# Accounting for Round-Off Errors When Using Gradient Minimization Methods
## 6 Examples of Numerical Experiments  
To demonstrate the capabilities of the proposed algorithm, we denote the following: (1) a matrix \(A\) of dimension \(M\times N\), with elements generated as random variables with a uniform distribution in the range \([0,1]\); (2) a model solution \(x^{model}\)--a column vector of dimension \(N\) whose elements correspond to the values of the sine on the interval \([0,2\pi]\). \[x^{model}_{n}=\sin\frac{2\pi(n-1)}{N-1},\quad n\in\overline{1,N}\]  
For the matrix \(A\) and the model solution \(x^{model}\), the right side \(b\) was calculated as follows: \(b=Ax^{model}\). To solve system (1) with this matrix and the right side, the considered variation of the conjugate gradient method with an improved criterion for stopping the iterative process was used. All calculations were performed in double precision (float64), i.e., \(\Delta=10^{-16}\). **Remark 3**. : _Note that all subsequent results may slightly differ in details when reproduced since the matrix \(A\) is given randomly._  
**Example 1**. : _Calculations were made for \(M=32\) and \(N=30\). These parameters were especially selected to better demonstrate the capabilities of the algorithm under consideration. Figure 1a shows a graph of \(\frac{\sigma_{s}^{2}\Delta^{2}}{\|r^{(s)}\|^{2}}\) depending on the iteration number \(s\). The classical stopping criterion for the iterative process would have stopped at the iteration with the number \(s=N\equiv 30\) (marked in the figure with the red dotted vertical line), but the improved criterion stopped the iterative process only at the iteration with the number \(s=48\), i.e., much later. Figure 1b shows the dependence of \(\|x^{(s)}-x^{model}\|\) on the iteration number \(s\). It is clearly seen from the graph that the classical criterion for stopping the iterative process gives a solution that is quite different from that of the model. Now, let us see what the approximate solutions look like in the case of using different criteria to stop the iterative process. Figure 2 shows an approximate solution--vector \(x^{\text{res}}\). It is perfectly clear that the approximate solution found by the classical method (marked in red on the graph) is quite different from the exact one (sine), even visually. In this case, the solution found using the improved criterion for stopping the iterative process no longer visually differs from the exact solution. That is, we managed to demonstrate the efficiency of the proposed algorithm with the use of such a simple example, although it was initially assumed that the method would work only when solving \"large\" problems._  
at the same time, it is important that the proposed variants of the formulas do not greatly increase the computational complexity of the algorithm. * Many systems of algebraic equations (both linear and non-linear) arise when solving ill-posed problems [23]. In this case, regularizing algorithms are constructed; in the application of these algorithms, one of the key points is the question of a reasonable choice for the regularization parameter. In this regard, the question remains as to how the rounding error could be correctly taken into account when automatically choosing the regularization parameter in accordance with the input data specification errors (for example, see the generalized residual principle in [23]). * The method considered in this paper assumes that the measure of inconsistency of the solved system of equations is sufficiently small. When solving problems with a sufficiently large measure of inconsistency, one should additionally monitor the iteration of the algorithm when the value of the discrepancy ceases to decrease. In this case, the iterative process must be interrupted without waiting for the value of the functional to enter the background of the rounding errors. * The formulas presented in this paper were derived from statistical considerations. Therefore, the reliability of the proposed algorithm is quite high when solving \"large\" problems with high-precision computation.