... | ... | @@ -18,8 +18,8 @@ systems of algebraic equations of the form: |
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```math
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\mathbf{A} \mathbf{x} = \mathbf{b},
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```
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where $\mathbf{A}$ is a square matrix of size $`N`$ (e.g. the stiffness matrix of the
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problem), $\mathbf{b}$ takes into account source terms and $\mathbf{x}$ is the
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where $`\mathbf{A}`$ is a square matrix of size $`N`$ (e.g. the stiffness matrix of the
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problem), $`\mathbf{b}`$ takes into account source terms and $`\mathbf{x}`$ is the
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vector of unknowns to be computed.
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For a general nonlinear problem, we want to find $`\mathbf{x}`$ solution to
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... | ... | @@ -34,7 +34,7 @@ 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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\quad k = 1, 2, \dots
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\quad k = 1, 2, ...
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```
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where
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```math
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... | ... | @@ -45,9 +45,10 @@ where |
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\frac{\partial\mathbf{F}_1}{\partial x_N} \\
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\vdots & \ddots & \vdots \\
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\frac{\partial\mathbf{F}_N}{\partial x_1} & \cdots &
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\frac{\partial\mathbf{F}_N}{\partial x_N} \right]
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\frac{\partial\mathbf{F}_N}{\partial x_N}
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\end{array} \right]
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```
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is the Jacobian matrix, i.e. $`\mathbf{J}_{ij} = \fraction{\partial\mathbf{F}_i}
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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:
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```math
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