## 3 Description of QLM-ChPs Matrix Approach
### The Main Algorithm  
To approximate the solutions of the linearized boundary value problems (13) and (14) via the QLM-ChPs technique, we assume that we have the approximation \(\mathbf{M}_{H}^{(e)}(x)\) at hand for the solution \(M_{e-1}(x)\) of (13) at the iteration \(e\geq 1\). In the next iteration, we take  
\[M_{e}(x)\approx M_{H}^{(e)}(x)=\sum_{h=0}^{H}c_{h}^{(e)}\,\mathcal{S}_{h}(x)= \mathbf{S}_{H}(x)\,\mathbf{C}_{H}^{(e)}=\mathbf{Y}_{H}(x)\,\mathbf{R}_{H}\,\mathbf{C}_{H}^{(e)}, \quad x\in[0,1], \tag{15}\]  
where \(\mathbf{C}_{H}^{(e)}= \begin{bmatrix}c_{0}^{(e)}&c_{1}^{(e)}&\ldots&c_{H}^{(e)} \end{bmatrix} ^{T}\) contains the vector of unknowns at the iteration number \(e=1,2,\ldots\). We also recall that the vector \(\mathbf{Y}_{H}(x)\) as well as the matrix \(\mathbf{R}_{H}\) are previously introduced in (5) and (6). We, moreover, need a set of collocation nodes on \([0,1]\). By uniform partitioning of this interval, we set these points as  
\[x_{e}:=e/H,\quad\epsilon=0,1,\ldots,H. \tag{16}\]  
By collocating the approximate solution \(M_{H}^{(e)}(x)\) in (15) at the points (16), we obtain the next matrix format  
\[\mathbf{N}_{e}=\mathbf{X}_{H}\,\mathbf{R}_{H}\,\mathbf{C}_{H}^{(e)}, \tag{17}\]  
where we have  
\[\mathbf{N}_{e}= \begin{pmatrix}M_{H}^{(e)}(x_{0})\\ M_{H}^{(e)}(x_{1})\\ \vdots\\ M_{H}^{(e)}(x_{H})\end{pmatrix} ,\quad\mathbf{X}_{H}= \begin{pmatrix}\mathbf{Y}_{H}(x_{0 })\\ \mathbf{Y}_{H}(x_{1})\\ \vdots\\ \mathbf{Y}_{H}(x_{H})\end{pmatrix} .\]  
In the next stage, we need to compute the first and second derivatives of \(M_{H}^{(e)}(x)\) in (15). A straightforward computation reveals that  
\[\frac{d}{dx}M_{H}^{(e)}(x)=\frac{d}{dx}\Big{(}\mathbf{Y}_{H}(x)\Big{)}\,\mathbf{R}_{H }\,\mathbf{C}_{H}^{(e)}.\]To calculate the derivative of \(\mathbf{Y}_{H}(x)\), we note that  
\[\frac{d}{dx}\mathbf{Y}_{H}(x)=\mathbf{Y}_{H}(x)\,\mathbf{Q}_{H},\quad\mathbf{Q}_{H}= \begin{pmatrix}0&1&0&\ldots&0\\ 0&0&2&\ldots&0\\ \vdots&\vdots&0&\vdots&\vdots\\ 0&0&0&\ddots&H\\ 0&0&0&\ldots&0\end{pmatrix} _{(H+1)\times(H+1)}.\]  
By combining two preceding relations, we arrive at  
\[\frac{d}{dx}M_{H}^{(e)}(x)=\mathbf{Y}_{H}(x)\,\mathbf{Q}_{H}\,\mathbf{R}_{H}\,\mathbf{C}_{H}^{ (e)}. \tag{18}\]  
Similarly, by repeating the differentiation once more, we obtain the next representation  
\[\frac{d^{2}}{dx^{2}}M_{H}^{(e)}(x)=\mathbf{Y}_{H}(x)\,\mathbf{Q}_{H}^{2}\,\mathbf{R}_{H}\, \mathbf{C}_{H}^{(e)}. \tag{19}\]  
To show the first and second derivatives of \(M_{H}^{(e)}(x)\) in (18) and (19) in a matrix representation form, we state the next lemma. Lemma 2: _By collocating \(\frac{d^{2}}{dx^{2}}M_{H}^{(e)}(x)\) for \(k=1,2\) at the points (16), we obtain the matrix formats_  
\[ \begin{cases}\dot{\mathbf{N}}_{e}=\mathbf{X}_{H}\,\mathbf{Q}_{H}\mathbf{R}_{H}\,\mathbf{C}_{H}^{(e )},\\ \ddot{\mathbf{N}}_{e}=\mathbf{X}_{H}\,\mathbf{Q}_{H}^{2}\mathbf{R}_{H}\,\mathbf{C}_{H}^{(e)},\end{cases}  \tag{20}\]  
_where two matrices \(\mathbf{Q}_{H}\) and \(\mathbf{R}_{H}\) are defined in (18) and (6). Additionally, the vector \(\mathbf{X}_{H}\) is given in (17). Moreover, we have used the following vectors_  
\[\dot{\mathbf{N}}_{e}= \begin{pmatrix}\frac{d}{dx}M_{H}^{(e)}(x_{0})\\ \frac{d}{dx}M_{H}^{(e)}(x_{1})\\ \vdots\\ \frac{d}{dx}M_{H}^{(e)}(x_{H})\end{pmatrix} ,\quad\dot{\mathbf{N}}_{e}= \begin{pmatrix} \frac{d^{2}}{dx^{2}}M_{H}^{(e)}(x_{0})\\ \frac{d^{2}}{dx^{2}}M_{H}^{(e)}(x_{1})\\ \vdots\\ \frac{d^{2}}{dx^{2}}M_{H}^{(e)}(x_{H})\end{pmatrix} .\]  
Proof. : The proof is easily obtained after placing the collocation points (16) into the relations (15) and (18), respectively. In the next step of our algorithm, we return to the linearized equations (13) and put the collocation nodes into them to obtain  
\[\kappa_{e-1,1}(x_{e})\,\frac{d^{2}}{dx^{2}}M_{e}(x_{e})+\kappa_{e-1,2}(x_{e}) \,\frac{d}{dx}M_{e}(x_{e})+\kappa_{e-1,3}(x_{e})\,M_{e}(x_{e})=\pi_{e-1}(x_{e}), \tag{21}\]  
for \(e=0,1,\ldots,H\), and \(e=1,2,\ldots\).