... | ... | @@ -52,7 +52,7 @@ is the Jacobian matrix, i.e. $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial\math |
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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 $`\mathbf{\delta x}_{k+1} := (\mathbf{x}_{k+1} - \mathbf{x}_k)`$:
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```math
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\mathbf{J}(\mathbf{x}_k) ( \mathbf{\delta x}_{k+1} ) = -\mathbf{F}(\mathbf{x}_k).
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\mathbf{J}(\mathbf{x}_k) \mathbf{\delta x}_{k+1} = -\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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... | ... | @@ -68,7 +68,7 @@ When the nonlinear function $`\mathbf{F}(\mathbf{x})`$ has the particular form |
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$`\mathbf{F}(\mathbf{x}) := \mathbf{A}(\mathbf{x}) \mathbf{x} - \mathbf{b}`$,
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the Newton-Raphson iteration becomes
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```math
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\mathbf{J}(\mathbf{x}_k) ( \mathbf{\delta x}_{k+1} )
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\mathbf{J}(\mathbf{x}_k) \mathbf{\delta x}_{k+1}
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= \mathbf{b} - \mathbf{A}(\mathbf{x}_k) \mathbf{x}_k
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```
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with $`\mathbf{J}(\mathbf{x})_{ij} = \frac{\partial(\mathbf{A}(\mathbf{x})\mathbf{x})_i}{\partial\mathbf{x}_j}`$. Equivalently, one can solve
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