## 3 Computational Complexity  
This section provides a computational complexity analysis of constructing the \(\alpha\)th-order FSGIM \({}^{E}\mathbf{Q}_{n}^{\alpha}\) and its generator \({}^{E}\mathbf{\hat{Q}}_{n}^{\alpha}\). The analysis is based on the key matrix operations involved in the construction process, which we analyze individually in the following:  
* The term \(\mathbf{z}_{M}^{\alpha\alpha}\) involves raising each element of an \((M+1)\)-dimensional vector to the power \(\alpha\). This operation requires \(O(M)\) operations. * Constructing \({}^{E}\mathbf{Q}_{n}^{\alpha}\) from \({}^{E}\mathbf{\hat{Q}}_{n}^{\alpha}\) involves a diagonal scaling by \(\text{diag}(\mathbf{z}_{M}^{\alpha\alpha})\), which requires another \(O(Mn)\) operations. * The matrix \({}^{E}\mathbf{\hat{Q}}_{n}^{\alpha}\) is constructed using several matrix multiplications and element-wise operations. For each entry of \(\mathbf{z}_{M}\), the dominant steps include the following:
* The computation of \(\hat{G}_{0:n}^{\lambda}\) using the three-term recurrence relation requires \(O(n)\) operations per point. Since the polynomial evaluation is required for polynomials up to degree \(n\), this requires \(O(n^{2})\) operations. * The quadrature approximation involves evaluating a polynomial at transformed nodes. The cost of calculating \(\mathbf{\hat{P}}\) depends on the chosen methods for computing factorials and the Gamma function, which can be considered a constant overhead. The computation of \(\hat{\mathbf{\hat{\lambda}}}_{n_{q}}^{\lambda_{q}^{\alpha\xi_{1}/\alpha}}\) involves raising each element of the column vector \(\hat{\mathbf{\hat{\lambda}}}_{n_{q}}^{\lambda_{q}}\) to the power \(1/\alpha\), which is linear in \((n_{q}+1)\). The cost of the matrix-vector multiplication is also linear in \(n_{q}+1\). Therefore, the computational cost of this step is \(O(n_{q})\) for each \(j\in\mathbb{J}_{n}^{+}\). The overall cost, considering all polynomial functions involved, is \(O(mn_{q})\). * The Hadamard product introduces another \(O(n^{2})\) operations. * The vector \(\hat{\mathbf{\hat{\lambda}}}_{0:n}^{\lambda}{}^{=}\left[1/\hat{\mathbf{\hat{ \lambda}}}_{0:1}^{\lambda}/1/\hat{\mathbf{\hat{\lambda}}}_{1:}^{\lambda}, \ldots,1/\hat{\mathbf{\hat{\lambda}}}_{n}^{\lambda}\right]^{\top}\) is the element-wise reciprocal of \(\hat{\mathbf{\hat{\lambda}}}_{0:n}^{\lambda}=[\hat{\mathbf{\hat{\lambda}}}_{0 :n}^{\lambda},\hat{\mathbf{\hat{\lambda}}}_{1:}^{\lambda},\ldots,\hat{\mathbf{ \hat{\lambda}}}_{n}^{\lambda}]^{\top}\), an \((n+1)\times 1\) column vector of coefficients. To evaluate \(\hat{\mathbf{\hat{\lambda}}}_{0:n}^{\lambda}\), we compute the reciprocal \(1/\hat{\mathbf{\hat{\lambda}}}_{j}^{\lambda}\) for each element \(\hat{\mathbf{\hat{\lambda}}}_{j}^{\lambda},j\in\mathbb{J}_{n}^{+}\). This requires one floating-point division per element, totaling \((n+1)\) divisions. Thus, the evaluation of \(\hat{\mathbf{\hat{\lambda}}}_{0:n}^{\lambda}{}^{\hat{\top}}\) requires \(O(n)\) operations, as the operation scales linearly with the vector length. * The product of \(\hat{\mathbf{\hat{\lambda}}}_{0:n}^{\lambda}{}^{\hat{\top}}\) by the result from the Hadamard product requires \(O(n^{2})\) operations. * The final diagonal scaling by \(\text{diag}\big{(}\hat{\mathbf{\hat{\sigma}}}_{0:n}^{\lambda}\big{)}\) contributes \(O(n)\). Summing the dominant terms, the overall computational complexity of constructing \({}^{E}\mathbf{\hat{Q}}_{n}^{\alpha}\) is \(O(n(n+n_{q}))\) per entry of \(\mathbf{z}_{M}\). Therefore, the total number of operations required to construct the matrix \({}^{E}\mathbf{Q}_{n}^{\alpha}\) for all entries of \(\mathbf{z}_{M}\) is \(O(Mn(n+n_{q}))\). Once the FSGIM is precomputed, applying it to compute the RLFI of a function requires only a matrix-vector multiplication with complexity \(O(Mn)\). The FSGIM's invariance for fixed points and parameters enables precomputation, making the GBFA method ideal for problems requiring repeated fractional integrations with varying \(\alpha\).