\gamma_{k+1}`$ can be introduced at each iteration, leading to a modified new
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(relaxed) iterate $`\tilde{\mathbf{x}}_{k+1} := \mathbf{x}_k$ + \gamma_{k+1}
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(\mathbf{x}_{k+1} - \mathbf{x}_k)`$.
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When the nonlinear function $`\mathbf{F}(\mathbf{x})`$ has the particular form
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$`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x}`$, the Jacobian
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matrix $`\mathbf{J}`$ writes:
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$`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$,
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the Newton-Raphson iteration becomes
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```math
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\mathbf{J}(\mathbf{x}_k) ( \mathbf{x}_{k+1} - \mathbf{x}_k ) =
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\mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k
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```
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with $`\mathbf{J}_{ij} = \frac{\partial(\mathbf{A}\mathbf{x})_i}{\partial\mathbf{x}_j}`$.
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