## 3 Description of QLM-ChPs Matrix Approach
### Theoretical Upper Bound for QLM-ChPs Approach  
In Theorem 1, we have obtained an upper bound for the error between \(M(x)\) as the true solution of the original model problem (1) and its approximation \(M_{H}(x)\) expanded in terms of ChPs in (8). This error is defined by  
\[E_{H}(x):=M(x)-M_{H}(x),\quad x\in[0,1]. \tag{27}\]  
Our aim is now to examine the error between \(M(x)\) and \(M_{H}^{(e)}(x)\) as the approximate solution of the associated linearized model problem (13) together with boundary conditions (14) at a fixed iteration number \(e\). Let this error be represented by \(E_{H}^{(e)}(x)\) and defined by  
\[E_{H}^{(e)}(x):=M(x)-M_{H}^{(e)}(x),\quad x\in[0,1]. \tag{28}\]  
The next theorem consists of an \(L_{2}\) error estimate for the QLM-ChPs solution.  
Theorem 2: _Let the assumptions of Theorem 1 hold. Then, the upper bound for the error is given by_  
\[\|E_{H}^{(e)}\|\leq\frac{L}{(H+1)!\sqrt{2H+3}}+(\log(H)+1)\|\mathbf{R}_{H}\|\,\| \mathbf{C}_{H}^{(e)}-\mathbf{C}_{H}\|,\quad L:=\max_{x\in[0,1]}|M^{(H+1)}(x)|.\]Proof.: Our proof starts with the following error decomposition  
\[E_{H}^{(e)}(x)=E_{H}(x)-\mathbb{E}(x), \tag{29}\]  
where \(\mathbb{E}(x)=M_{H}^{(e)}(x)-M_{H}(x)\) denotes the difference between the approximate solutions of (13) and (1) when solving with the ChPs collocation procedures. One can immediately write the following conclusion using the triangle inequality  
\[\|E_{H}^{(e)}\|\leq\|E_{H}\|+\|\mathbb{E}\|. \tag{30}\]  
For the first term, we apply the result of Theorem 1, which asserted that  
\[\|E_{H}\|\leq\frac{L}{(H+1)!\,\sqrt{2H+3}}. \tag{31}\]  
We now need to obtain an upper bound for the second term \(\|\mathbb{E}\|\). We recall that the approximate solutions \(M_{H}(x)\) and \(M_{H}^{(e)}(x)\) can be stated as  
\[M_{H}(x)=\boldsymbol{Y}_{H}(x)\,\boldsymbol{R}_{H}\,\boldsymbol{C}_{H},\quad M _{H}^{(e)}(x)=\boldsymbol{Y}_{H}(x)\,\boldsymbol{R}_{H}\,\boldsymbol{C}_{H}^{ (e)},\]  
in accordance to relations (8) and (15), respectively. Here, we have the corresponding coefficients obtained by the direct ChPs and QLM-ChPs collocation matrix procedures as  
\[\boldsymbol{C}_{H}:=[c_{0}\quad c_{1}\quad\ldots\quad c_{H}]^{T},\quad \boldsymbol{C}_{H}^{(e)}=\Big{[}c_{0}^{(e)}\quad c_{1}^{(e)}\quad\ldots\quad c _{H}^{(e)}\Big{]}^{T}.\]  
The consequence is that  
\[\|\mathbb{E}\|\leq\|\boldsymbol{Y}_{H}\|\,\|\boldsymbol{R}_{H}\|\,\| \boldsymbol{C}_{H}^{(e)}-\boldsymbol{C}_{H}\|.\]  
For the first term, we further have  
\[\|\boldsymbol{Y}_{H}\|^{2}=\int_{0}^{1}|\boldsymbol{Y}_{H}(x)|^{2}dx=\int_{0} ^{1}\sum_{h=0}^{H}|x^{h}|^{2}dx=\sum_{h=0}^{H}\frac{1}{2h+1}=:S_{H}.\]  
By comparing \(S_{H}\) to the corresponding harmonic series, we can find that \(S_{H}<\log(H)+\gamma\), where \(\gamma\approx 0.5772\) is the Euler constant. Now, by inserting the upper bounds for \(\|E_{H}\|\) and \(\|\mathbb{E}\|\) in the inequity (30), we have completed the proof.  
It seems that the obtained upper bound for the error term \(\mathbb{E}\) is not sufficiently small in practical problems. However, we will show that this upper bound is quite tight when computed for the actual experimental results.