We call our structural analysis the

When the DAE is

A simple pendulum in Cartesian coordinate system is \begin{align*} 0 = f_1 &= x'' + x\lambda\\[1ex] 0 = f_2 &= y'' + y\lambda - G \\[1ex] 0 = f_3 &= x^2 + y^2 - L^2 \end{align*} Here $x, y, \lambda$ are state variables; $G$ is gravity, and $L>0$ is the length of the pendulum.

The signature matrix of this DAE is \[\begin{array}{r} \\f_1 \\ \Sigma=\,\, f_2 \\ f_3 \\ \end{array} \begin{aligned} \begin{array}{ccc} \,\,\,\,x\,\,\, & \,\,\,\,y\,\,\,\, & \,\,\,\,\lambda\,\,\,\, \end{array} \\ \left( \begin{array}{ccc} {\color{green}2} & -\infty & {\color{blue}0} \\ -\infty & {\color{blue}2} & {\color{green}0} \\ {\color{blue}0} & {\color{green}0} & -\infty \end{array} \right) \end{aligned} \] There are two HVTs in $\Sigma$: one in $\color{green}{green}$, the other in $\color{blue}{blue}$. The value of $\Sigma$ is $\text{Val}(\Sigma)=2$. The canonical offsets are $\mathbf{c}=(0,0,2)$ and $\mathbf{d}=(2,2,0)$.

Critical to the numerical solution method is a

The Jacobian of a subsystem in stages $k\ge 0$ is exactly the System Jacobian $\mathbf{J}$ of the DAE. If $\mathbf{J}$ is nonsingular and the derivatives $x_j^{(r)}$, for all $0\le r\le d_j$ and $j=1:n$, are uniquely determined in stages $k=k_d:0$, we say the $\Sigma$-method

In the success case, we use the canonical offsets $\mathbf{c},\mathbf{d}$ to determine the

**Theory review of the Sigma method**

- Specifying a DAE in a Matlab function
- A simple call
**daeSA()**to perform structural analysis (SA) - Visualizing structure of a DAE by
**showStruct()** - Obtaining more SA result
- Identifying constraints and initialization data
- Block solution scheme for a Taylor series method
**A simple tutorial of SA theory**- Improvement on Structural Analysis of DAEs (to edit)

**tgn3000.com (Gary G. Tan's webpage)**