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docs/general_idea.md

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+# Homotopy Continuation
+
+### Introduction
+
+The aim of a homotopy method consists in solving a system of N nonlinear
+equations in N variables \[1, p.1\]:
+
+$$
+F(\bm{x}) = 0, \hspace{5mm} F: \mathbb{R}^N \rightarrow \mathbb{R}^N
+$$
+
+This is achieved by defining a homotopy, or deformation,
+$H : \mathbb{R}^N \rightarrow \mathbb{R}^N$ such that
+
+$$
+H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x},1) = F(\bm{x}),
+$$
+
+where $G: \mathbb{R}^N \rightarrow \mathbb{R}^N$ is trivial, having known zero
+points. One can then trace an implicitly defined curve
+$\bm{c}(s) \in H^{-1}(0)$ from $(\bm{x}_1,0)$ to $(\bm{x}_2, 1)$ [2, p.122].
+Typically, a convex homotopy such as
+
+$$
+H(\bm{x}, t) := tG(\bm{x}) + (1-t)F(\bm{x})
+$$
+
+is chosen [1, p.2].
+
+### Implicit Definition of Solution Curve
+
+"In the construction of the homotopy H(x, t) = 0, it may seem natural to use t
+as the designated parameter for the solution curve [...]. However, this
+parametrization has a severe limitation, as t cannot be used as a smooth
+parameter in certain situations. For example, [...] at points where ∂H/∂x is
+singular the solution curve of H(x, t) = 0 cannot be parametrized by t
+directly. Actually, one may avoid such difficulties by considering both x and t
+as independent variables and parametrize the smooth curve γ by the arc-length."
+[2, p. 125-126].
+
+We can show that the solution curve can be implicitly defined as
+
+$$
+\begin{align}
+DH(\bm{y}(s))\cdot \dot{\bm{y}}(s) = 0 \\[.5em]
+\text{det}\left(\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)\end{array}\right) = \sigma_0 \\[.5em]
+\lVert \dot{\bm{y}}(s) \rVert = 1 \\[.5em]
+\bm{y}(0) = (\bm{x}_0, 0)
+.
+\end{align}
+$$
+
+where $DH(y)$ is the Jacobian of $H(y)$ [2, p.127] [1, p.9]. Equation (1)
+corresponds to $\dot{y}(s)$ being tangent to the solution curve, the purpose of
+(2) is to fix the direction in which the curve is traced.
+
+### Tracing solution curves
+
+"In principle, any of the available ODE solvers capable of integrating the
+above system can be used to trace the curve [...]. However, due to numerical
+stability concerns, a more preferable method to trace the curve is the
+'prediction-correction scheme'. [...] Among many different choices for the
+'predictors' as well as 'correctors', we shall present here a commonly used
+combination of the (generalized) Euler’s method for the predictor and Newton’s
+method for the corrector." [2, p.127]
+
+#### Euler's Predictor
+
+$$
+\hat{\bm{y}} = \bm{y}_0 + \Delta s \cdot \sigma \cdot \Delta \bm{y},
+$$
+
+with $\Delta s$ denoting the step size and $\Delta y$ the numerical
+approximation of $\dot{y}(s)$ [2, p.128].
+
+#### Newton's corrector
+
+$$
+\bm{y} = \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm} \mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} - (DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}}).
+$$
+
+" [...] the shrinking distance
+$\lVert \mathcal{N}^j(\hat{\bm{y}}) - \mathcal{N}^{j-1} (\hat{\bm{y}})\rVert$
+between successive points produced by the iterations can be used as a criterion
+for convergence. Of course, if the iterations fail to converge, one must go
+back to adjust the step size for the Euler’s predictor." [2, p.130]
+
+______________________________________________________________________
+
+## References
+
+\[1\]: E. L. Allgower and K. Georg, Introduction to numerical continuation
+methods. in Classics in applied mathematics, no. 45. Philadelphia, Pa: Society
+for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6,
+Philadelphia, PA 19104), 2003. doi: 10.1137/1.9780898719154.
+
+\[2\]: T. Chen and T.-Y. Li, “Homotopy continuation method for solving systems
+of nonlinear and polynomial equations,” Communications in Information and
+Systems, vol. 15, no. 2, pp. 119–307, 2015, doi: 10.4310/CIS.2015.v15.n2.a1.