## 5 Numerical Experiments and Conclusions  
The main purpose of this section is to highlight the convergence properties of the proposed parallel Broyden's algorithms in the case of a synchronous model. The communication costs between processors, scheduling and loads balancing, optimal processors assignment, and so on, both for synchronous and asynchronous models, are important issues in themselves. However they are not the subject of the present study. The behavior of the sequence \(\{x_{n}\}\) obtained by the proposed algorithms and for various linear or nonlinear cases is shown in our experiments by the graph of \(ln\varepsilon_{n}\), where \(\varepsilon_{n}\) is the error at step \(n\), \(\varepsilon_{n}=\|F(x_{n})\|\) or \(\varepsilon_{n}=\|x_{n}-x^{*}\|\) (\(n\) on horizontal axis and \(ln\varepsilon_{n}\) on vertical axis). The convergence of the process entails a decreasing graph until negative values (for example, \(ln\varepsilon_{n}=-30\) means \(\varepsilon_{n}\approx 10^{-15}\)). If the convergence is linear (\(r=1\)) then \(ln\varepsilon_{n}\approx ln\eta+ln\varepsilon_{0}\); \(ln\varepsilon_{n}\) depends linearly on \(n\) and the graph of \(ln\varepsilon_{n}\) is close to a straight line. For comparison reasons, the proposed parallel [PERSON]'s method is compared with the parallel Newton method and the parallel Cimmino method. We consider the following parallel Newton method [8]:  
\[x_{n+1}=x_{n}-D(x_{n})^{-1}F(x_{n}),\;D(x_{n})=dg(J(x_{n})),\]  
where \(J(x_{n})\) is the Jacobian of \(F\). Note that in the case of a linear system, by applying this form of [PERSON] method, the direct computation of \(x^{*}\) (in one iteration) is lost and the convergence is linear. Consequently, the graph of \(ln(\varepsilon_{n})\) is a straight line. The block Cimmino method is considered only for the linear case, \(F(x)=Ax+b\), and it can be described as follows (see, for example, [9]). The system is partitioned into \(p\) strips of rows, \(A^{i}\) and \(b^{i}\), \(1\leq i\leq p\). Supposing that \(A^{i}\) has full row rank, the Moore-Penrose pseudo inverse of \(A^{i}\) is defined by \(A^{i^{+}}=A^{i^{T}}(A^{i}A^{i^{T}})^{-1}\). The block Cimmino algorithm is  
Step 1: Compute in parallel \(Q_{n}^{i}=A^{i^{+}}(A^{i}x_{n}+b^{i});\)  
Step 2: \(x_{n+1}=x_{n}+\omega\sum_{i=1}^{p}Q_{n}^{i},\)  
where \(\omega\) is a relaxation parameter. **Experiment 1. **  
We applied Algorithm 3 to linear systems of medium dimensions, \(m=20,50,100,500,600\), with different values of condition numbers. A program \"genA\" generates a square, sparse matrix \(A\) of dimension \(m\) with random elements \(a_{ij}\in(-1,1),\;i\
eq j\) and \(a_{ii}\in(d1,d2)\), where \(d1,d2\) are given (positive) numbers. The percent of nonzero elements is an entry data of \"genA\" and in this experiment we considered it to be about 50%. The free term \(b\) and the starting point \(x_{0}\) were also randomly generated with elements between some limits. Thus the function \(F\) is randomly defined by \(F(x)=Ax-b\). For every considered system, we applied Algorithm 3, the parallel Newton method and the parallel Cimmino method, and we drew the resulting graphs in the same plot; for the graph of Algorithm 3 we used a continuous line, for the parallel Newton method a dashed line, and for the parallel Cimmino  
Figure 1: The graphs of \(ln(\varepsilon_{n})\) generated by Algorithm 3, parallel Newton method and parallel Cimmino method in the linear case; (**a**) system well conditioned; (**b**) system medium conditioned; (**c**) system poorly conditioned. method a dotted line. The number of partitions and the dimensions of every partition are entry data of the routine \"diag(A)\".