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Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamar product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_2=F(x_0, y_0)$, and $y_2 = G(y_0, x_0)$: $$x_1 = F^{1/2}(x_0, y_0), y_1=G^{1/2}(y_0, x_0)$$ $$x_2 = F^{1/2}(x_1, y_1), y_2 = G^{1/2}(y_1, x_1)$$

Any thoughts are appreciated.

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamard product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_2=F(x_0, y_0)$, and $y_2 = G(y_0, x_0)$: $$x_1 = F^{1/2}(x_0, y_0), y_1=G^{1/2}(y_0, x_0)$$ $$x_2 = F^{1/2}(x_1, y_1), y_2 = G^{1/2}(y_1, x_1)$$

Any thoughts are appreciated.

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamar product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_1=F(x_0, y_0)$, and $y_1 = G(y_0, x_0)$: $$x^'_1 = F^{1/2}(x_0, y_0), y^{'}_1=G^{1/2}(y_0, x_0)$$ $$x_1 = F^{1/2}(x^'_1, y^'_1), y_1 = G^{1/2}(y^'_1, x^'_1)$$

Any thoughts are appreciated.

EDIT: prime ($'$) is not a differentiation.

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamar product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_2=F(x_0, y_0)$, and $y_2 = G(y_0, x_0)$: $$x_1 = F^{1/2}(x_0, y_0), y_1=G^{1/2}(y_0, x_0)$$ $$x_2 = F^{1/2}(x_1, y_1), y_2 = G^{1/2}(y_1, x_1)$$

Any thoughts are appreciated.

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamar product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_1=F(x_0, y_0)$, and $y_1 = G(y_0, x_0)$: $$x^'_1 = F^{1/2}(x_0, y_0), y^{'}_1=G^{1/2}(y_0, x_0)$$ $$x_1 = F^{1/2}(x^'_1, y^'_1), y_1 = G^{1/2}(y^'_1, x^'_1)$$

Any thoughts are appreciated.

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are some matrices, $\circ$ is component-wise product (Hadamar product). My question is how to build (find) functions $F^{1/2}$ and $G^{1/2}$ such that given $x_1=F(x_0, y_0)$, and $y_1 = G(y_0, x_0)$: $$x^'_1 = F^{1/2}(x_0, y_0), y^{'}_1=G^{1/2}(y_0, x_0)$$ $$x_1 = F^{1/2}(x^'_1, y^'_1), y_1 = G^{1/2}(y^'_1, x^'_1)$$

Any thoughts are appreciated.

EDIT: prime ($'$) is not a differentiation.