## 5 Extension to a Family of the Sixth-Order Methods for Nonlinear Systems of Equations  
Let \(f:\mathbf{D}\subset\mathbb{C}^{\mathbf{d}}\rightarrow\mathbb{C}^{\mathbf{d}}\) with \(d\in\mathbb{N}\) have a zero \(\alpha\in\mathbf{D}\) and be holomorphic in a neighborhood of \(\alpha\). [PERSON] expansion of \(f(x_{n})\) about \(\alpha\) easily gives:  
\[f(x^{(n)})=f^{\prime}(\alpha)(e^{(n)}+c_{2}{e^{(n)}}^{2}+\cdots+c_{m}{e^{(n)}} ^{m})+O(||{e^{(n)}}^{m+1}||),n=0,1,\cdots, \tag{30}\]  
where \(e^{(n)}=x^{(n)}-\alpha\) and \(c_{j}=\frac{1}{j! }f^{\prime}(\alpha)^{-1}f^{(j)}(\alpha)\) for \(j\geq 2\). For notational convenience, we drop the subscript \(n\) of \(e^{(n)}\) and \(x^{(n)}\) for the time being. We observe that \(f^{\prime}(\alpha)\) and \(f^{\prime}(\alpha)^{-1}\) are \(d\times d\) matrices,  
with \(c_{j}e^{(n)}{}^{j}\in\mathbb{C}^{d}\). From (30), we find that the truncated \(f(x)\) defines a polynomial in \(e\) with matrix coefficients (independent of \(x\)). Hence, it is easily seen that  
\[f^{\prime}(x)=f^{\prime}(\alpha)(I+2c_{2}e+3c_{3}e^{2}+\cdots+mc_{m}e^{m-1})+O( ||e||^{m}),n=0,1,\cdots, \tag{31}\]  
where \(I\) is the \(d\times d\) identity matrix. The inverse of \(f^{\prime}(x^{(n)})\) can be found by identifying \(B=-(2c_{2}e+3c_{3}e^{2}+\cdots+mc_{m}e^{m-1}\) from the relation  
\[(I-B)^{-1}=I+B+B^{2}+B^{3}+\cdots,\text{ with }||B||<1. \tag{32}\]  
Consequently, we find that:  
\[f^{\prime}(x)^{-1}=(I+X_{1}e+X_{2}e^{2}+X_{3}e^{3}+X_{4}e^{4}+X_{5}e^{5}+X_{6} e^{6})f^{\prime}(\alpha)^{-1}+O(||e^{7}||), \tag{33}\]  
where \(X_{1}=-2c_{2},X_{2}=(4c_{2}^{2}-3c_{3}),X_{3}=-8c_{3}^{3}-4c_{4}+6(c_{2}c_{3}+ c_{3}c_{2}),X_{4}=16c_{2}^{4}+9c_{3}^{2}-5c_{5}+8c_{2}c_{4}+8c_{4}c_{2}-12(c_{2}^{2}c_ {3}+c_{3}c_{2}^{2}+c_{2}c_{3}c_{2}),X_{5}=-32c_{2}^{5}-6c_{6}+2(5c_{2}c_{5}+6c_ {3}c_{4}+6c_{4}c_{3}+5c_{5}c_{2})+24(c_{2}^{3}c_{3}+c_{3}c_{2}^{3}+c_{2}c_{3}c _{2}^{2}+c_{2}c_{3}c_{2})-2(9c_{2}c_{3}^{2}+8c_{2}^{2}c_{4}+9c_{3}^{2}c_{2}+8c_ {4}c_{2}^{2}+8c_{2}c_{4}c_{2}+9c_{3}c_{2}c_{3}),X_{6}=64c_{2}^{4}+16c_{4}^{4}+12 c_{2}c_{6}-20c_{2}^{2c_{5}}+15c_{3}c_{5}+15c_{5}c_{3}+12c_{6}c_{2}-48(c_{2}^{2}c_ {3}+c_{3}c_{4}^{2}+c_{2}c_{3}c_{2}^{2}+c_{2}^{2}c_{3}c_{2})-24(c_{2}c_{3}c_{4} +c_{3}c_{2}c_{4})-3(9c_{3}^{2}+8c_{2}c_{4}c_{3}+8c_{4}c_{2}c_{3})-4(5c_{5}c_{5}^ {2}+5c_{5}c_{5}c_{2}+6(c_{3}c_{4}c_{2}+c_{4}c_{3}c_{2}))+4(9c_{2}^{2}c_{3}^{2}+8 c_{3}^{2}c_{4}+9c_{3}^{2}c_{2}+8c_{4}c_{2}^{2}+9c_{2}c_{3}^{2}c_{3}+9c_{3}c_{2}c_{3} ).\]  
Additional computations show that:  
\(f^{\prime}(x)^{-1}f(x)=e-c_{2}e^{2}+(2c_{2}^{2}-2c_{3})e^{3}+(-4c_{2}^{3}-3c _{4}+4c_{2}c_{3}+3c_{3}c_{2})e^{4}+\sum_{j=5}^{6}\mathcal{A}_{j}e^{j}+O(||e|| ^{7})\) with \(\mathcal{A}_{j}=\mathcal{A}_{j}(c_{2},c_{3},c_{4},c_{5},c_{6})\). \(e_{y}=y-\alpha=x-\alpha-\gamma f^{\prime}(x)^{-1}f(x)=e(1-\gamma)+\gamma c_{2} e^{2}-2\gamma(c_{2}^{2}-c_{3})e^{3}+\gamma(4c_{2}^{3}+3c_{4}-4c_{2}c_{3}-3c_{3}c_{2})e^{4} +\sum_{j=5}^{6}\mathcal{B}_{j}e^{j}+O(||e||^{7})\) with \(\mathcal{B}_{j}=\mathcal{B}_{j}(\gamma,c_{2},c_{3},c_{4},c_{5},c_{6})\). We find \(f^{\prime}(y^{(n)})=f^{\prime}(x)\big{|}_{e->e_{y}}\) and \(s=f^{\prime}(x)^{-1}f^{\prime}(y)=I-2\gamma c_{2}e+3\gamma(2c_{2}^{2}+c_{3}( \gamma-2))e^{2}+\sum_{j=3}^{5}\mathcal{C}_{j}e^{j}+O(||e||^{6})\) with \(\mathcal{C}_{j}=\mathcal{C}_{j}(\gamma,c_{2},c_{3},c_{4},c_{5},c_{6})\). \((s-I)^{2}=\gamma^{2}\big{[}4c_{2}^{2}e^{2}+e^{3}(-24c_{2}^{3}-6(-2+\gamma)c_{2} c_{3}-6(-2+\gamma)c_{3}c_{2}))e^{3}+\sum_{j=4}^{5}\mathcal{A}_{j}e^{j}\big{]}+O(||e|| ^{6})\) with \(\mathcal{D}_{j}=\mathcal{D}_{j}(\gamma,c_{2},c_{3},c_{4},c_{5},c_{6})\). \((s-I)^{3}=2\gamma^{2}\big{[}-4c_{2}^{3}e^{3}\gamma+(12(1+2\gamma)c_{2}^{4}+6(- 2+\gamma)\gamma c_{2}^{2}c_{3}+6(-2+\gamma)\gamma c_{3}c_{2}^{2}+6(-2+\gamma) \gamma\gamma c_{2}c_{3}c_{2})e^{4}+\mathcal{C}_{5}e^{5}\big{]}\big{)}+O(||e|| ^{6})\) with \(\mathcal{E}_{5}=\mathcal{C}_{j}(\gamma,c_{2},c_{3},c_{4},c_{5},c_{6})\). Theorem 1 suggests us to use \(T_{f}\) and \(L_{f}\) as at most third-degree matrix polynomials in \((s-I)\).