## 2 Methods
The measurement data obtained by all small-aperture beams at all sampling positions can be expressed in matrix form, as follows:  
\[ \begin{bmatrix}\partial Z_{\text{\tiny{1}}}/\partial x&\partial Z_{\text{\tiny{ 12}}}/\partial x&\cdots&\partial Z_{\text{\tiny{1 mm}}}/\partial x&1&\cdots&0&0& 0&\cdots&0&0\\ \partial Z_{\text{\tiny{21}}}/\partial x&\partial Z_{\text{\tiny{21}}}/ \partial x&\cdots&\partial Z_{\text{\tiny{2 mm}}}/\partial x&1&\cdots&0&0&0& \cdots&0&0\\ \vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots&0&\vdots&\cdots&\vdots&0\\ \partial Z_{\text{\tiny{s1}}}/\partial x&\partial Z_{\text{\tiny{s2}}}/ \partial x&\cdots&\partial Z_{\text{\tiny{s,norm}}}/\partial x&0&\cdots&0&1&0& \cdots&0&0\\ \partial Z_{\text{\tiny{11}}}/\partial y&\partial Z_{\text{\tiny{12}}}/ \partial y&\cdots&\partial Z_{\text{\tiny{1 mm}}}/\partial y&0&\cdots&0&0&1& \cdots&0&0\\ \partial Z_{\text{\tiny{s1}}}/\partial y&\partial Z_{\text{\tiny{21}}}/ \partial y&\cdots&\partial Z_{\text{\tiny{2 mm}}}/\partial y&0&\cdots&0&0&1& \cdots&0&0\\ \partial Z_{\text{\tiny{s1}}}/\partial y&\partial Z_{\text{\tiny{s2}}}/ \partial y&\cdots&\partial Z_{\text{\tiny{2 mm}}}/\partial y&0&\cdots&0&0&1& \cdots&0&0\\ \vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots&0&\vdots&\cdots&\vdots& \vdots\\ \partial Z_{\text{\tiny{s1}}}/\partial y&\partial Z_{\text{\tiny{s2}}}/ \partial y&\cdots&\partial Z_{\text{\tiny{s,norm}}}/\partial y&0&\cdots&0&0&0& \cdots&0&1\end{bmatrix}\begin{bmatrix}M_{\text{\tiny{s1}}}\\ M_{\text{\tiny{s2}}}\\ \vdots\\ M_{\text{\tiny{s,norm}}}\\ M_{\text{\tiny{s,2}}}\\ \vdots\\ M_{\text{\tiny{s,norm}}}\\ M_{\text{\tiny{s,2}}}\\ \vdots\\ M_{\text{\tiny{s,norm}}}\end{bmatrix}  \tag{5}\]  
In Equation (5), the symbol '_Coff_' represents the Zernike coefficients of the wavefront aberration for the optical system, and the symbol '_T_' represents the slope associated with '\(\Delta\)'. By applying the least squares method to solve Equation (5), the values of '_T_' and '_Coff_' can be accurately calculated. If the magnitude of '_E_' is too large to be ignored, the '_Coff_' obtained from Equation (5) will be incorrect with significant astigmatism. Therefore, this paper proposes an iterative method to first obtain the multi-beam pointing error and then compensate it into the test data to eliminate the influence of '_E_' on the reconstructed wavefront. For example, let us consider a large-aperture optical system with an RMS value of aberration better than 0.08 \(\lambda\) and a focal length of 8 m. In this system, a 50 mm sized sub-aperture aberration at any position causes the centroid coordinate shift '_D_' at the focal plane to be less than 0.5 \(\upmu\)m, which is much smaller than the quantities of '_E_' and '\(\Delta\)'. Therefore, the '_D_' in Equation (1) can be temporarily ignored, and the centroid data obtained for all sub-apertures are written in matrix form, as follows:  
\[ \begin{bmatrix}\Delta_{\text{\tiny{1T}}}\\ \Delta_{\text{\tiny{2x}}}\\ \vdots\\ \Delta_{\text{\tiny{1 yr}}}\\ \Delta_{\text{\tiny{2y}}}\\ \vdots\\ \Delta_{\text{\tiny{2y}}}\\ \vdots\\ 0\end{bmatrix} \] (6) \[\begin{bmatrix}1&\ldots&0&0&0&\ldots&0&0&0&\ldots&0&0&\ldots&1&0\\ 1&\ldots&0&0&0&\ldots&0&0&\ldots&0&0&\ldots&0&1\\ \vdots&\ldots&\vdots&0&\vdots&\ldots&\vdots&\vdots&\ldots&\vdots&0&\vdots& \ldots&0&0\\ 0&\ldots&0&1&0&\ldots&0&0&\ldots&0&1&0&\ldots&0&0\\ 0&\ldots&0&0&1&\ldots&0&0&0&\ldots&0&1&\ldots&0&0\\ 0&\ldots&0&0&1&\ldots&0&0&1&\ldots&0&By applying the least squares method to solve Equation (6), the value of \(E\) can be determined. However, since the calculation process neglects \(D\), the obtained value for \(E\) in this instance is denoted as \(E_{near}\), which is an approximation to \(E\). The accuracy of the computed value of \(E\) can be improved through the following iterative calculations. The iterative process is illustrated in Figure 2. The notation with the subscript \"near\" represents an approximation with errors, while the notation without subscripts represents the actual values. The iterative steps are as follows:  
1.