## 1 Introduction  
Let \(F:\mathcal{R}^{m}\rightarrow\mathcal{R}^{m}\) be a (nonlinear) mapping and let \(F(x)=0\) be the corresponding system of equations. In [1], [PERSON] proposed the following partially asynchronous block quasi-Newton method for such a nonlinear system. Let \(\mathcal{P}\) be a partition of \(F\) and \(x\) in \(p\) partitions, \(F=(F_{1},...,F_{p})\), \(x=~{}(x_{1},...,x_{p})\), where \(F_{i}\) and \(x_{i}\) are block functions and block variables, respectively, of the same dimensions \(m_{i}\). The variable \(x\) is the common memory of the multiprocessor system, every processor \(i\) can read the content of \(x\) and can write its new update of \(x_{i}\). Let \(B_{i}\), \(i=1,...,p\), be square matrices of dimension \(m_{i}\). The main steps of the process \(i\) are :```
0:\(x,\;B_{i}\)
0:\(x\)
1. Compute \(x_{i}^{+}:=x_{i}-B_{i}^{-1}F_{i}(x);\)
2. Get \(x^{+}\) from \(x\) by replacing \(x_{i}\) with \(x_{i}^{+};\)
3. Compute \(s=x_{i}^{+}-x_{i}\;\) and \(y:=F_{i}(x^{+})-F_{i}(x);\)
4. Update \(B_{i},\;B_{i}:=B_{i}+\dfrac{(y-B_{i}s)s^{T}}{s^{T}s};\)
5. \(x_{i}:=x_{i}^{+},\;x:=x^{+};\)
```

**Algorithm 1**. Applying the Sherman-Morrison lemma (Step 4, Algorithm 1) to inverse the matrix \(B_{i}\), the polynomial complexity of order three is reduced to polynomial complexity of order two. More recently, in [2,3] [PERSON] and [PERSON] proposed several preconditioners for the block Broyden's method and discussed the implementation process. The main steps of parallel variant of [PERSON]'s method considered in [2,3] (Algorithm 2 below) are (\(B_{0}\) is a block diagonal matrix, \(B_{k}^{i}\) is the \(i^{th}\) diagonal block of \(B_{k}\)):

```
0:\(x_{0},\;B_{0},\;kmax\)
0:\(x_{k}\)
1. **For**\(k=0,1,..\) Until\(convergence\) Or\(k>kmax\) Do
2. \(x_{k+1}=x_{k}-B_{k}^{-1}F(x_{k});\)
3. \(s=x_{k+1}-x_{k}\);
4. **For**\(i=1,...,p\)
5. \(B_{k}^{i}=B_{k}^{i}+\dfrac{F_{i}(x_{k+1})s_{i}^{T}}{s^{T}s}.\)
6. **End**
7. **End**
```

**Algorithm 2**. In the case of a linear system, the sequential Broyden's method has global convergence provided that the system matrix is invertible [4], _i.e._, the sequence generated by this method converges to the solution of the system for any starting point \(x_{0}\) and for any starting matrix \(B_{0}\) (or \(H_{0}\)). Moreover, in this case the convergence is finite [5], and the solution is obtained in at most \(2n\) steps; conditions in which this method requires a full \(2n\) steps to converge are also given. In the same paper [5], it is shown that [PERSON]'s method has \(2n\)-step Q-quadratic local convergence for the nonlinear case (the initial starting point \(x_{0}\) and the initial starting matrix \(B_{0}\) must be close to the solution point of the system and to the Jacobian matrix in the solution point, respectively). Note that \(2n\)-_step Q-quadratic convergence_ means that the subsequence \(\{x_{2n}\}\) of \(\{x_{n}\}\) has Q-quadratic convergence. In this paper we propose some new variants of parallel [PERSON]'s method that are suitable both for linear and nonlinear systems. We prove the convergence of our basic algorithm in the linear case. Numerical experiments are presented for linear and nonlinear cases, and some remarks are made concerning the behavior of the generated sequence. ## 2 The Parallel Variants and Preliminaries

In the case of a linear mapping, \(F(x)=Ax-b\), the [PERSON]'s \"good formula\" [6, 7], \(B^{+}=B+\theta(y-Bs)s^{T}/s^{T}s\) (\(B\) and \(B^{+}\) are the current and the next iterates, respectively, and \(\theta\) is a positive number chosen such that \(B^{+}\) is invertible [4]) can be written in the following form: \(B^{+}=B-\theta(B-A)ss^{T}/s^{T}s\). Using the notation \(E=B-A\), the [PERSON]'s method becomes \(x^{+}=x-(E+A)^{-1}F(x),\ E^{+}=E-\theta Ess^{T}/s^{T}s\). The main idea of the proposed variants is to use instead of \(A\) the block diagonal matrix of \(A\), denoted by \(D\) throughout the paper, partitioned according to \(\mathcal{P}\).