## 2 An Efficient Method and Its Acceleration
### Accelerating the Initial Phase  
Another useful motivation of the present paper is processed here. The iterative scheme:  
\[X_{k+1}=X_{k}\left((\beta+1)I-\beta AX_{k}\right),\;1\leq\beta\leq 2 \tag{13}\]  
was applied in [22] to achieve the convergence phase in the main (modified Householder) method much more rapidly and to accelerate the beginning of the process. In the second iteration phase, it is sufficient to apply the introduced fast and efficient modified Householder method, which then reaches its own full speed of convergence [22].  
In the same vein, the iterative expression Equation (13) can be rewritten in the following equivalent, but more practical form:  
\[X_{k+1}=X_{k}\left((1-\beta)I+\beta\sum_{i=0}^{1}R_{k}^{i}\right),\ \ R_{k}=I-AX_{k},\ 1 \leq\beta\leq 2 \tag{14}\]  
One can now observe that Equation (14) is the particular case (\(p=2\)) of the following new scheme:  
\[\begin{split} X_{k+1}&=X_{k}\left(I+\beta\left((p- 1)I+\sum_{i=1}^{p-1}(-1)^{i}\binom{p}{i+1}(AX_{k})^{i}\right)\right)\\ &=X_{k}\left((1-\beta)I+\beta\left(\sum_{i=0}^{p-1}(-1)^{i} \binom{p}{i+1}(AX_{k})^{i}\right)\right)\\ &=X_{k}\left((1-\beta)I+\beta\sum_{i=0}^{p-1}R_{k}^{i}\right),\ \ R_{k}=I-AX_{k},\ 1 \leq\beta\leq 2\end{split} \tag{15}\]  
Therefore, Equation (15) can be considered as an important acceleration of the Schulz-type method Equation (2) in the initial phase, before the convergence rate is practically achievable. We note that such accelerations are useful for large matrices, whereas the iterative methods require too much iterations to converge. Particularly, by following Equation (15), one can immediately notice that the accelerating in the initial phase of iteration Equation (9) is of the form:  
\[X_{k+1}=X_{k}\left((1-\beta)I+\beta\left(I+(R_{k}+R_{k}^{2})(I+R_{k}^{2}+R_{k} ^{4})\left(I+(R_{k}^{2}+R_{k}^{8})(R_{k}^{4}+R_{k}^{16})\right)\right)\right) \tag{16}\]  
**Remark 1**.: _The particular choice \(\beta=1\) in Equation (15) reduces these iterations to the usual hyper-power family of the iterative methods possessing the order \(p\geq 2\):_  
\[\begin{split} X_{k+1}&=X_{k}\left(pI-\binom{p}{2} AX_{k}+\dots+(-1)^{p-1}\binom{p}{p}(AX_{k})^{p-1}\right)\\ &=X_{k}\,\sum_{i=0}^{p-1}R_{k}^{i},\ \ R_{k}=I-AX_{k}\end{split} \tag{17}\]  
_The choice \(p=2\) in Equation (15) leads to the scaled Schulz matrix iteration considered recently in [23], and the choice \(p=2\), \(\beta=1\) produces the original Schulz matrix iteration, originated in [12]._  
Finally, a hybrid algorithm may be written now by incorporating Equations (9) and (16) as follows.  
```
1:The input is given \(X_{0}\in\mathbb{C}^{n\times m}\).
2:Use Equation (16) until \(\|X_{l+1}-X_{l}\|<\delta\) (for an inner loop counter \(l\) or \(\epsilon<\delta\)).
3:set \(X_{0}=X_{l}\)
4:for \(k=0,1,\dots\) until convergence (\(\|X_{k+1}-X_{k}\|<\epsilon\)), use Equation (9) to converge with high order.
5:end for
```  
**Algorithm 1** The new hybrid method for computing generalized inverses.  
Instead of the hybrid Algorithm 1, based on the usage of Equation (16) in the initial phase and Equation (9) in the final stage, our third result here is to define a unique iterative method, which can be derived by applying variable acceleration parameter \(\beta=1+\beta_{k}\), \(0\leq\beta_{k}\leq 1\). This approach yields scaled hyper-power iterations of the general form:  
\[X_{0} =\alpha G \tag{18}\] \[X_{k+1} =X_{k}\left(-\beta_{k}I+(1+\beta_{k})\sum_{i=0}^{p-1}R_{k}^{i} \right),\;0\leq\beta_{k}\leq 1,\;k\geq 0\] \[=X_{k}\left(I+(1+\beta_{k})\sum_{i=1}^{p-1}R_{k}^{i}\right)\]  
where the initial approximation \(X_{0}=\alpha G\) is chosen according to Equation (11).  
Furthermore, it is possible to propose various modifications of \(\beta_{k}\) in a manner that guarantees \(1+\beta_{k}\to 1\). For example:  
\[\beta_{0}=1,\;\;\;\beta_{k+1}=\frac{\beta_{k}}{2},\;k\geq 0 \tag{19}\]