The following theorem is proved in [@minoids].

The multiplication map ${\Lambda {\mathrm{SU}}(2)_\rho}\times {\Lambda_+^{\mathbb{R}}{\mathrm{SL}}(2,{\mathbb{C}})}_\rho \longmapsto {\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}$ is a smooth diffeomorphism between Banach manifolds. Its inverse map is called “Iwasawa decomposition” and is denoted for $\Phi\in{\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}$: $${\mathop \mathrm{Iwa}}(\Phi) = \left( {\mathop \mathrm{Uni}}(\Phi), {\mathop \mathrm{Pos}}(\Phi) \right).$$

#### The ingredients. {#sectionIngredients}

Let $H>1$, $q={\mathop \mathrm{arcoth}}H>0$ and $\rho>e^q$. The DPW method takes for input data:

-   A Riemann surface $\Sigma$.

-   A holomorphic $1$-form on $\Sigma$ with values in ${{\Lambda {\mathfrak{sl}}(2,{\mathbb{C}})}_\rho}$ of the following form: $$\label{eqPotentielXialphabetagamma}
            \xi = \begin{pmatrix}
            \alpha & {\lambda}^{-1}\beta \\
            \gamma & -\alpha
            \end{pmatrix}$$ where $\alpha$, $\beta$, $\gamma$ are holomorphic $1$-forms on $\Sigma$ with values in $\Lambda{\mathbb{C}}_{\rho}^{\geq 0}$. The $1$-form $\xi$ is called “the potential”.

-   A base point $z_0\in\Sigma$.

-   An initial condition $\phi\in{\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}$.

#### The recipe.

The DPW method consists in the following steps:

1.  Let $\widetilde{z}_0$ be any point above $z_0$ in the universal cover $\widetilde{\Sigma}$ of $\Sigma$. Solve on $\widetilde{\Sigma}$ the following Cauchy problem: $$\label{eqCauchyProblem}
            \left\{
            \begin{array}{l}
            d\Phi = \Phi \xi \\
            \Phi(\widetilde{z}_0) = \phi.
            \end{array}
            \right.$$ Then $\Phi: {\widetilde{\Sigma}}\longrightarrow {\Lambda {\mathrm{SL}}(2,{\mathbb{C}})_\rho}$ is called “the holomorphic frame”.

2.  Compute pointwise on ${\widetilde{\Sigma}}$ the Iwasawa decomposition of $\Phi$: $$(F(z),B(z)) := {\mathop \mathrm{Iwa}}\Phi(z).$$ The unitary part $F$ 