## Appendix C Dimensional Analysis of Matrix Expressions in RLFI Approximations  
This appendix provides a detailed dimensional analysis of the matrix expressions within Equations (15) and (16), addressing the mechanics of operations such as the Hadamard product \(\odot\) and Kronecker product \(\otimes\). The following clarifications ensure clarity regarding the shapes of intermediate matrices and vectors, using notation defined in Table 1.  
In Equation (15), the expression inside the brackets, \(\operatorname{trp}\left(\hat{\lambda}_{0:n}^{\lambda}\hat{\cdot}\right)\left( \mathcal{I}_{1}^{(y)}\hat{G}_{0:n}^{\lambda}[t(1-y^{1/\alpha})\mathbf{1}_{n+1}] \odot\hat{G}_{0:n}^{\lambda}[\hat{\mathbf{t}}_{n}^{\lambda}]\right)\operatorname {diag}\left(\hat{\omega}_{0:n}^{\lambda}\right)\), is a \(1\times(n+1)\) row vector. Here, \(\hat{G}_{0:n}^{\lambda}=[\hat{G}_{0}^{\lambda},\dots,\hat{G}_{n}^{\lambda}]^{\top}\) is an \((n+1)\times 1\) column vector of SG polynomials, as defined in Table 1. The argument \(t(1-y^{1/\alpha})\mathbf{1}_{n+1}\), where \(\mathbf{1}_{n+1}=[1,\dots,1]^{\top}\) is \((n+1)\times 1\), is an \((n+1)\times 1\) column vector with all elements equal to \(t(1-y^{1/\alpha})\). Evaluating \(\hat{G}_{0:n}^{\lambda}[t(1-y^{1/\alpha})\mathbf{1}_{n+1}]\) yields an \((n+1)\times(n+1)\) matrix, where each row \(j+1\) (for \(j\in\mathbb{J}_{n}^{+}\)) contains \(\hat{G}_{j}^{\lambda}(t(1-y^{1/\alpha}))\):  
\[ \begin{bmatrix}\hat{G}_{0}^{\lambda}(t(1-y^{1/\alpha}))&\hat{G}_{0}^{\lambda} (t(1-y^{1/\alpha}))&\cdots&\hat{G}_{0}^{\lambda}(t(1-y^{1/\alpha}))\\ \hat{G}_{1}^{\lambda}(t(1-y^{1/\alpha}))&\hat{G}_{1}^{\lambda}(t(1-y^{1/\alpha }))&\cdots&\hat{G}_{1}^{\lambda}(t(1-y^{1/\alpha}))\\ \vdots&\vdots&\ddots&\vdots\\ \hat{G}_{n}^{\lambda}(t(1-y^{1/\alpha}))&\hat{G}_{n}^{\lambda}(t(1-y^{1/\alpha }))&\cdots&\hat{G}_{n}^{\lambda}(t(1-y^{1/\alpha}))\\ \end{bmatrix} _{(n+1)\times(n+1)}.\]  
Applying the integral \(\mathcal{I}_{1}^{(y)}=\int_{0}^{1}dy\) element-wise, as per Table 1, produces an \((n+1)\times(n+1)\) matrix where each row \(j+1\) is filled with \(\int_{0}^{1}\hat{G}_{j}^{\lambda}(t(1-y^{1/\alpha}))\,dy\):  
\[\mathcal{I}_{1}^{(y)}\hat{G}_{0:n}^{\lambda}[t(1-y^{1/\alpha})\mathbf{1}_{n+1}]= \begin{bmatrix}a_{0}&a_{0}&\cdots&a_{0}\\ a_{1}&a_{1}&\cdots&a_{1}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n}&a_{n}&\cdots&a_{n}\\ \end{bmatrix} _{(n+1)\times(n+1)},\]  
where \(a_{j}=\int_{0}^{1}\hat{G}_{j}^{\lambda}(t(1-y^{1/\alpha}))\,dy\) for \(j\in\mathbb{J}_{n}^{+}\). This matrix is equivalent to \([a_{0},\dots,a_{n}]^{\top}\mathbf{1}_{n+1}^{\top}\), where \(\mathbf{1}_{n+1}^{\top}\) is the transpose of \(\mathbf{1}_{n+1}\). The matrix \(\hat{G}_{0:n}^{\lambda}[\hat{\mathbf{t}}_{n}^{\lambda}]\), where \(\hat{\mathbf{t}}_{n}^{\lambda}=[\hat{t}_{n,0}^{\lambda},\dots,\hat{t}_{n,n}^{ \lambda}]^{\top}\) is \((n+1)\times(n+1)\), is also \((n+1)\times(n+1)\), with elements \(\hat{G}_{j}^{\lambda}(\hat{t}_{n,k}^{\lambda})\) for \(j,k\in\mathbb{J}_{n}^{+}\). The Hadamard product \(\odot\) multiplies these matrices element-wise, yielding an \((n+1)\times(n+1)\) matrix. The row vector \(\operatorname{trp}\left(\hat{\lambda}_{0:n}^{\lambda}\hat{\cdot}\right)\), of size \(1\times(n+1)\), multiplies this matrix to produce a \(1\times(n+1)\) row vector, which is then multiplied by the \((n+1)\times(n+1)\) diagonal matrix \(\operatorname{diag}\left(\hat{\omega}_{0:n}^{\lambda}\right)\), resulting in a \(1\times(n+1)\) row vector. This vector is multiplied by \(f_{0:n}\), an \((n+1)\times 1\) vector, to compute the RLFI approximation at point \(t\). Likewise, in Equation (16), the expression inside the brackets, \(\operatorname{trp}\left(\hat{\lambda}_{0:n}^{\lambda}\hat{\cdot}\right)\left( \hat{G}_{0:n}^{\lambda}\mathbf{1}_{n+1}\right)\odot\hat{G}_{0:n}^{\lambda}[\mathbf{1} _{M+1}\otimes\hat{\mathbf{t}}_{n}^{\lambda}]\right)\left(\mathbf{1}_{M+1}\otimes \operatorname{diag}\left(\hat{\omega}_{0:n}^{\lambda}\right)\right)\), is a \(1\times(M+1)(n+1)\) row vector.