... | ... | @@ -49,17 +49,18 @@ where |
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\end{array} \right]
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```
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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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{\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:
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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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```
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in terms of a the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_{k})`$. Equivalently, we
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in terms of a the unknown $`(\mathbf{x}_{k+1} - \mathbf{x}_{k})`$. Equivalently, one
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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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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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