# A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations
## 2 Development of Seventh-Order Method and Convergence Analysis
### Special Cases  
**Case 1.**_We take \(G(T^{(k)})\) as a polynomial function:_  
\[G(T^{(k)})=a_{0}I+a_{1}T^{(k)}+a_{2}(T^{(k)})^{2}+a_{3}(T^{(k)})^{3},\]  
_and other weight functions \(H(T^{(k)})\) and \(V(U^{(k)})\) as rational functions:_  
\[H(T^{(k)})=[b_{0}I+b_{1}T^{(k)}+b_{2}(T^{(k)})^{2}]^{-1},\]  
\[V(U^{(k)})=(c_{1}U^{(k)}+I)^{-1}(c_{0}I),\]  
_where_  
\[a_{0}=1,\;a_{1}=2,\;a_{2}=-1,\;a_{3}=6,\]  
\[b_{0}=1,b_{1}=-2,b_{2}=4,\]  
\[c_{0}=1,c_{1}=-1.\]  
_Then for multidimensional case, a new seventh-order scheme is obtained by using the above weight functions in the scheme (4):_  
\[Y^{(k)} = X^{(k)}-(F^{\prime}(X^{(k)}))^{-1}F(X^{(k)}),\] \[Z^{(k)} = Y^{(k)}-(a_{0}I+a_{1}T^{(k)}+a_{2}(T^{(k)})^{2}+a_{3}(T^{(k)})^{ 3})(F^{\prime}(X^{(k)}))^{-1}F(Y^{(k)}),\] \[X^{(k+1)} = Z^{(k)}-[b_{0}I+b_{1}T^{(k)}+b_{2}(T^{(k)})^{2}]^{-1}(c_{1}U^{(k )}+I)^{-1}(c_{0}I) \tag{22}\] \[(F^{\prime}(X^{(k)}))^{-1}F(Z^{(k)}),\]  
_where_  
\[a_{0} = 1,\;a_{1}=2,\;a_{2}=-1,\;a_{3}=6,\] \[b_{0} = 1,b_{1}=-2,b_{2}=4,\] \[c_{0} = 1,c_{1}=-1.\]_For this case, the scheme is represented as \(HM_{1}\)._  
**Case 2.**: _By taking weight functions \(G(T^{(k)})\), \(H(T^{(k)})\) and \(V(U^{(k)})\) as rational functions such that_  
\[G(T^{(k)})=\left[a_{2}I+a_{3}T^{(k)}\right]^{-1}(I+a_{0}T^{(k)}+a_{1}(T^{(k)})^ {2}),\]  
\[H(T^{(k)})=[I+b_{1}T^{(k)}+b_{2}(T^{(k)})^{2}]^{-1}(I+b_{0}T^{(k)}),\]  
_and_  
\[V(U^{(k)})=[c_{0}I+c_{1}U^{(k)}]^{-1},\]  
_where_  
\[a_{0}=8,a_{1}=11,a_{2}=1,a_{3}=6,\]  
\[b_{0} = 3,b_{1}=1,b_{2}=-2,\] \[c_{0} = 1,c_{1}=-1.\]  
_Then, by substituting these weight functions in (4), another scheme having seventh-order convergence is obtained for a multidimensional system:_  
\[Y^{(k)} = X^{(k)}-F^{\prime}(X^{(k)})^{-1}F(X^{(k)}),\] \[Z^{(k)} = Y^{(k)}-\left[a_{2}I+a_{3}T^{(k)}\right]^{-1}(I+a_{0}T^{(k)}+a_{1 }(T^{(k)})^{2})F^{\prime}(X^{(k)})^{-1}F(Y^{(k)}),\] \[X^{(k+1)} = Z^{(k)}-[I+b_{1}T^{(k)}+b_{2}(T^{(k)})^{2}]^{-1}(I+b_{0}T^{(k)})[ c_{0}I+c_{1}U^{(k)}]^{-1} \tag{23}\] \[F^{\prime}(X^{(k)})^{-1}F(Z^{(k)}).\]  
_The above scheme is represented as \(HM_{2}\), where_  
\[a_{0} = 8,a_{1}=11,a_{2}=1,a_{3}=6,\] \[b_{0} = 3,b_{1}=1,b_{2}=-2,\] \[c_{0} = 1,c_{1}=-1.\]  
**Case 3.**: _Choosing weight function \(G(T^{(k)})\) as a rational function,_  
\[G(T^{(k)})=\left[a_{1}I+a_{2}T^{(k)}+a_{3}\Big{(}T^{(k)}\Big{)}^{2}\right]^{-1 }\Big{(}I+a_{0}T^{(k)}\Big{)},\]  
_and weight functions \(H(T^{(k)})\) and \(V(U^{(k)})\) as the polynomial functions such that_  
\[L(T^{(k)})=b_{0}I+b_{1}T^{(k)}+b_{2}(T^{(k)})^{2},\]  
\[K(U^{(k)})=c_{0}I+c_{1}U^{(k)},\]  
_where_  
\[a_{0}=\frac{18}{5},a_{1}=1,a_{2}=\frac{8}{5},a_{3}=\frac{-11}{5}\]  
\[b_{0}=1,b_{1}=2,b_{2}=0,c_{0}=1,c_{1}=1.\]_Then, we obtain another seventh-order multivariate scheme, i.e., \(HM_{3}\), by substituting \(G(T^{(k)})\), \(H(T^{(k)})\) and \(V(U^{(k)})\) in the scheme (4) such as_  
\[Y^{(k)} = X^{(k)}-F^{\prime}(X^{(k)})F(X^{(k)}),\] \[Z^{(k)} = Y^{(k)}-\left[a_{1}I+a_{2}T^{(k)}+a_{3}\Big{(}T^{(k)}\Big{)}^{2} \right]^{-1}\Big{(}I+a_{0}T^{(k)}\Big{)}F^{\prime}(X^{(k)})F(Y^{(k)}),\] \[X^{(k+1)} = Z^{(k)}-(b_{0}I+b_{1}T^{(k)}+b_{2}(T^{(k)})^{2})(c_{0}I+c_{1}U^{( k)}) \tag{24}\] \[F^{\prime}(X^{(k)})F(Z^{(k)}),\]  
_where_  
\[a_{0} = \frac{18}{5},a_{1}=1,a_{2}=\frac{8}{5},a_{3}=\frac{-11}{5},\] \[b_{0} = 1,b_{1}=2,b_{2}=0,\] \[c_{0} = 1,c_{1}=1.\]